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Handbook of mathematics for engineers and scientists:

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Bibliographic Details
Main Authors:	Poljanin, Andrej D. 1951- (Author), Manžirov, Aleksandr V. (Author)
Format:	Book
Language:	English
Published:	Boca Raton, Fla. [u.a.] Chapman & Hall/CRC 2007
Subjects:	Mathematik
Online Access:	Inhaltsverzeichnis
Physical Description:	XXXII, 1509 S.
ISBN:	1584885025 9781584885023

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adam_text	CONTENTS Authors .................................................................. xxv Preface ...................................................................xxvii Main Notation ............................................................. xxix Part I. Definitions, Formulas, Methods, and Theorems 1 1. Arithmetic and Elementary Algebra ......................................... 3 1.1. Real Numbers ........................................................... 3 1.1.1. Integer Numbers ................................................... 3 1.1.2. Real, Rational, and Irrational Numbers ................................. 4 1.2. Equalities and Inequalities. Arithmetic Operations. Absolute Value ................ 5 1.2.1. Equalities and Inequalities ........................................... 5 1.2.2. Addition and Multiplication of Numbers ................................ 6 1.2.3. Ratios and Proportions .............................................. 6 1.2.4. Percentage ....................................................... 7 1.2.5. Absolute Value of a Number (Modulus of a Number) ...................... 8 1.3. Powers and Logarithms ................................................... 8 1.3.1. Powers and Roots .................................................. 8 1.3.2. Logarithms ....................................................... 9 1.4. Binomial Theorem and Related Formulas ..................................... 10 1.4.1. Factorials. Binomial Coefficients. Binomial Theorem ..................... 10 1.4.2. Related Formulas .................................................. 10 1.5. Arithmetic and Geometric Progressions. Finite Sums and Products ................. 11 1.5.1. Arithmetic and Geometric Progressions ................................ 11 1.5.2. Finite Series and Products ........................................... 12 1.6. Mean Values and Inequalities of General Form ................................. 13 1.6.1. Arithmetic Mean, Geometric Mean, and Other Mean Values. Inequalities for Mean Values ...................................................... 13 1.6.2. Inequalities of General Form ......................................... 14 1.7. Some Mathematical Methods .............................................. 15 1.7.1. Proof by Contradiction .............................................. 15 1.7.2. Mathematical Induction ............................................. 16 1.7.3. Proof by Counterexample ........................................... 17 1.7.4. Method of Undetermined Coefficients .................................. 17 References for Chapter 1 ...................................................... 18 2. Elementary Functions ..................................................... 19 2.1. Power, Exponential, and Logarithmic Functions ................................ 19 2.1.1. Power Function: y = xa ............................................ 19 2.1.2. Exponential Function: y = ax ........................................ 21 2.1.3. Logarithmic Function: y = loga χ ..................................... 22 2.2. Trigonometric Functions .................................................. 24 2.2.1. Trigonometric Circle. Definition of Trigonometric Functions ............... 24 2.2.2. Graphs of Trigonometric Functions .................................... 25 í 2.2.3. Properties of Trigonometric Functions ...........,,., ...і. ............ 27 vi Contents 2.3. Inverse Trigonometrie Functions............................................ 30 2.3.1. Definitions. Graphs of Inverse Trigonometric Functions ................... 30 2.3.2. Properties of Inverse Trigonometric Functions ........................... 33 2.4. Hyperbolic Functions ..................................................... 34 2.4.1. Definitions. Graphs of Hyperbolic Functions ............................ 34 2.4.2. Properties of Hyperbolic Functions .................................... 36 2.5. Inverse Hyperbolic Functions .............................................. 39 2.5.1. Definitions. Graphs of Inverse Hyperbolic Functions ...................... 39 2.5.2. Properties of Inverse Hyperbolic Functions .............................. 41 References for Chapter 2 ...................................................... 42 3. Elementary Geometry .................................................... 43 3.1. Plane Geometry ......................................................... 43 3.1.1. Triangles ......................................................... 43 3.1.2. Polygons ......................................................... 51 3.1.3. Circle ........................................................... 56 3.2. Solid Geometry ......................................................... 59 3.2.1. Straight Lines, Planes, and Angles in Space ............................. 59 3.2.2. Polyhedra ........................................................ 61 3.2.3. Solids Formed by Revolution of Lines ................................. 65 3.3. Spherical Trigonometry ................................................... 70 3.3.1. Spherical Geometry ................................................ 70 3.3.2. Spherical Triangles ................................................ 71 References for Chapter 3 ...................................................... 75 4. Analytic Geometry ....................................................... 77 4.1. Points, Segments, and Coordinates on Line and Plane ........................... 77 4.1.1. Coordinates on Line ................................................ 77 4.1.2. Coordinates on Plane ............................................... 78 4.1.3. Points and Segments on Plane ........................................ 81 4.2. Curves on Plane ......................................................... 84 4.2.1. Curves and Their Equations .......................................... 84 4.2.2. Main Problems of Analytic Geometry for Curves ......................... 88 4.3. Straight Lines and Points on Plane .......................................... 89 4.3.1. Equations of Straight Lines on Plane ................................... 89 4.3.2. Mutual Arrangement of Points and Straight Lines ........................ 93 4.4. Second-Order Curves ..................................................... 97 4.4.1. Circle ..............................................._ 97 4.4.2. Ellipse .......................................................... 98 4.4.3. Hyperbola ........................................................ q АЛА. Parabola .............................................. jO4 4.4.5. Transformation of Second-Order Curves to Canonical Form ................ 107 4.5. Coordinates, Vectors, Curves, and Surfaces in Space ............................ 113 4.5.1. Vectors. Cartesian Coordinate System ................................. 113 4.5.2. Coordinate Systems ................................................ 114 4.5.3. Vectors. Products of Vectors ......................................... 120 4.5.4. Curves and Surfaces in Space ........................................ 123 Contents vii 4.6. Line and Plane in Space................................................... 124 4.6.1. Plane in Space .................................................... 124 4.6.2. Line in Space ..................................................... 131 4.6.3. Mutual Arrangement of Points, Lines, and Planes ........................ 135 4.7. Quadric Surfaces (Quadrics) ............................................... 143 4.7.1. Quadrics (Canonical Equations) ...................................... 143 4.7.2. Quadrics (General Theory) .......................................... 148 References for Chapter 4 ...................................................... 153 5. Algebra ................................................................. 155 5.1. Polynomials and Algebraic Equations ........................................ 155 5.1.1. Polynomials and Their Properties ..................................... 155 5.1.2. Linear and Quadratic Equations ....................................... 157 5.1.3. Cubic Equations ................................................... 158 5.1.4. Fourth-Degree Equation ............................................. 159 5.1.5. Algebraic Equations of Arbitrary Degree and Their Properties .............. 161 5.2. Matrices and Determinants ................................................ 167 5.2.1. Matrices ......................................................... 167 5.2.2. Determinants ..................................................... 175 5.2.3. Equivalent Matrices. Eigenvalues ..................................... 180 5.3. Linear Spaces ........................................................... 187 5.3.1. Concept of a Linear Space. Its Basis and Dimension ...................... 187 5.3.2. Subspaces of Linear Spaces .......................................... 190 5.3.3. Coordinate Transformations Corresponding to Basis Transformations in a Linear Space ........................................................... 191 5.4. Euclidean Spaces ........................................................ 192 5.4.1. Real Euclidean Space ............................................... 192 5.4.2. Complex Euclidean Space (Unitary Space) .............................. 195 5.4.3. Banach Spaces and Hubert Spaces .................................... 196 5.5. Systems of Linear Algebraic Equations ....................................... 197 5.5.1. Consistency Condition for a Linear System ............................. 197 5.5.2. Finding Solutions of a System of Linear Equations ....................... 198 5.6. Linear Operators ........................................................ 204 5.6.1. Notion of a Linear Operator. Its Properties .............................. 204 5.6.2. Linear Operators in Matrix Form ...................................... 208 5.6.3. Eigenvectors and Eigenvalues of Linear Operators ........................ 209 5.7. Bilinear and Quadratic Forms .............................................. 213 5.7.1. Linear and Sesquilinear Forms ....................................... 213 5.7.2. Bilinear Forms .................................................... 214 5.7.3. Quadratic Forms ................................................... 216 5.7.4. Bilinear and Quadratic Forms in Euclidean Space ........................ 219 5.7.5. Second-Order Hypersurfaces ......................................... 220 5.8. Some Facts from Group Theory ............................................ 225 5.8.1. Groups and Their Basic Properties .................................... 225 5.8.2. Transformation Groups ............................................. 228 5.8.3. Group Representations .............................................. 230 References for Chapter 5 ...................................................... 233 viii Contents 6. Limits and Derivatives .................................................... 235 6.1. Basic Concepts of Mathematical Analysis .................................... 235 6.1.1. Number Sets. Functions of Real Variable ............................... 235 6.1.2. Limit of a Sequence ................................................ 237 6.1.3. Limit of a Function. Asymptotes ...................................... 240 6.1.4. Infinitely Small and Infinitely Large Functions ........................... 242 6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind ....... 243 6.1.6. Convex and Concave Functions ....................................... 245 6.1.7. Functions of Bounded Variation ...................................... 246 6.1.8. Convergence of Functions ........................................... 249 6.2. Differential Calculus for Functions of a Single Variable .......................... 250 6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning ......... 250 6.2.2. Table of Derivatives and Differentiation Rules ........................... 252 6.2.3. Theorems about Differentiable Functions. L Hospital Rule ................. 254 6.2.4. Higher-Order Derivatives and Differentials. Taylor s Formula ............... 255 6.2.5. Extremal Points. Points of Inflection ................................... 257 6.2.6. Qualitative Analysis of Functions and Construction of Graphs .............. 259 6.2.7. Approximate Solution of Equations (Root-Finding Algorithms for Continuous Functions) ....................................................... 260 6.3. Functions of Several Variables. Partial Derivatives .............................. 263 6.3.1. Point Sets. Functions. Limits and Continuity ............................ 263 6.3.2. Differentiation of Functions of Several Variables ......................... 264 6.3.3. Directional Derivative. Gradient. Geometrical Applications ................ 267 6.3.4. Extremal Points of Functions of Several Variables ........................ 269 6.3.5. Differential Operators of the Field Theory .............................. 272 References for Chapter 6 ...................................................... 272 7. Integrals ................................................................ 273 7.1. Indefinite Integral ........................................................ 273 7.1.1. Antiderivative. Indefinite Integral and Its Properties ....................... 273 7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. Integration Examples ........................................................ 274 7.1.3. Integration of Rational Functions ..................................... 276 7.1.4. Integration of Irrational Functions ..................................... 279 7.1.5. Integration of Exponential and Trigonometric Functions ................... 281 7.1.6. Integration of Polynomials Multiplied by Elementary Functions ............. 283 7.2. Definite Integral ......................................................... 286 7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the Definite Integral ....................................;.............. 286 7.2.2. Properties of Definite Integrals and Useful Formulas ...................... 287 7.2.3. General Reduction Formulas for the Evaluation of Integrals ................ 289 7.2.4. General Asymptotic Formulas for the Calculation of Integrals ............... 290 7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities. Arithmetic Mean and Geometric Mean of Functions ...................... 295 7.2.6. Geometric and Physical Applications of the Definite Integral ............... 299 7.2.7. Improper Integrals with Infinite Integration Limit ......................... 301 7.2.8. General Reduction Formulas for the Calculation of Improper Integrals ........ 304 7.2.9. General Asymptotic Formulas for the Calculation of Improper Integrals ....... 307 7.2.10. Improper Integrals of Unbounded Functions ............................ 308 7.2.11. Cauchy-Type Singular Integrals ...................................... 310 Contents ix 7.2.12. Stieltjes Integral .................................................. 312 7.2.13. Square Integrable Functions ........................................ 314 7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals .... 315 7.3. Double and Triple Integrals ................................................ 317 7.3.1. Definition and Properties of the Double Integral .......................... 317 7.3.2. Computation of the Double Integral ................................... 319 7.3.3. Geometric and Physical Applications of the Double Integral ................ 323 7.3.4. Definition and Properties of the Triple Integral ........................... 324 7.3.5. Computation of the Triple Integral. Some Applications. Iterated Integrals and Asymptotic Formulas ............................................... 325 7.4. Line and Surface Integrals ................................................. 329 7.4.1. Line Integral of the First Kind ........................................ 329 7.4.2. Line Integral of the Second Kind ...................................... 330 7.4.3. Surface Integral of the First Kind ..................................... 332 7.4.4. Surface Integral of the Second Kind ................................... 333 7.4.5. Integral Formulas of Vector Calculus .................................. 334 References for Chapter 7 ...................................................... 335 8. Series .................................................................. 337 8.1. Numerical Series and Infinite Products ....................................... 337 8.1.1. Convergent Numerical Series and Their Properties. Cauchy s Criterion ....... 337 8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms .......... 338 8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional Convergence ...................................................... 341 8.1.4. Multiplication of Series. Some Inequalities ............................. 343 8.1.5. Summation Methods. Convergence Acceleration ......................... 344 8.1.6. Infinite Products ................................................... 346 8.2. Functional Series ........................................................ 348 8.2.1. Pointwise and Uniform Convergence of Functional Series .................. 348 8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent Series ........................................................... 349 8.3. Power Series ............................................................ 350 8.3.1. Radius of Convergence of Power Series. Properties of Power Series .......... 350 8.3.2. Taylor and Maclaurin Power Series .................................... 352 8.3.3. Operations with Power Series. Summation Formulas for Power Series ........ 354 8.4. Fourier Series ........................................................... 357 8.4.1. Representation of гтг-Регккііс Functions by Fourier Series. Main Results ..... 357 8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions ....... 359 8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series ......... 361 8.4.4. Summation Formulas for Trigonometric Series ........................... 362 8.5. Asymptotic Series ....................................................... 363 8.5.1. Asymptotic Series of Poincaré Type. Formulas for the Coefficients ........... 363 8.5.2. Operations with Asymptotic Series .................................... 364 References for Chapter 8 ...................................................... 366 9. Differential Geometry ..................................................... 367 9.1. Theory of Curves ........................................................ 367 9.1.1. Plane Curves ..................................................... 367 9.1.2. Space Curves ..................................................... 379 x Contents 9.2. Theory of Surfaces ....................................................... 386 9.2.1. Elementary Notions in Theory of Surfaces .............................. 386 9.2.2. Curvature of Curves on Surface ....................................... 392 9.2.3. Intrinsic Geometry of Surface ........................................ 395 References for Chapter 9 ...................................................... 397 10. Functions of Complex Variable ............................................ 399 10.1. Basic Notions .......................................................... 399 10.1.1. Complex Numbers. Functions of Complex Variable ..................... 399 10.1.2. Functions of Complex Variable ..................................... 401 10.2. Main Applications ...................................................... 419 10.2.1. Conformai Mappings ............................................. 419 10.2.2. Boundary Value Problems ......................................... 427 References for Chapter 10 ..................................................... 433 11. Integral Transforms ...................................................... 435 11.1. General Form of Integral Transforms. Some Formulas .......................... 435 11.1.1. Integral Transforms and Inversion Formulas ........................... 435 11.1.2. Residues. Jordan Lemma .......................................... 435 11.2. Laplace Transform ...................................................... 436 11.2.1. Laplace Transform and the Inverse Laplace Transform .................. 436 11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some Functions ...................................................... 437 11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series and Asymptotic Expansions ........................................ 440 11.3. Mellin Transform ....................................................... 441 11.3.1. Mellin Transform and the Inversion Formula .......................... 441 11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin, Laplace, and Fourier Transforms .................................... 442 11.4. Various Forms of the Fourier Transform ..................................... 443 11.4.1. Fourier Transform and the Inverse Fourier Transform ................... 443 11.4.2. Fourier Cosine and Sine Transforms ................................. 445 11.5. Other Integral Transforms ................................................ 446 11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified Bessel Functions ................................................ 446 11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral Transforms ..................................................... 448 References for Chapter 11 .............·.·....................................... 451 12. Ordinary Differential Equations ........................................... 453 12.1. First-Order Differential Equations .......................................... 453 12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 453 12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration ___ 456 12.1.3. Exact Differential Equations. Integrating Factor ........................ 458 12.1.4. Riccati Equation ................................................. 460 12.1.5. Abel Equations of the First Kind .................................... 462 12.1.6. Abel Equations of the SecondKind .................................. 464 12.1.7. Equations Not Solved for the Derivative .............................. 465 12.1.8. Contact Transformations .......................................... 468 12.1.9. Approximate Analytic Methods for Solution of Equations ................ 469 12.1.10. Numerical Integration of Differential Equations ....................... 471 Contents xi 12.2. Second-Order Linear Differential Equations.................................. 472 12.2.1. Formulas for the General Solution. Some Transformations ............... 472 12.2.2. Representation of Solutions as a Series in the Independent Variable ........ 475 12.2.3. Asymptotic Solutions ............................................. 477 12.2.4. Boundary Value Problems ......................................... 480 12.2.5. Eigenvalue Problems ............................................. 482 12.2.6. Theorems on Estimates and Zeros of Solutions ......................... 487 12.3. Second-Order Nonlinear Differential Equations ............................... 488 12.3.1. Form of the General Solution. Cauchy Problem ........................ 488 12.3.2. Equations Admitting Reduction of Order ............................. 489 12.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable ....................................................... 492 12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations. Painlevé Transcendents ........................................... 494 12.3.5. Perturbation Methods of Mechanics and Physics ....................... 499 12.3.6. Galerkin Method and Its Modifications (Projection Methods) ............. 508 12.3.7. Iteration and Numerical Methods ................................... 511 12.4. Linear Equations of Arbitrary Order ........................................ 514 12.4.1. Linear Equations with Constant Coefficients .......................... 514 12.4.2. Linear Equations with Variable Coefficients ........................... 518 12.4.3. Asymptotic Solutions of Linear Equations ............................ 522 12.4.4. Collocation Method and Its Convergence ............................. 523 12.5. Nonlinear Equations of Arbitrary Order ..................................... 524 12.5.1. Structure of the General Solution. Cauchy Problem ..................... 524 12.5.2. Equations Admitting Reduction of Order ............................. 525 12.6. Linear Systems of Ordinary Differential Equations ............................ 528 12.6.1. Systems of Linear Constant-Coefficient Equations ...................... 528 12.6.2. Systems of Linear Variable-Coefficient Equations ...................... 539 12.7. Nonlinear Systems of Ordinary Differential Equations .......................... 542 12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems ......... 542 12.7.2. Integrable Combinations. Autonomous Systems of Equations ............. 545 12.7.3. Elements of Stability Theory ....................................... 546 References for Chapter 12 ..................................................... 550 13. First-Order Partial Differential Equations ................................... 553 13.1. Linear and Quasilinear Equations .......................................... 553 13.1.1. Characteristic System. General Solution .............................. 553 13.1.2. Cauchy Problem. Existence and Uniqueness Theorem ................... 556 13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations .. 558 13.1.4. Quasilinear Equations of General Form. Generalized Solution, Jump Condition, and Stability Condition .................................. 567 13.2. Nonlinear Equations ..................................................... 570 13.2.1. Solution Methods ................................................ 570 13.2.2. Cauchy Problem. Existence and Uniqueness Theorem ................... 576 13.2.3. Generalized Viscosity Solutions and Their Applications ................. 579 References for Chapter 13 ..................................................... 584 xii Contents 14. Linear Partial Differential Equations....................................... 585 14.1. Classification of Second-Order Partial Differential Equations .................... 585 14.1.1. Equations with Two Independent Variables ............................ 585 14.1.2. Equations with Many Independent Variables .......................... 589 14.2. Basic Problems of Mathematical Physics .................................... 590 14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 590 14.2.2. First, Second, Third, and Mixed Boundary Value Problems ............... 593 14.3. Properties and Exact Solutions of Linear Equations ............................ 594 14.3.1. Homogeneous Linear Equations and Their Particular Solutions ............ 594 14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions ......... 598 14.3.3. General Solutions of Some Hyperbolic Equations ...................... 600 14.4. Method of Separation of Variables (Fourier Method) ........................... 602 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution ....................................................... 602 14.4.2. Problems for Parabolic Equations: Final Stage of Solution ............... 605 14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution .............. 607 14.4.4. Solution of Boundary Value Problems for Elliptic Equations .............. 609 14.5. Integral Transforms Method .............................................. 611 14.5.1. Laplace Transform and Its Application in Mathematical Physics ........... 611 14.5.2. Fourier Transform and Its Application in Mathematical Physics ........... 614 14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution .. 615 14.6.1. Cauchy Problem for Parabolic Equations ............................. 615 14.6.2. Cauchy Problem for Hyperbolic Equations ............................ 617 14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green s Function .............................................................. 618 14.7.1. Representation of Solutions via the Green s Function .................... 618 14.7.2. Problems for Equation s(x)^ = -^ [p(x)^]-q(x)w + Ф(х, t) ........... 620 14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green s Function. Goursat Problem ............................................... 623 14.8.1. Representation of Solutions via the Green s Function .................... 623 14.8.2. Problems for Equation s(x)Š$- = -J^ [p(x)^]-q(x)w + Ф(х, t) .......... 624 14.8.3. Problems for Equation ^ + a(i)-f^ = ò(í){ -¿fe [pix)%¿] - q{x)w) + Ф(ж, ť) 626 14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve ..... 627 14.8.5. Goursat Problem (a Problem with Initial Data of Characteristics) .......... 629 14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables ......... 631 14.9.1. Problems and the Green s Functions for Equation u™ _i_ vw _l ij^õw j л/пъ „„ — rb//. «л ¿ro i ■ ^v· » У J .......................... Ό j L 14.9.2. Representation of Solutions to Boundary Value Problems via the Green s Functions ...................................................... 633 14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green s Function ................................................ 634 14.10.1. Problems for Parabolic Equations ................................. 634 14.10.2. Problems for Hyperbolic Equations ................................ 636 14.10.3. Problems for Elliptic Equations ................................... 637 14.10.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types ...................................... 638 Contents xiii 14.11. Construction of the Green s Functions. General Formulas and Relations .......... 639 14.11.1. Green s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains ...................................... 639 14.11.2. Green s Functions Admitting Incomplete Separation of Variables ........ 640 14.11.3. Construction of Green s Functions via Fundamental Solutions .......... 642 14.12. Duhamel s Principles in Nonstationary Problems ............................. 646 14.12.1. Problems for Homogeneous Linear Equations ....................... 646 14.12.2. Problems for Nonhomogeneous Linear Equations .................... 648 14.13. Transformations Simplifying Initial and Boundary Conditions .................. 649 14.13.1. Transformations That Lead to Homogeneous Boundary Conditions ...... 649 14.13.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions ................................................... 650 References for Chapter 14 ..................................................... 650 15. Nonlinear Partial Differential Equations .................................... 653 15.1. Classification of Second-Order Nonlinear Equations ........................... 653 15.1.1. Classification of Semilinear Equations in Two Independent Variables ....... 653 15.1.2. Classification of Nonlinear Equations in Two Independent Variables ........ 653 15.2. Transformations of Equations of Mathematical Physics ......................... 655 15.2.1. Point Transformations: Overview and Examples ....................... 655 15.2.2. Hodograph Transformations (Special Point Transformations) ............. 657 15.2.3. Contact Transformations. Legendre and Euler Transformations ............ 660 15.2.4. Bäcklund Transformations. Differential Substitutions ................... 663 15.2.5. Differential Substitutions .......................................... 666 15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. Similarity Method ...................................................... 667 15.3.1. Preliminary Remarks ............................................. 667 15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations ..... 667 15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations ................................................. 669 15.3.4. Equations Invariant Under Combinations of Translation and Scaling Transformations, and Their Solutions ................................ 674 15.3.5. Generalized Self-Similar Solutions .................................. 677 15.4. Exact Solutions with Simple Separation of Variables ........................... 678 15.4.1. Multiplicative and Additive Separable Solutions ....................... 678 15.4.2. Simple Separation of Variables in Nonlinear Partial Differential Equations ... 678 15.4.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . 679 15.5. Method of Generalized Separation of Variables ............................... 681 15.5.1. Structure of Generalized Separable Solutions .......................... 681 15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions ........................................... 683 15.5.3. Solution of Functional Differential Equations by Differentiation ........... 684 15.5.4. Solution of Functional-Differential Equations by Splitting ................ 688 15.5.5. Titov-Galaktionov Method ........................................ 693 15.6. Method of Functional Separation of Variables ................................ 697 15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities .............................. 697 15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions ...................................................... 697 xiv Contents 15.6.3. Differentiation Method ........................................... 700 15.6.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications ............................................... 704 15.7. Direct Method of Symmetry Reductions of Nonlinear Equations .................. 708 15.7.1. Clarkson-Kruskal Direct Method ................................... 708 15.7.2. Some Modifications and Generalizations ............................. 712 15.8. Classical Method of Studying Symmetries of Differential Equations ............... 716 15.8.1. One-Parameter Transformations and Their Local Properties .............. 716 15.8.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition .... 719 15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions ...................................................... 724 15.8.4. Some Generalizations. Higher-Order Equations ........................ 730 15.9. Nonclassical Method of Symmetry Reductions ................................ 732 15.9.1. Description of the Method. Invariant Surface Condition ................. 732 15.9.2. Examples: The Newell-Whitehead Equation and a Nonlinear Wave Equation 733 15.10. Differential Constraints Method .......................................... 737 15.10.1. Description of the Method ....................................... 737 15.10.2. First-Order Differential Constraints ................................ 739 15.10.3. Second- and Higher-Order Differential Constraints ................... 744 15.10.4. Connection Between the Differential Constraints Method and Other Methods ..................................................... 746 15.11. Painlevé Test for Nonlinear Equations of Mathematical Physics ................. 748 15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method Description ................................................... 748 15.11.2. Examples of Performing the Painlevé Test and Truncated Expansions for Studying Nonlinear Equations .................................... 750 15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painlevé Test, Using Truncated Expansions ................................. 753 15.12. Methods of the Inverse Scattering Problem (Soliton Theory) .................... 755 15.12.1. Method Based on Using Lax Pairs ................................. 755 15.12.2. Method Based on a Compatibility Condition for Systems of Linear Equations .................................................... 757 15.12.3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760 15.13. Conservation Laws and Integrals of Motion ................................. 766 15.13.1. Basic Definitions and Examples .................................. 766 15.13.2. Equations Admitting Variational Formulation. Noetherian Symmetries ... 767 15.14. Nonlinear Systems of Partial Differential Equations ........................... 770 15.14.1. Overdetermined Systems of Two Equations ......................... 770 15.14.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems ..................................................... 772 15.14.3. Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction .......................................... 775 15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type ........................... 780 15.14.5. Systems of Second-Order Equations of Reaction-Diffusion Type ........ 796 References for Chapter 15..................................................... 798 Contents xv 16. Integral Equations ...................................................... 801 16.1. Linear Integral Equations of the First Kind with Variable Integration Limit ......... 801 16.1.1. Volterra Equations of the First Kind ................................. 801 16.1.2. Equations with Degenerate Kernel: K(x,t) = g (x)hi(t) + ■■■ + gn(x)hn(t) . . 802 16.1.3. Equations with Difference Kernel: K(x, t) = K(x -ť) ................... 804 16.1.4. Reduction of Volterra Equations of the First Kind to Volterra Equations of the SecondKind .................................................... 807 16.1.5. Method of Quadratures ........................................... 808 16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit ....... 810 16.2.1. Volterra Equations of the Second Kind ............................... 810 16.2.2. Equations with Degenerate Kernel: K(x, t) = g (x)h (t) + · · · + gn(x)hn(t) .. 811 16.2.3. Equations with Difference Kernel: K(x, t) = K(x -ť) ................... 813 16.2.4. Construction of Solutions of Integral Equations with Special Right-Hand Side 815 16.2.5. Method of Model Solutions ........................................ 818 16.2.6. Successive Approximation Method .................................. 822 16.2.7. Method of Quadratures ........................................... 823 16.3. Linear Integral Equations of the First Kind with Constant Limits of Integration ...... 824 16.3.1. Fredholm Integral Equations of the First Kind ......................... 824 16.3.2. Method of Integral Transforms ..................................... 825 16.3.3. Regularizaron Methods ........................................... 827 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration .... 829 16.4.1. Fredholm Integral Equations of the Second Kind. Resolvent .............. 829 16.4.2. Fredholm Equations of the Second Kind with Degenerate Kernel .......... 830 16.4.3. Solution as a Power Series in the Parameter. Method of Successive Approximations ................................................. 832 16.4.4. Fredholm Theorems and the Fredholm Alternative ...................... 834 16.4.5. Fredholm Integral Equations of the Second Kind with Symmetric Kernel .... 835 16.4.6. Methods of Integral Transforms .................................... 841 16.4.7. Method of Approximating a Kernel by a Degenerate One ................ 844 16.4.8. Collocation Method .............................................. 847 16.4.9. Method of Least Squares .......................................... 849 16.4.10. Bubnov-Galerkin Method ........................................ 850 16.4.11. Quadrature Method ............................................. 852 16.4.12. Systems of Fredholm Integral Equations of the Second Kind ............. 854 16.5. Nonlinear Integral Equations .............................................. 856 16.5.1. Nonlinear Volterra and Urysohn Integral Equations ..................... 856 16.5.2. Nonlinear Volterra Integral Equations ................................ 856 16.5.3. Equations with Constant Integration Limits ........................... 863 References for Chapter 16 ..................................................... 871 17. Difference Equations and Other Functional Equations ........................ 873 17.1. Difference Equations of Integer Argument ................................... 873 17.1.1. First-Order Linear Difference Equations of Integer Argument ............. 873 17.1.2. First-Order Nonlinear Difference Equations of Integer Argument .......... 874 17.1.3. Second-Order Linear Difference Equations with Constant Coefficients ...... 877 17.1.4. Second-Order Linear Difference Equations with Variable Coefficients ...... 879 17.1.5. Linear Difference Equations of Arbitrary Order with Constant Coefficients .. 881 17.1.6. Linear Difference Equations of Arbitrary Order with Variable Coefficients ... 882 17.1.7. Nonlinear Difference Equations of Arbitrary Order ..................... 884 xvi Contents 17.2. Linear Difference Equations with a Single Continuous Variable .................. 885 17.2.1. First-Order Linear Difference Equations .............................. 885 17.2.2. Second-Order Linear Difference Equations with Integer Differences ........ 894 17.2.3. Linear mth-Order Difference Equations with Integer Differences .......... 898 17.2.4. Linear mth-Order Difference Equations with Arbitrary Differences ........ 904 17.3. Linear Functional Equations .............................................. 907 17.3.1. Iterations of Functions and Their Properties ........................... 907 17.3.2. Linear Homogeneous Functional Equations ........................... 910 17.3.3. Linear Nonhomogeneous Functional Equations ........................ 912 17.3.4. Linear Functional Equations Reducible to Linear Difference Equations with Constant Coefficients ............................................. 916 17.4. Nonlinear Difference and Functional Equations with a Single Variable ............. 918 17.4.1. Nonlinear Difference Equations with a Single Variable .................. 918 17.4.2. Reciprocal (Cyclic) Functional Equations ............................. 919 17.4.3. Nonlinear Functional Equations Reducible to Difference Equations ........ 921 17.4.4. Power Series Solution of Nonlinear Functional Equations ................ 922 17.5. Functional Equations with Several Variables .................................. 922 17.5.1. Method of Differentiation in a Parameter ............................. 922 17.5.2. Method of Differentiation in Independent Variables ..................... 925 17.5.3. Method of Substituting Particular Values of Independent Arguments ....... 926 17.5.4. Method of Argument Elimination by Test Functions .................... 928 17.5.5. Bilinear Functional Equations and Nonlinear Functional Equations Reducible to Bilinear Equations ............................................. 930 References for Chapter 17 ..................................................... 935 18. Special Functions and Their Properties ..................................... 937 18.1. Some Coefficients, Symbols, and Numbers ................................... 937 18.1.1. Binomial Coefficients ............................................ 937 18.1.2. Pochhammer Symbol ............................................. 938 18.1.3. Bernoulli Numbers ............................................... 938 18.1.4. Euler Numbers .................................................. 939 18.2. Error Functions. Exponential and Logarithmic Integrals ........................ 939 18.2.1. Error Function and Complementary Error Function ..................... 939 18.2.2. Exponential Integral .............................................. 940 18.2.3. Logarithmic Integral ............................................. 941 18.3. Sine Integral and Cosine Integral. Fresnel Integrals ............................ 941 18.3.1. Sine Integral .................................................... 941 18.3.2. Cosine Integral .................................................. 942 18.3.3. Fresnel Integrals ................................................. 942 18.4. Gamma Function, Psi Function, and Beta Function ............................ 943 18.4.1. Gamma Function ................................................ 943 18.4.2. Psi Function (Digamma Function) ................................... 944 18.4.3. Beta Function ................................................... 945 18.5. Incomplete Gamma and Beta Functions ..................................... 946 18.5.1. Incomplete Gamma Function ....................................... 946 18.5.2. Incomplete Beta Function ......................................... 947 Contents xvii 18.6. Bessel Functions (Cylindrical Functions) .................................... 947 18.6.1. Definitions and Basic Formulas ..................................... 947 18.6.2. Integral Representations and Asymptotic Expansions .................... 949 18.6.3. Zeros and Orthogonality Properties of Bessel Functions ................. 951 18.6.4. Hankel Functions (Bessel Functions of the Third Kind) .................. 952 18.7. Modified Bessel Functions ................................................ 953 18.7.1. Definitions. Basic Formulas ....................................... 953 18.7.2. Integral Representations and Asymptotic Expansions .................... 954 18.8. Airy Functions ......................................................... 955 18.8.1. Definition and Basic Formulas ...................................... 955 18.8.2. Power Series and Asymptotic Expansions ............................. 956 18.9. Degenerate Hypergeometric Functions (Kummer Functions) ..................... 956 18.9.1. Definitions and Basic Formulas ..................................... 956 18.9.2. Integral Representations and Asymptotic Expansions .................... 959 18.9.3. Whittaker Functions .............................................. 960 18.10. Hypergeometric Functions ............................................... 960 18.10.1. Various Representations of the Hypergeometric Function .............. 960 18.10.2. Basic Properties ............................................... 960 18.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions ... 962 18.11.1. Legendre Polynomials and Legendre Functions ...................... 962 18.11.2. Associated Legendre Functions with Integer Indices and Real Argument .. 964 18.11.3. Associated Legendre Functions. General Case ....................... 965 18.12. Parabolic Cylinder Functions ............................................. 967 18.12.1. Definitions. Basic Formulas ..................................... 967 18.12.2. Integral Representations, Asymptotic Expansions, and Linear Relations ... 968 18.13. Elliptic Integrals ....................................................... 969 18.13.1. Complete Elliptic Integrals ...................................... 969 18.13.2. Incomplete Elliptic Integrals (Elliptic Integrals) ...................... 970 18.14. Elliptic Functions ...................................................... 972 18.14.1. Jacobi Elliptic Functions ........................................ 972 18.14.2. Weierstrass Elliptic Function ..................................... 976 18.15. Jacobi Theta Functions ................................................. 978 18.15.1. Series Representation of the Jacobi Theta Functions. Simplest Properties .. 978 18.15.2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions 978 18.16. Mathieu Functions and Modified Mathieu Functions .......................... 980 18.16.1. Mathieu Functions ............................................. 980 18.16.2. Modified Mathieu Functions ..................................... 982 18.17. Orthogonal Polynomials ................................................ 982 18.17.1. Laguerre Polynomials and Generalized Laguerre Polynomials ........... 982 18.17.2. Chebyshev Polynomials and Functions ............................. 983 18.17.3. Hermite Polynomials ........................................... 985 18.17.4. Jacobi Polynomials and Gegenbauer Polynomials .................... 986 18.18. Nonorthogonal Polynomials ............................................. 988 18.18.1. Bernoulli Polynomials .......................................... 988 18.18.2. Euler Polynomials ............................................. 989 References for Chapter 18 ..................................................... 990 xviii ________ Contents _____________________________ 19. Calculus of Variations and Optimization ................. .................. 991 19.1. Calculus of Variations and Optimal Control .................................. 991 19.1.1. Some Definitions and Formulas ..................................... 991 19.1.2. Simplest Problem of Calculus of Variations ........................... 993 19.1.3. Isoperimetric Problem ............................................1002 19.1.4. Problems with Higher Derivatives ...................................1006 19.1.5. Lagrange Problem ...............................................1008 19.1.6. Pontryagin Maximum Principle .....................................1010 19.2. Mathematical Programming ...............................................1012 19.2.1. Linear Programming .............................................1012 19.2.2. Nonlinear Programming ...........................................1027 References for Chapter 19 .....................................................1028 20. Probability Theory ......................................................1031 20.1. Simplest Probabilistic Models .............................................1031 20.1.1. Probabilities of Random Events .....................................1031 20.1.2. Conditional Probability and Simplest Formulas ........................1035 20.1.3. Sequences of Trials ..............................................1037 20.2. Random Variables and Their Characteristics ..................................1039 20.2.1. One-Dimensional Random Variables .................................1039 20.2.2. Characteristics of One-Dimensional Random Variables ..................1042 20.2.3. Main Discrete Distributions ........................................1047 20.2.4. Continuous Distributions ..........................................1051 20.2.5. Multivariate Random Variables .....................................1057 20.3. Limit Theorems ........................................................1068 20.3.1. Convergence of Random Variables ..................................1068 20.3.2. Limit Theorems .................................................1069 20.4. Stochastic Processes .....................................................1071 20.4.1. Theory of Stochastic Processes .....................................1071 20.4.2. Models of Stochastic Processes .....................................1074 References for Chapter 20.....................................................1079 21. Mathematical Statistics ..................................................1081 21.1. Introduction to Mathematical Statistics ......................................1081 21.1.1. Basic Notions and Problems of Mathematical Statistics ..................1081 21.1.2. Simplest Statistical Transformations .................................1082 21.1.3. Numerical Characteristics of Statistical Distribution ....................1087 21.2. Statistical Estimation ....................................................1088 21.2.1. Estimators and Their Properties .....................................1088 21.2.2. Estimation Methods for Unknown Parameters .........................1091 21.2.3. Interval Estimators (Confidence Intervals) ............................1093 21.3. Statistical Hypothesis Testing .............................................1094 21.3.1. Statistical Hypothesis. Test ........................................1094 21.3.2. Goodness-of-Fit Tests ............................................1098 21.3.3. Problems Related to Normal Samples ................................1101 References for Chapter 21 .............................................. П09 Contents xix Partii. Mathematical Tables 1111 Tl. Finite Sums and Infinite Series ............................................1113 TU. Finite Sums ...........................................................1113 Tl.l.l. Numerical Sum .................................................1113 Tl.1.2. Functional Sums ................................................1116 T1.2. Infinite Series ..........................................................1118 Tl.2.1. Numerical Series ................................................1118 Tl.2.2. Functional Series ................................................1120 References for Chapter Tl .....................................................1127 T2. Integrals ...............................................................1129 T2.1. Indefinite Integrals ......................................................1129 T2.1.1. Integrals Involving Rational Functions ...............................1129 T2. T2. T2. T2. T2. T2. .2. Integrals Involving Irrational Functions ...............................1134 .3. Integrals Involving Exponential Functions ............................1137 .4. Integrals Involving Hyperbolic Functions .............................1137 .5. Integrals Involving Logarithmic Functions ............................1140 .6. Integrals Involving Trigonometric Functions ...........................1142 .7. Integrals Involving Inverse Trigonometric Functions ....................1147 T2.2. Tables of Definite Integrals ...............................................1147 T2.2.1. Integrals Involving Power-Law Functions .............................1147 T2.2.2. Integrals Involving Exponential Functions ............................1150 T2.2.3. Integrals Involving Hyperbolic Functions .............................1152 T2.2.4. Integrals Involving Logarithmic Functions ............................1152 T2.2.5. Integrals Involving Trigonometric Functions ...........................1153 References for Chapter T2 .....................................................1155 T3. Integral Transforms .....................................................1157 T3.1. Tables of Laplace Transforms .............................................1157 T3.1.1. General Formulas ................................................1157 T3.1.2. Expressions with Power-Law Functions ..............................1159 T3.1.3. Expressions with Exponential Functions ..............................1159 T3.1.4. Expressions with Hyperbolic Functions ..............................1160 T3. 1.5. Expressions with Logarithmic Functions ..............................1161 T3.1.6. Expressions with Trigonometric Functions ............................1161 T3.1.7. Expressions with Special Functions ..................................1163 T3.2. Tables of Inverse Laplace Transforms .......................................1164 T3.2.1. General Formulas ................................................1164 T3.2.2. Expressions with Rational Functions .................................1166 T3.2.3. Expressions with Square Roots .....................................1170 T3.2.4. Expressions with Arbitrary Powers ..................................1172 T3.2.5. Expressions with Exponential Functions ..............................1172 T3.2.6. Expressions with Hyperbolic Functions ..............................1174 T3.2.7. Expressions with Logarithmic Functions ..............................1174 T3.2.8. Expressions with Trigonometric Functions ............................1175 T3.2.9. Expressions with Special Functions ..................................1176 xx Contents ТЗ.З. Tables of Fourier Cosine Transforms ........................................1177 T3.3.1. General Formulas ................................................1177 T3.3.2. Expressions with Power-Law Functions ..............................1177 T3.3.3. Expressions with Exponential Functions ..............................1178 T3.3.4. Expressions with Hyperbolic Functions ..............................1179 T3.3.5. Expressions with Logarithmic Functions ..............................1179 T3.3.6. Expressions with Trigonometric Functions ............................1180 T3.3.7. Expressions with Special Functions ..................................1181 T3.4. Tables of Fourier Sine Transforms ..........................................1182 T3.4.1. General Formulas ................................................1182 T3.4.2. Expressions with Power-Law Functions ..............................1182 T3.4.3. Expressions with Exponential Functions ..............................1183 T3.4.4. Expressions with Hyperbolic Functions ..............................1184 T3.4.5. Expressions with Logarithmic Functions ..............................1184 T3.4.6. Expressions with Trigonometric Functions ............................1185 T3.4.7. Expressions with Special Functions ..................................1186 T3.5. Tables of Mellin Transforms ..............................................1187 ТЗ.о.і. General Formulas ................................................1187 T3.5.2. Expressions with Power-Law Functions ..............................1188 T3.5.3. Expressions with Exponential Functions ..............................1188 T3.5.4. Expressions with Logarithmic Functions ..............................1189 T3.5.5. Expressions with Trigonometric Functions ............................1189 T3.5.6. Expressions with Special Functions ..................................1190 T3.6. Tables of Inverse Mellin Transforms ........................................1190 T3.6.1. Expressions with Power-Law Functions ..............................1190 T3.6.2. Expressions with Exponential and Logarithmic Functions ................1191 T3.6.3. Expressions with Trigonometric Functions ............................1192 T3.6.4. Expressions with Special Functions ..................................1193 References for Chapter T3 .....................................................1194 T4. Orthogonal Curvilinear Systems of Coordinate ..............................1195 T4.1. Arbitrary Curvilinear Coordinate Systems ...................................1195 T4.1.1. General Nonorthogonal Curvilinear Coordinates .......................1195 T4.1.2. General Orthogonal Curvilinear Coordinates ..........................1196 T4.2. Special Curvilinear Coordinate Systems .....................................1198 T4.2.1. Cylindrical Coordinates ...........................................1198 T4.2.2. Spherical Coordinates ............................................1199 T4.2.3. Coordinates of a Prolate Ellipsoid of Revolution .......................1200 T4.2.4. Coordinates of an Oblate Ellipsoid of Revolution .......................1201 T4.2.5. Coordinates of an Elliptic Cylinder ..................................1202 T4.2.6. Conical Coordinates ..............................................1202 T4.2.7. Parabolic Cylinder Coordinates .....................................1203 T4.2.8. Parabolic Coordinates ............................................1203 T4.2.9. Bicylindrical Coordinates .........................................1204 T4.2.10. Bipolar Coordinates (in Space) ....................................1204 T4.2.1 1. Toroidal Coordinates ............................................1205 References for Chapter T4 ...........................,......................... 1205 Contents xxi Т5. Ordinary Differential Equations ...........................................1207 T5.1. First-Order Equations ....................................................1207 T5.2. Second-Order Linear Equations ............................................1212 T5.2.1. Equations Involving Power Functions ................................1213 T5.2.2. Equations Involving Exponential and Other Functions ...................1220 T5.2.3. Equations Involving Arbitrary Functions ..............................1222 T5.3. Second-Order Nonlinear Equations .........................................1223 ТЅ.ЗЛ. Equations ofthe Form y%x = f(x,y) .................................1223 T5.3.2. Equations ofthe Form f(x, y)y x x = g(x, y,y x) .........................1225 References for Chapter T5 .....................................................1228 T6. Systems of Ordinary Differential Equations .................................1229 T6. 1. Linear Systems of Two Equations ..........................................1229 Тб.і.і. Systems of First-Order Equations ...................................1229 T6.1.2. Systems of Second-Order Equations .................................1232 T6.2. Linear Systems of Three and More Equations .................................1237 T6.3. Nonlinear Systems of Two Equations .......................................1239 T6.3.1. Systems of First-Order Equations ...................................1239 T6.3.2. Systems of Second-Order Equations .................................1240 T6.4. Nonlinear Systems of Three or More Equations ...............................1244 References for Chapter T6 .....................................................1246 T7. First-Order Partial Differential Equations ..................................1247 T7.1. Linear Equations .......................................................1247 T7.1.1. Equations of the Form f{x,y)^ + д(.х,у)^~ =0......................1247 T7.1.2. Equations of the Form f(x,y)^ + д(х,у)Щ; = h(x,y) .................1248 T7.1.3. Equations ofthe Form f(x,y)^ + д(х,у)щ = h(x,y)w + r(x,y) ........1250 T7.2. Quasilinear Equations ...................................................1252 T7.2.1. Equations of the Form f(x,y)^ + д(х,у)щт - h(x,y,w) ...............1252 T7.2.2. Equations ofthe Form f^ + f(x,y,w)^ = 0.........................1254 T7.2.3. Equations ofthe Form ^ + f(x, y, w)^- = g(x, y,w) ..................1256 T7.3. Nonlinear Equations .....................................................1258 T7.3.1. Equations Quadratic in One Derivative ...............................1258 T7.3.2. Equations Quadratic in Two Derivatives ..............................1259 T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives ...................1261 References for Chapter T7 .....................................................1265 T8. Linear Equations and Problems of Mathematical Physics ......................1267 T8.1. Parabolic Equations .....................................................1267 Τδ.Ι.Ι. Heat Equation ff = af^- .........................................1267 T8.1.2. Nonhomogeneous Heat Equation ^ = a^f + Ф(х,і) ..................1268 T8.1.3. Equation ofthe Form ff = aţ^ + fcf^ + cw + Φ(χ,ί) ................. 1270 T8.1.4. Heat Equation with Axial Symmetry ff = o(^f + 7^7)...............1270 T8.1.5. Equation ofthe Form ff = a(f^r + ff^) +Ф(г, ť) ...................1271 T8.1.6. Heat Equation with Central Symmetry ^f = a(\|^ + ^^) .............1272 18.1.7. Equation ofthe Form ff =a(fe + r If) +ф(г>ѓ> .................... 1273 T8.1.8. Equation ofthe Form ^f = -fgr + ^ψ-^ ...........................1274 xxii Contents 78.1.9. Equations of the Diffusion (Thermal) Boundary Layer ...................1276 TS.l.lO. Schrödinger Equation ¿ñff = -^^r + U(x)w .....................1276 T8.2. Hyperbolic Equations ....................................................1278 T8.2.1. Wave Equation ^ = a2ţ^ ......................................1278 T8.2.2. Equation of the Form ţ^=a2ţ^- + Ф(х, t) ..........................1279 T8.2.3. Klein-Gordon Equation ţg- = a2ţŞ--bw ...........................1280 T8.2.4. Equation of the Form ţ^ = a2ţ$- -bw + Ф(х, t) ......................1281 T8.2.5. Equation of the Form ^= α2 (^Ѕ + 1^)+Ф(г, í) ..................1282 T8.2.6. Equation of the Form ţţr = a (íz + r If) +ф(г > .................. 1283 T8.2.7. Equations of the Form ţ^ + к ff = a2 f^ + òf^ + cw + Φ(ζ, ί) .........1284 Τ8.3. Elliptic Equations .......................................................1284 T8.3.1. Laplace Equation Aw = 0 .........................................1284 T8.3.2. Poisson Equation Aw + Φ(χ) = 0 ...................................1287 T8.3.3. Helmholtz Equation Aw + w = -Ф(х) ..............................1289 T8.4. Fourth-Order Linear Equations ............................................1294 T8.4.1. Equation of the Form ξ$- + a2ţŞ- =0 ..............................1294 T8.4.2. Equation of the Form ţg- + a2\|^r = Φ(χ,ί) ..........................1295 T8.4.3. Biharmonic Equation ΔΔ«; =0 ....................................1297 T8.4.4. Nonhomogeneous Biharmonic Equation A Aw = Ф(х, у) ................ 1298 References for Chapter T8 .....................................................1299 T9. Nonlinear Mathematical Physics Equations .................................1301 T9.1. Parabolic Equations .....................................................1301 TO. 1.1. Nonlinear Heat Equations of the Form ^ = ^f + f(w) ................1301 T9.1.2. Equations of the Form ^ = £[f(w)^] + g(w) ......................1303 T9. 1.3. Burgers Equation and Nonlinear Heat Equation in Radial Symmetric Cases . . 1307 T9.1.4. Nonlinear Schrödinger Equations ...................................1309 T9.2. Hyperbolic Equations ....................................................1312 T9.2.1. Nonlinear Wave Equations of the Form &$- = af^ + f(w) ..............1312 T9.2.2. Other Nonlinear Wave Equations ....................................1316 T9.3. Elliptic Equations .......................................................1318 T9.3.1. Nonlinear Heat Equations of the Form ţ^- + ţş- = f (w) ................1318 T9.3.2. Equations of the Form £ [f(x)fé + % [g(y)%] = f(w) ..............1321 T9.3.3. Equations of the Form £ [f(w)fê] + £ [íK«)^] = Kw) ..............1322 T9.4. Other Second-Order Equations ................... ;___....................1324 T9.4.1. Equations of Transonic Gas Flow ...................................1324 T9.4.2. Monge-Ampère Equations ........................................1326 T9.5. Higher-Order Equations ..................................................1327 T9.5.1. Third-Order Equations ............................................1327 T9.5.2. Fourth-Order Equations ...........................................1332 References for Chapter T9 .....................................................1335 T10. Systems of Partial Differential Equations ..................................1337 TlO.l. Nonlinear Systems of Two First-Order Equations ..___.......................1337 T10.2. Linear Systems of Two Second-Order Equations .............................1341 Contents xxiii Т1О.З. Nonlinear Systems of Two Second-Order Equations ..........................1343 T10.3.1. Systems of the Form ff = α0 + F(u,w), ^f = òf^ + G(u,w) ......1343 TIO.3.2. Systems of the Form ^ = ^k-§^{xn^) + F(u,w), öt x71 ox ox / v * ................... .... T10.3.3. Systems of the Form Au = F(u, w), Aw = G(u, w) ..................1364 Т10.3.4. Systems of the Form ţg- - рг^(жп\|\|) + F(u,w), д2и> _±l/rnft»K η(η. „л 1 -ifrQ dt2 ~ xn дх У дх J V w> .................................. 1J ° Т10.3.5. Other Systems ................................................1373 T10.4. Systems of General Form ................................................1374 T10.4.1. Linear Systems ................................................1374 T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t . . 1374 T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378 T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in í . 1381 References for Chapter T10 ....................................................1382 Til. Integral Equations .....................................................1385 Tl 1.1. Linear Equations of the First Kind with Variable Limit of Integration .............1385 T11.2. Linear Equations of the Second Kind with Variable Limit of Integration ...........1391 Tl 1.3. Linear Equations of the First Kind with Constant Limits of Integration ............1396 T 11.4. Linear Equations of the Second Kind with Constant Limits of Integration .........1401 References for Chapter Til ....................................................1406 T12. Functional Equations ...................................................1409 Tl 2.1. Linear Functional Equations in One Independent Variable ......................1409 T 12.1.1. Linear Difference and Functional Equations Involving Unknown Function with Two Different Arguments ...................................1409 T12.1.2. Other Linear Functional Equations ................................1421 T12.2. Nonlinear Functional Equations in One Independent Variable ...................1428 T12.2.1. Functional Equations with Quadratic Nonlinearity ....................1428 T12.2.2. Functional Equations with Power Nonlinearity .......................1433 T12.2.3. Nonlinear Functional Equation of General Form .....................1434 T12.3. Functional Equations in Several Independent Variables ........................1438 T12.3.1. Linear Functional Equations .....................................1438 T12.3.2. Nonlinear Functional Equations ..................................1443 References for Chapter Tl 2....................................................1450 Supplement. Some Useful Electronic Mathematical Resources .....................1451 Index ....................................................................1453
adam_txt	CONTENTS Authors . xxv Preface .xxvii Main Notation . xxix Part I. Definitions, Formulas, Methods, and Theorems 1 1. Arithmetic and Elementary Algebra . 3 1.1. Real Numbers . 3 1.1.1. Integer Numbers . 3 1.1.2. Real, Rational, and Irrational Numbers . 4 1.2. Equalities and Inequalities. Arithmetic Operations. Absolute Value . 5 1.2.1. Equalities and Inequalities . 5 1.2.2. Addition and Multiplication of Numbers . 6 1.2.3. Ratios and Proportions . 6 1.2.4. Percentage . 7 1.2.5. Absolute Value of a Number (Modulus of a Number) . 8 1.3. Powers and Logarithms . 8 1.3.1. Powers and Roots . 8 1.3.2. Logarithms . 9 1.4. Binomial Theorem and Related Formulas . 10 1.4.1. Factorials. Binomial Coefficients. Binomial Theorem . 10 1.4.2. Related Formulas . 10 1.5. Arithmetic and Geometric Progressions. Finite Sums and Products . 11 1.5.1. Arithmetic and Geometric Progressions . 11 1.5.2. Finite Series and Products . 12 1.6. Mean Values and Inequalities of General Form . 13 1.6.1. Arithmetic Mean, Geometric Mean, and Other Mean Values. Inequalities for Mean Values . 13 1.6.2. Inequalities of General Form . 14 1.7. Some Mathematical Methods . 15 1.7.1. Proof by Contradiction . 15 1.7.2. Mathematical Induction . 16 1.7.3. Proof by Counterexample . 17 1.7.4. Method of Undetermined Coefficients . 17 References for Chapter 1 . 18 2. Elementary Functions . 19 2.1. Power, Exponential, and Logarithmic Functions . 19 2.1.1. Power Function: y = xa . 19 2.1.2. Exponential Function: y = ax . 21 2.1.3. Logarithmic Function: y = loga χ . 22 2.2. Trigonometric Functions . 24 2.2.1. Trigonometric Circle. Definition of Trigonometric Functions . 24 2.2.2. Graphs of Trigonometric Functions . 25 í 2.2.3. Properties of Trigonometric Functions .,,., .і. . 27 vi Contents 2.3. Inverse Trigonometrie Functions. 30 2.3.1. Definitions. Graphs of Inverse Trigonometric Functions . 30 2.3.2. Properties of Inverse Trigonometric Functions . 33 2.4. Hyperbolic Functions . 34 2.4.1. Definitions. Graphs of Hyperbolic Functions . 34 2.4.2. Properties of Hyperbolic Functions . 36 2.5. Inverse Hyperbolic Functions . 39 2.5.1. Definitions. Graphs of Inverse Hyperbolic Functions . 39 2.5.2. Properties of Inverse Hyperbolic Functions . 41 References for Chapter 2 . 42 3. Elementary Geometry . 43 3.1. Plane Geometry . 43 3.1.1. Triangles . 43 3.1.2. Polygons . 51 3.1.3. Circle . 56 3.2. Solid Geometry . 59 3.2.1. Straight Lines, Planes, and Angles in Space . 59 3.2.2. Polyhedra . 61 3.2.3. Solids Formed by Revolution of Lines . 65 3.3. Spherical Trigonometry . 70 3.3.1. Spherical Geometry . 70 3.3.2. Spherical Triangles . 71 References for Chapter 3 . 75 4. Analytic Geometry . 77 4.1. Points, Segments, and Coordinates on Line and Plane . 77 4.1.1. Coordinates on Line . 77 4.1.2. Coordinates on Plane . 78 4.1.3. Points and Segments on Plane . 81 4.2. Curves on Plane . 84 4.2.1. Curves and Their Equations . 84 4.2.2. Main Problems of Analytic Geometry for Curves . 88 4.3. Straight Lines and Points on Plane . 89 4.3.1. Equations of Straight Lines on Plane . 89 4.3.2. Mutual Arrangement of Points and Straight Lines . 93 4.4. Second-Order Curves . 97 4.4.1. Circle ._ 97 4.4.2. Ellipse . 98 4.4.3. Hyperbola . \q\ АЛА. Parabola . jO4 4.4.5. Transformation of Second-Order Curves to Canonical Form . 107 4.5. Coordinates, Vectors, Curves, and Surfaces in Space . 113 4.5.1. Vectors. Cartesian Coordinate System . 113 4.5.2. Coordinate Systems . 114 4.5.3. Vectors. Products of Vectors . 120 4.5.4. Curves and Surfaces in Space . 123 Contents vii 4.6. Line and Plane in Space. 124 4.6.1. Plane in Space . 124 4.6.2. Line in Space . 131 4.6.3. Mutual Arrangement of Points, Lines, and Planes . 135 4.7. Quadric Surfaces (Quadrics) . 143 4.7.1. Quadrics (Canonical Equations) . 143 4.7.2. Quadrics (General Theory) . 148 References for Chapter 4 . 153 5. Algebra . 155 5.1. Polynomials and Algebraic Equations . 155 5.1.1. Polynomials and Their Properties . 155 5.1.2. Linear and Quadratic Equations . 157 5.1.3. Cubic Equations . 158 5.1.4. Fourth-Degree Equation . 159 5.1.5. Algebraic Equations of Arbitrary Degree and Their Properties . 161 5.2. Matrices and Determinants . 167 5.2.1. Matrices . 167 5.2.2. Determinants . 175 5.2.3. Equivalent Matrices. Eigenvalues . 180 5.3. Linear Spaces . 187 5.3.1. Concept of a Linear Space. Its Basis and Dimension . 187 5.3.2. Subspaces of Linear Spaces . 190 5.3.3. Coordinate Transformations Corresponding to Basis Transformations in a Linear Space . 191 5.4. Euclidean Spaces . 192 5.4.1. Real Euclidean Space . 192 5.4.2. Complex Euclidean Space (Unitary Space) . 195 5.4.3. Banach Spaces and Hubert Spaces . 196 5.5. Systems of Linear Algebraic Equations . 197 5.5.1. Consistency Condition for a Linear System . 197 5.5.2. Finding Solutions of a System of Linear Equations . 198 5.6. Linear Operators . 204 5.6.1. Notion of a Linear Operator. Its Properties . 204 5.6.2. Linear Operators in Matrix Form . 208 5.6.3. Eigenvectors and Eigenvalues of Linear Operators . 209 5.7. Bilinear and Quadratic Forms . 213 5.7.1. Linear and Sesquilinear Forms . 213 5.7.2. Bilinear Forms . 214 5.7.3. Quadratic Forms . 216 5.7.4. Bilinear and Quadratic Forms in Euclidean Space . 219 5.7.5. Second-Order Hypersurfaces . 220 5.8. Some Facts from Group Theory . 225 5.8.1. Groups and Their Basic Properties . 225 5.8.2. Transformation Groups . 228 5.8.3. Group Representations . 230 References for Chapter 5 . 233 viii Contents 6. Limits and Derivatives . 235 6.1. Basic Concepts of Mathematical Analysis . 235 6.1.1. Number Sets. Functions of Real Variable . 235 6.1.2. Limit of a Sequence . 237 6.1.3. Limit of a Function. Asymptotes . 240 6.1.4. Infinitely Small and Infinitely Large Functions . 242 6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind . 243 6.1.6. Convex and Concave Functions . 245 6.1.7. Functions of Bounded Variation . 246 6.1.8. Convergence of Functions . 249 6.2. Differential Calculus for Functions of a Single Variable . 250 6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning . 250 6.2.2. Table of Derivatives and Differentiation Rules . 252 6.2.3. Theorems about Differentiable Functions. L'Hospital Rule . 254 6.2.4. Higher-Order Derivatives and Differentials. Taylor's Formula . 255 6.2.5. Extremal Points. Points of Inflection . 257 6.2.6. Qualitative Analysis of Functions and Construction of Graphs . 259 6.2.7. Approximate Solution of Equations (Root-Finding Algorithms for Continuous Functions) . 260 6.3. Functions of Several Variables. Partial Derivatives . 263 6.3.1. Point Sets. Functions. Limits and Continuity . 263 6.3.2. Differentiation of Functions of Several Variables . 264 6.3.3. Directional Derivative. Gradient. Geometrical Applications . 267 6.3.4. Extremal Points of Functions of Several Variables . 269 6.3.5. Differential Operators of the Field Theory . 272 References for Chapter 6 . 272 7. Integrals . 273 7.1. Indefinite Integral . 273 7.1.1. Antiderivative. Indefinite Integral and Its Properties . 273 7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. Integration Examples . 274 7.1.3. Integration of Rational Functions . 276 7.1.4. Integration of Irrational Functions . 279 7.1.5. Integration of Exponential and Trigonometric Functions . 281 7.1.6. Integration of Polynomials Multiplied by Elementary Functions . 283 7.2. Definite Integral . 286 7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the Definite Integral .;. 286 7.2.2. Properties of Definite Integrals and Useful Formulas . 287 7.2.3. General Reduction Formulas for the Evaluation of Integrals . 289 7.2.4. General Asymptotic Formulas for the Calculation of Integrals . 290 7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities. Arithmetic Mean and Geometric Mean of Functions . 295 7.2.6. Geometric and Physical Applications of the Definite Integral . 299 7.2.7. Improper Integrals with Infinite Integration Limit . 301 7.2.8. General Reduction Formulas for the Calculation of Improper Integrals . 304 7.2.9. General Asymptotic Formulas for the Calculation of Improper Integrals . 307 7.2.10. Improper Integrals of Unbounded Functions . 308 7.2.11. Cauchy-Type Singular Integrals . 310 Contents ix 7.2.12. Stieltjes Integral . 312 7.2.13. Square Integrable Functions . 314 7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals . 315 7.3. Double and Triple Integrals . 317 7.3.1. Definition and Properties of the Double Integral . 317 7.3.2. Computation of the Double Integral . 319 7.3.3. Geometric and Physical Applications of the Double Integral . 323 7.3.4. Definition and Properties of the Triple Integral . 324 7.3.5. Computation of the Triple Integral. Some Applications. Iterated Integrals and Asymptotic Formulas . 325 7.4. Line and Surface Integrals . 329 7.4.1. Line Integral of the First Kind . 329 7.4.2. Line Integral of the Second Kind . 330 7.4.3. Surface Integral of the First Kind . 332 7.4.4. Surface Integral of the Second Kind . 333 7.4.5. Integral Formulas of Vector Calculus . 334 References for Chapter 7 . 335 8. Series . 337 8.1. Numerical Series and Infinite Products . 337 8.1.1. Convergent Numerical Series and Their Properties. Cauchy's Criterion . 337 8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms . 338 8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional Convergence . 341 8.1.4. Multiplication of Series. Some Inequalities . 343 8.1.5. Summation Methods. Convergence Acceleration . 344 8.1.6. Infinite Products . 346 8.2. Functional Series . 348 8.2.1. Pointwise and Uniform Convergence of Functional Series . 348 8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent Series . 349 8.3. Power Series . 350 8.3.1. Radius of Convergence of Power Series. Properties of Power Series . 350 8.3.2. Taylor and Maclaurin Power Series . 352 8.3.3. Operations with Power Series. Summation Formulas for Power Series . 354 8.4. Fourier Series . 357 8.4.1. Representation of гтг-Регккііс Functions by Fourier Series. Main Results . 357 8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions . 359 8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series . 361 8.4.4. Summation Formulas for Trigonometric Series . 362 8.5. Asymptotic Series . 363 8.5.1. Asymptotic Series of Poincaré Type. Formulas for the Coefficients . 363 8.5.2. Operations with Asymptotic Series . 364 References for Chapter 8 . 366 9. Differential Geometry . 367 9.1. Theory of Curves . 367 9.1.1. Plane Curves . 367 9.1.2. Space Curves . 379 x Contents 9.2. Theory of Surfaces . 386 9.2.1. Elementary Notions in Theory of Surfaces . 386 9.2.2. Curvature of Curves on Surface . 392 9.2.3. Intrinsic Geometry of Surface . 395 References for Chapter 9 . 397 10. Functions of Complex Variable . 399 10.1. Basic Notions . 399 10.1.1. Complex Numbers. Functions of Complex Variable . 399 10.1.2. Functions of Complex Variable . 401 10.2. Main Applications . 419 10.2.1. Conformai Mappings . 419 10.2.2. Boundary Value Problems . 427 References for Chapter 10 . 433 11. Integral Transforms . 435 11.1. General Form of Integral Transforms. Some Formulas . 435 11.1.1. Integral Transforms and Inversion Formulas . 435 11.1.2. Residues. Jordan Lemma . 435 11.2. Laplace Transform . 436 11.2.1. Laplace Transform and the Inverse Laplace Transform . 436 11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some Functions . 437 11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series and Asymptotic Expansions . 440 11.3. Mellin Transform . 441 11.3.1. Mellin Transform and the Inversion Formula . 441 11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin, Laplace, and Fourier Transforms . 442 11.4. Various Forms of the Fourier Transform . 443 11.4.1. Fourier Transform and the Inverse Fourier Transform . 443 11.4.2. Fourier Cosine and Sine Transforms . 445 11.5. Other Integral Transforms . 446 11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified Bessel Functions . 446 11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral Transforms . 448 References for Chapter 11 .·.·. 451 12. Ordinary Differential Equations . 453 12.1. First-Order Differential Equations . 453 12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 453 12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration _ 456 12.1.3. Exact Differential Equations. Integrating Factor . 458 12.1.4. Riccati Equation . 460 12.1.5. Abel Equations of the First Kind . 462 12.1.6. Abel Equations of the SecondKind . 464 12.1.7. Equations Not Solved for the Derivative . 465 12.1.8. Contact Transformations . 468 12.1.9. Approximate Analytic Methods for Solution of Equations . 469 12.1.10. Numerical Integration of Differential Equations . 471 Contents xi 12.2. Second-Order Linear Differential Equations. 472 12.2.1. Formulas for the General Solution. Some Transformations . 472 12.2.2. Representation of Solutions as a Series in the Independent Variable . 475 12.2.3. Asymptotic Solutions . 477 12.2.4. Boundary Value Problems . 480 12.2.5. Eigenvalue Problems . 482 12.2.6. Theorems on Estimates and Zeros of Solutions . 487 12.3. Second-Order Nonlinear Differential Equations . 488 12.3.1. Form of the General Solution. Cauchy Problem . 488 12.3.2. Equations Admitting Reduction of Order . 489 12.3.3. Methods of Regular Series Expansions with Respect to the Independent Variable . 492 12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations. Painlevé Transcendents . 494 12.3.5. Perturbation Methods of Mechanics and Physics . 499 12.3.6. Galerkin Method and Its Modifications (Projection Methods) . 508 12.3.7. Iteration and Numerical Methods . 511 12.4. Linear Equations of Arbitrary Order . 514 12.4.1. Linear Equations with Constant Coefficients . 514 12.4.2. Linear Equations with Variable Coefficients . 518 12.4.3. Asymptotic Solutions of Linear Equations . 522 12.4.4. Collocation Method and Its Convergence . 523 12.5. Nonlinear Equations of Arbitrary Order . 524 12.5.1. Structure of the General Solution. Cauchy Problem . 524 12.5.2. Equations Admitting Reduction of Order . 525 12.6. Linear Systems of Ordinary Differential Equations . 528 12.6.1. Systems of Linear Constant-Coefficient Equations . 528 12.6.2. Systems of Linear Variable-Coefficient Equations . 539 12.7. Nonlinear Systems of Ordinary Differential Equations . 542 12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems . 542 12.7.2. Integrable Combinations. Autonomous Systems of Equations . 545 12.7.3. Elements of Stability Theory . 546 References for Chapter 12 . 550 13. First-Order Partial Differential Equations . 553 13.1. Linear and Quasilinear Equations . 553 13.1.1. Characteristic System. General Solution . 553 13.1.2. Cauchy Problem. Existence and Uniqueness Theorem . 556 13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . 558 13.1.4. Quasilinear Equations of General Form. Generalized Solution, Jump Condition, and Stability Condition . 567 13.2. Nonlinear Equations . 570 13.2.1. Solution Methods . 570 13.2.2. Cauchy Problem. Existence and Uniqueness Theorem . 576 13.2.3. Generalized Viscosity Solutions and Their Applications . 579 References for Chapter 13 . 584 xii Contents 14. Linear Partial Differential Equations. 585 14.1. Classification of Second-Order Partial Differential Equations . 585 14.1.1. Equations with Two Independent Variables . 585 14.1.2. Equations with Many Independent Variables . 589 14.2. Basic Problems of Mathematical Physics . 590 14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 590 14.2.2. First, Second, Third, and Mixed Boundary Value Problems . 593 14.3. Properties and Exact Solutions of Linear Equations . 594 14.3.1. Homogeneous Linear Equations and Their Particular Solutions . 594 14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions . 598 14.3.3. General Solutions of Some Hyperbolic Equations . 600 14.4. Method of Separation of Variables (Fourier Method) . 602 14.4.1. Description of the Method of Separation of Variables. General Stage of Solution . 602 14.4.2. Problems for Parabolic Equations: Final Stage of Solution . 605 14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution . 607 14.4.4. Solution of Boundary Value Problems for Elliptic Equations . 609 14.5. Integral Transforms Method . 611 14.5.1. Laplace Transform and Its Application in Mathematical Physics . 611 14.5.2. Fourier Transform and Its Application in Mathematical Physics . 614 14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution . 615 14.6.1. Cauchy Problem for Parabolic Equations . 615 14.6.2. Cauchy Problem for Hyperbolic Equations . 617 14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green's Function . 618 14.7.1. Representation of Solutions via the Green's Function . 618 14.7.2. Problems for Equation s(x)^ = -^ [p(x)^]-q(x)w + Ф(х, t) . 620 14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green's Function. Goursat Problem . 623 14.8.1. Representation of Solutions via the Green's Function . 623 14.8.2. Problems for Equation s(x)Š$- = -J^ [p(x)^]-q(x)w + Ф(х, t) . 624 14.8.3. Problems for Equation ^ + a(i)-f^ = ò(í){ -¿fe [pix)%¿] - q{x)w) + Ф(ж, ť) 626 14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve . 627 14.8.5. Goursat Problem (a Problem with Initial Data of Characteristics) . 629 14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables . 631 14.9.1. Problems and the Green's Functions for Equation u™ _i_ vw _l ij^õw j л/пъ\„„ — rb//. «л ¿ro i ■ ^v·'» У J . Ό j L 14.9.2. Representation of Solutions to Boundary Value Problems via the Green's Functions . 633 14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green's Function . 634 14.10.1. Problems for Parabolic Equations . 634 14.10.2. Problems for Hyperbolic Equations . 636 14.10.3. Problems for Elliptic Equations . 637 14.10.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types . 638 Contents xiii 14.11. Construction of the Green's Functions. General Formulas and Relations . 639 14.11.1. Green's Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains . 639 14.11.2. Green's Functions Admitting Incomplete Separation of Variables . 640 14.11.3. Construction of Green's Functions via Fundamental Solutions . 642 14.12. Duhamel's Principles in Nonstationary Problems . 646 14.12.1. Problems for Homogeneous Linear Equations . 646 14.12.2. Problems for Nonhomogeneous Linear Equations . 648 14.13. Transformations Simplifying Initial and Boundary Conditions . 649 14.13.1. Transformations That Lead to Homogeneous Boundary Conditions . 649 14.13.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions . 650 References for Chapter 14 . 650 15. Nonlinear Partial Differential Equations . 653 15.1. Classification of Second-Order Nonlinear Equations . 653 15.1.1. Classification of Semilinear Equations in Two Independent Variables . 653 15.1.2. Classification of Nonlinear Equations in Two Independent Variables . 653 15.2. Transformations of Equations of Mathematical Physics . 655 15.2.1. Point Transformations: Overview and Examples . 655 15.2.2. Hodograph Transformations (Special Point Transformations) . 657 15.2.3. Contact Transformations. Legendre and Euler Transformations . 660 15.2.4. Bäcklund Transformations. Differential Substitutions . 663 15.2.5. Differential Substitutions . 666 15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions. Similarity Method . 667 15.3.1. Preliminary Remarks . 667 15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations . 667 15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling Transformations . 669 15.3.4. Equations Invariant Under Combinations of Translation and Scaling Transformations, and Their Solutions . 674 15.3.5. Generalized Self-Similar Solutions . 677 15.4. Exact Solutions with Simple Separation of Variables . 678 15.4.1. Multiplicative and Additive Separable Solutions . 678 15.4.2. Simple Separation of Variables in Nonlinear Partial Differential Equations . 678 15.4.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . 679 15.5. Method of Generalized Separation of Variables . 681 15.5.1. Structure of Generalized Separable Solutions . 681 15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System of Coordinate Functions . 683 15.5.3. Solution of Functional Differential Equations by Differentiation . 684 15.5.4. Solution of Functional-Differential Equations by Splitting . 688 15.5.5. Titov-Galaktionov Method . 693 15.6. Method of Functional Separation of Variables . 697 15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities . 697 15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave Solutions . 697 xiv Contents 15.6.3. Differentiation Method . 700 15.6.4. Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications . 704 15.7. Direct Method of Symmetry Reductions of Nonlinear Equations . 708 15.7.1. Clarkson-Kruskal Direct Method . 708 15.7.2. Some Modifications and Generalizations . 712 15.8. Classical Method of Studying Symmetries of Differential Equations . 716 15.8.1. One-Parameter Transformations and Their Local Properties . 716 15.8.2. Symmetries of Nonlinear Second-Order Equations. Invariance Condition . 719 15.8.3. Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions . 724 15.8.4. Some Generalizations. Higher-Order Equations . 730 15.9. Nonclassical Method of Symmetry Reductions . 732 15.9.1. Description of the Method. Invariant Surface Condition . 732 15.9.2. Examples: The Newell-Whitehead Equation and a Nonlinear Wave Equation 733 15.10. Differential Constraints Method . 737 15.10.1. Description of the Method . 737 15.10.2. First-Order Differential Constraints . 739 15.10.3. Second- and Higher-Order Differential Constraints . 744 15.10.4. Connection Between the Differential Constraints Method and Other Methods . 746 15.11. Painlevé Test for Nonlinear Equations of Mathematical Physics . 748 15.11.1. Solutions of Partial Differential Equations with a Movable Pole. Method Description . 748 15.11.2. Examples of Performing the Painlevé Test and Truncated Expansions for Studying Nonlinear Equations . 750 15.11.3. Construction of Solutions of Nonlinear Equations That Fail the Painlevé Test, Using Truncated Expansions . 753 15.12. Methods of the Inverse Scattering Problem (Soliton Theory) . 755 15.12.1. Method Based on Using Lax Pairs . 755 15.12.2. Method Based on a Compatibility Condition for Systems of Linear Equations . 757 15.12.3. Solution of the Cauchy Problem by the Inverse Scattering Problem Method 760 15.13. Conservation Laws and Integrals of Motion . 766 15.13.1. Basic Definitions and Examples . 766 15.13.2. Equations Admitting Variational Formulation. Noetherian Symmetries . 767 15.14. Nonlinear Systems of Partial Differential Equations . 770 15.14.1. Overdetermined Systems of Two Equations . 770 15.14.2. Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems . 772 15.14.3. Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction . 775 15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type . 780 15.14.5. Systems of Second-Order Equations of Reaction-Diffusion Type . 796 References for Chapter 15. 798 Contents xv 16. Integral Equations . 801 16.1. Linear Integral Equations of the First Kind with Variable Integration Limit . 801 16.1.1. Volterra Equations of the First Kind . 801 16.1.2. Equations with Degenerate Kernel: K(x,t) = g\(x)hi(t) + ■■■ + gn(x)hn(t) . . 802 16.1.3. Equations with Difference Kernel: K(x, t) = K(x -ť) . 804 16.1.4. Reduction of Volterra Equations of the First Kind to Volterra Equations of the SecondKind . 807 16.1.5. Method of Quadratures . 808 16.2. Linear Integral Equations of the Second Kind with Variable Integration Limit . 810 16.2.1. Volterra Equations of the Second Kind . 810 16.2.2. Equations with Degenerate Kernel: K(x, t) = g\(x)h\(t) + · · · + gn(x)hn(t) . 811 16.2.3. Equations with Difference Kernel: K(x, t) = K(x -ť) . 813 16.2.4. Construction of Solutions of Integral Equations with Special Right-Hand Side 815 16.2.5. Method of Model Solutions . 818 16.2.6. Successive Approximation Method . 822 16.2.7. Method of Quadratures . 823 16.3. Linear Integral Equations of the First Kind with Constant Limits of Integration . 824 16.3.1. Fredholm Integral Equations of the First Kind . 824 16.3.2. Method of Integral Transforms . 825 16.3.3. Regularizaron Methods . 827 16.4. Linear Integral Equations of the Second Kind with Constant Limits of Integration . 829 16.4.1. Fredholm Integral Equations of the Second Kind. Resolvent . 829 16.4.2. Fredholm Equations of the Second Kind with Degenerate Kernel . 830 16.4.3. Solution as a Power Series in the Parameter. Method of Successive Approximations . 832 16.4.4. Fredholm Theorems and the Fredholm Alternative . 834 16.4.5. Fredholm Integral Equations of the Second Kind with Symmetric Kernel . 835 16.4.6. Methods of Integral Transforms . 841 16.4.7. Method of Approximating a Kernel by a Degenerate One . 844 16.4.8. Collocation Method . 847 16.4.9. Method of Least Squares . 849 16.4.10. Bubnov-Galerkin Method . 850 16.4.11. Quadrature Method . 852 16.4.12. Systems of Fredholm Integral Equations of the Second Kind . 854 16.5. Nonlinear Integral Equations . 856 16.5.1. Nonlinear Volterra and Urysohn Integral Equations . 856 16.5.2. Nonlinear Volterra Integral Equations . 856 16.5.3. Equations with Constant Integration Limits . 863 References for Chapter 16 . 871 17. Difference Equations and Other Functional Equations . 873 17.1. Difference Equations of Integer Argument . 873 17.1.1. First-Order Linear Difference Equations of Integer Argument . 873 17.1.2. First-Order Nonlinear Difference Equations of Integer Argument . 874 17.1.3. Second-Order Linear Difference Equations with Constant Coefficients . 877 17.1.4. Second-Order Linear Difference Equations with Variable Coefficients . 879 17.1.5. Linear Difference Equations of Arbitrary Order with Constant Coefficients . 881 17.1.6. Linear Difference Equations of Arbitrary Order with Variable Coefficients . 882 17.1.7. Nonlinear Difference Equations of Arbitrary Order . 884 xvi Contents 17.2. Linear Difference Equations with a Single Continuous Variable . 885 17.2.1. First-Order Linear Difference Equations . 885 17.2.2. Second-Order Linear Difference Equations with Integer Differences . 894 17.2.3. Linear mth-Order Difference Equations with Integer Differences . 898 17.2.4. Linear mth-Order Difference Equations with Arbitrary Differences . 904 17.3. Linear Functional Equations . 907 17.3.1. Iterations of Functions and Their Properties . 907 17.3.2. Linear Homogeneous Functional Equations . 910 17.3.3. Linear Nonhomogeneous Functional Equations . 912 17.3.4. Linear Functional Equations Reducible to Linear Difference Equations with Constant Coefficients . 916 17.4. Nonlinear Difference and Functional Equations with a Single Variable . 918 17.4.1. Nonlinear Difference Equations with a Single Variable . 918 17.4.2. Reciprocal (Cyclic) Functional Equations . 919 17.4.3. Nonlinear Functional Equations Reducible to Difference Equations . 921 17.4.4. Power Series Solution of Nonlinear Functional Equations . 922 17.5. Functional Equations with Several Variables . 922 17.5.1. Method of Differentiation in a Parameter . 922 17.5.2. Method of Differentiation in Independent Variables . 925 17.5.3. Method of Substituting Particular Values of Independent Arguments . 926 17.5.4. Method of Argument Elimination by Test Functions . 928 17.5.5. Bilinear Functional Equations and Nonlinear Functional Equations Reducible to Bilinear Equations . 930 References for Chapter 17 . 935 18. Special Functions and Their Properties . 937 18.1. Some Coefficients, Symbols, and Numbers . 937 18.1.1. Binomial Coefficients . 937 18.1.2. Pochhammer Symbol . 938 18.1.3. Bernoulli Numbers . 938 18.1.4. Euler Numbers . 939 18.2. Error Functions. Exponential and Logarithmic Integrals . 939 18.2.1. Error Function and Complementary Error Function . 939 18.2.2. Exponential Integral . 940 18.2.3. Logarithmic Integral . 941 18.3. Sine Integral and Cosine Integral. Fresnel Integrals . 941 18.3.1. Sine Integral . 941 18.3.2. Cosine Integral . 942 18.3.3. Fresnel Integrals . 942 18.4. Gamma Function, Psi Function, and Beta Function . 943 18.4.1. Gamma Function . 943 18.4.2. Psi Function (Digamma Function) . 944 18.4.3. Beta Function . 945 18.5. Incomplete Gamma and Beta Functions . 946 18.5.1. Incomplete Gamma Function . 946 18.5.2. Incomplete Beta Function . 947 Contents xvii 18.6. Bessel Functions (Cylindrical Functions) . 947 18.6.1. Definitions and Basic Formulas . 947 18.6.2. Integral Representations and Asymptotic Expansions . 949 18.6.3. Zeros and Orthogonality Properties of Bessel Functions . 951 18.6.4. Hankel Functions (Bessel Functions of the Third Kind) . 952 18.7. Modified Bessel Functions . 953 18.7.1. Definitions. Basic Formulas . 953 18.7.2. Integral Representations and Asymptotic Expansions . 954 18.8. Airy Functions . 955 18.8.1. Definition and Basic Formulas . 955 18.8.2. Power Series and Asymptotic Expansions . 956 18.9. Degenerate Hypergeometric Functions (Kummer Functions) . 956 18.9.1. Definitions and Basic Formulas . 956 18.9.2. Integral Representations and Asymptotic Expansions . 959 18.9.3. Whittaker Functions . 960 18.10. Hypergeometric Functions . 960 18.10.1. Various Representations of the Hypergeometric Function . 960 18.10.2. Basic Properties . 960 18.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions . 962 18.11.1. Legendre Polynomials and Legendre Functions . 962 18.11.2. Associated Legendre Functions with Integer Indices and Real Argument . 964 18.11.3. Associated Legendre Functions. General Case . 965 18.12. Parabolic Cylinder Functions . 967 18.12.1. Definitions. Basic Formulas . 967 18.12.2. Integral Representations, Asymptotic Expansions, and Linear Relations . 968 18.13. Elliptic Integrals . 969 18.13.1. Complete Elliptic Integrals . 969 18.13.2. Incomplete Elliptic Integrals (Elliptic Integrals) . 970 18.14. Elliptic Functions . 972 18.14.1. Jacobi Elliptic Functions . 972 18.14.2. Weierstrass Elliptic Function . 976 18.15. Jacobi Theta Functions . 978 18.15.1. Series Representation of the Jacobi Theta Functions. Simplest Properties . 978 18.15.2. Various Relations and Formulas. Connection with Jacobi Elliptic Functions 978 18.16. Mathieu Functions and Modified Mathieu Functions . 980 18.16.1. Mathieu Functions . 980 18.16.2. Modified Mathieu Functions . 982 18.17. Orthogonal Polynomials . 982 18.17.1. Laguerre Polynomials and Generalized Laguerre Polynomials . 982 18.17.2. Chebyshev Polynomials and Functions . 983 18.17.3. Hermite Polynomials . 985 18.17.4. Jacobi Polynomials and Gegenbauer Polynomials . 986 18.18. Nonorthogonal Polynomials . 988 18.18.1. Bernoulli Polynomials . 988 18.18.2. Euler Polynomials . 989 References for Chapter 18 . 990 xviii _ Contents _ 19. Calculus of Variations and Optimization . . 991 19.1. Calculus of Variations and Optimal Control . 991 19.1.1. Some Definitions and Formulas . 991 19.1.2. Simplest Problem of Calculus of Variations . 993 19.1.3. Isoperimetric Problem .1002 19.1.4. Problems with Higher Derivatives .1006 19.1.5. Lagrange Problem .1008 19.1.6. Pontryagin Maximum Principle .1010 19.2. Mathematical Programming .1012 19.2.1. Linear Programming .1012 19.2.2. Nonlinear Programming .1027 References for Chapter 19 .1028 20. Probability Theory .1031 20.1. Simplest Probabilistic Models .1031 20.1.1. Probabilities of Random Events .1031 20.1.2. Conditional Probability and Simplest Formulas .1035 20.1.3. Sequences of Trials .1037 20.2. Random Variables and Their Characteristics .1039 20.2.1. One-Dimensional Random Variables .1039 20.2.2. Characteristics of One-Dimensional Random Variables .1042 20.2.3. Main Discrete Distributions .1047 20.2.4. Continuous Distributions .1051 20.2.5. Multivariate Random Variables .1057 20.3. Limit Theorems .1068 20.3.1. Convergence of Random Variables .1068 20.3.2. Limit Theorems .1069 20.4. Stochastic Processes .1071 20.4.1. Theory of Stochastic Processes .1071 20.4.2. Models of Stochastic Processes .1074 References for Chapter 20.1079 21. Mathematical Statistics .1081 21.1. Introduction to Mathematical Statistics .1081 21.1.1. Basic Notions and Problems of Mathematical Statistics .1081 21.1.2. Simplest Statistical Transformations .1082 21.1.3. Numerical Characteristics of Statistical Distribution .1087 21.2. Statistical Estimation .1088 21.2.1. Estimators and Their Properties .1088 21.2.2. Estimation Methods for Unknown Parameters .1091 21.2.3. Interval Estimators (Confidence Intervals) .1093 21.3. Statistical Hypothesis Testing .1094 21.3.1. Statistical Hypothesis. Test .1094 21.3.2. Goodness-of-Fit Tests .1098 21.3.3. Problems Related to Normal Samples .1101 References for Chapter 21 . П09 Contents xix Partii. Mathematical Tables 1111 Tl. Finite Sums and Infinite Series .1113 TU. Finite Sums .1113 Tl.l.l. Numerical Sum .1113 Tl.1.2. Functional Sums .1116 T1.2. Infinite Series .1118 Tl.2.1. Numerical Series .1118 Tl.2.2. Functional Series .1120 References for Chapter Tl .1127 T2. Integrals .1129 T2.1. Indefinite Integrals .1129 T2.1.1. Integrals Involving Rational Functions .1129 T2. T2. T2. T2. T2. T2. .2. Integrals Involving Irrational Functions .1134 .3. Integrals Involving Exponential Functions .1137 .4. Integrals Involving Hyperbolic Functions .1137 .5. Integrals Involving Logarithmic Functions .1140 .6. Integrals Involving Trigonometric Functions .1142 .7. Integrals Involving Inverse Trigonometric Functions .1147 T2.2. Tables of Definite Integrals .1147 T2.2.1. Integrals Involving Power-Law Functions .1147 T2.2.2. Integrals Involving Exponential Functions .1150 T2.2.3. Integrals Involving Hyperbolic Functions .1152 T2.2.4. Integrals Involving Logarithmic Functions .1152 T2.2.5. Integrals Involving Trigonometric Functions .1153 References for Chapter T2 .1155 T3. Integral Transforms .1157 T3.1. Tables of Laplace Transforms .1157 T3.1.1. General Formulas .1157 T3.1.2. Expressions with Power-Law Functions .1159 T3.1.3. Expressions with Exponential Functions .1159 T3.1.4. Expressions with Hyperbolic Functions .1160 T3. 1.5. Expressions with Logarithmic Functions .1161 T3.1.6. Expressions with Trigonometric Functions .1161 T3.1.7. Expressions with Special Functions .1163 T3.2. Tables of Inverse Laplace Transforms .1164 T3.2.1. General Formulas .1164 T3.2.2. Expressions with Rational Functions .1166 T3.2.3. Expressions with Square Roots .1170 T3.2.4. Expressions with Arbitrary Powers .1172 T3.2.5. Expressions with Exponential Functions .1172 T3.2.6. Expressions with Hyperbolic Functions .1174 T3.2.7. Expressions with Logarithmic Functions .1174 T3.2.8. Expressions with Trigonometric Functions .1175 T3.2.9. Expressions with Special Functions .1176 xx Contents ТЗ.З. Tables of Fourier Cosine Transforms .1177 T3.3.1. General Formulas .1177 T3.3.2. Expressions with Power-Law Functions .1177 T3.3.3. Expressions with Exponential Functions .1178 T3.3.4. Expressions with Hyperbolic Functions .1179 T3.3.5. Expressions with Logarithmic Functions .1179 T3.3.6. Expressions with Trigonometric Functions .1180 T3.3.7. Expressions with Special Functions .1181 T3.4. Tables of Fourier Sine Transforms .1182 T3.4.1. General Formulas .1182 T3.4.2. Expressions with Power-Law Functions .1182 T3.4.3. Expressions with Exponential Functions .1183 T3.4.4. Expressions with Hyperbolic Functions .1184 T3.4.5. Expressions with Logarithmic Functions .1184 T3.4.6. Expressions with Trigonometric Functions .1185 T3.4.7. Expressions with Special Functions .1186 T3.5. Tables of Mellin Transforms .1187 ТЗ.о.і. General Formulas .1187 T3.5.2. Expressions with Power-Law Functions .1188 T3.5.3. Expressions with Exponential Functions .1188 T3.5.4. Expressions with Logarithmic Functions .1189 T3.5.5. Expressions with Trigonometric Functions .1189 T3.5.6. Expressions with Special Functions .1190 T3.6. Tables of Inverse Mellin Transforms .1190 T3.6.1. Expressions with Power-Law Functions .1190 T3.6.2. Expressions with Exponential and Logarithmic Functions .1191 T3.6.3. Expressions with Trigonometric Functions .1192 T3.6.4. Expressions with Special Functions .1193 References for Chapter T3 .1194 T4. Orthogonal Curvilinear Systems of Coordinate .1195 T4.1. Arbitrary Curvilinear Coordinate Systems .1195 T4.1.1. General Nonorthogonal Curvilinear Coordinates .1195 T4.1.2. General Orthogonal Curvilinear Coordinates .1196 T4.2. Special Curvilinear Coordinate Systems .1198 T4.2.1. Cylindrical Coordinates .1198 T4.2.2. Spherical Coordinates .1199 T4.2.3. Coordinates of a Prolate Ellipsoid of Revolution .1200 T4.2.4. Coordinates of an Oblate Ellipsoid of Revolution .1201 T4.2.5. Coordinates of an Elliptic Cylinder .1202 T4.2.6. Conical Coordinates .1202 T4.2.7. Parabolic Cylinder Coordinates .1203 T4.2.8. Parabolic Coordinates .1203 T4.2.9. Bicylindrical Coordinates .1204 T4.2.10. Bipolar Coordinates (in Space) .1204 T4.2.1 1. Toroidal Coordinates .1205 References for Chapter T4 .,. 1205 Contents xxi Т5. Ordinary Differential Equations .1207 T5.1. First-Order Equations .1207 T5.2. Second-Order Linear Equations .1212 T5.2.1. Equations Involving Power Functions .1213 T5.2.2. Equations Involving Exponential and Other Functions .1220 T5.2.3. Equations Involving Arbitrary Functions .1222 T5.3. Second-Order Nonlinear Equations .1223 ТЅ.ЗЛ. Equations ofthe Form y%x = f(x,y) .1223 T5.3.2. Equations ofthe Form f(x, y)y'x'x = g(x, y,y'x) .1225 References for Chapter T5 .1228 T6. Systems of Ordinary Differential Equations .1229 T6. 1. Linear Systems of Two Equations .1229 Тб.і.і. Systems of First-Order Equations .1229 T6.1.2. Systems of Second-Order Equations .1232 T6.2. Linear Systems of Three and More Equations .1237 T6.3. Nonlinear Systems of Two Equations .1239 T6.3.1. Systems of First-Order Equations .1239 T6.3.2. Systems of Second-Order Equations .1240 T6.4. Nonlinear Systems of Three or More Equations .1244 References for Chapter T6 .1246 T7. First-Order Partial Differential Equations .1247 T7.1. Linear Equations .1247 T7.1.1. Equations of the Form f{x,y)^ + д(.х,у)^~ =0.1247 T7.1.2. Equations of the Form f(x,y)^ + д(х,у)Щ; = h(x,y) .1248 T7.1.3. Equations ofthe Form f(x,y)^ + д(х,у)щ = h(x,y)w + r(x,y) .1250 T7.2. Quasilinear Equations .1252 T7.2.1. Equations of the Form f(x,y)^ + д(х,у)щт - h(x,y,w) .1252 T7.2.2. Equations ofthe Form f^ + f(x,y,w)^ = 0.1254 T7.2.3. Equations ofthe Form ^ + f(x, y, w)^- = g(x, y,w) .1256 T7.3. Nonlinear Equations .1258 T7.3.1. Equations Quadratic in One Derivative .1258 T7.3.2. Equations Quadratic in Two Derivatives .1259 T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives .1261 References for Chapter T7 .1265 T8. Linear Equations and Problems of Mathematical Physics .1267 T8.1. Parabolic Equations .1267 Τδ.Ι.Ι. Heat Equation ff = af^- .1267 T8.1.2. Nonhomogeneous Heat Equation ^ = a^f + Ф(х,і) .1268 T8.1.3. Equation ofthe Form ff = aţ^ + fcf^ + cw + Φ(χ,ί) . 1270 T8.1.4. Heat Equation with Axial Symmetry ff = o(^f + 7^7).1270 T8.1.5. Equation ofthe Form ff = a(f^r + ff^) +Ф(г, ť) .1271 T8.1.6. Heat Equation with Central Symmetry ^f = a(\|^ + ^^) .1272 18.1.7. Equation ofthe Form ff =a(fe + r If) +ф(г>ѓ> . 1273 T8.1.8. Equation ofthe Form ^f = -fgr + ^ψ-^ .1274 xxii Contents 78.1.9. Equations of the Diffusion (Thermal) Boundary Layer .1276 TS.l.lO. Schrödinger Equation ¿ñff = -^^r + U(x)w .1276 T8.2. Hyperbolic Equations .1278 T8.2.1. Wave Equation ^ = a2ţ^ .1278 T8.2.2. Equation of the Form ţ^=a2ţ^- + Ф(х, t) .1279 T8.2.3. Klein-Gordon Equation ţg- = a2ţŞ--bw .1280 T8.2.4. Equation of the Form ţ^ = a2ţ$- -bw + Ф(х, t) .1281 T8.2.5. Equation of the Form ^= α2 (^Ѕ + 1^)+Ф(г, í) .1282 T8.2.6. Equation of the Form ţţr = a (íz + r If) +ф(г'> . 1283 T8.2.7. Equations of the Form ţ^ + к ff = a2 f^ + òf^ + cw + Φ(ζ, ί) .1284 Τ8.3. Elliptic Equations .1284 T8.3.1. Laplace Equation Aw = 0 .1284 T8.3.2. Poisson Equation Aw + Φ(χ) = 0 .1287 T8.3.3. Helmholtz Equation Aw + \w = -Ф(х) .1289 T8.4. Fourth-Order Linear Equations .1294 T8.4.1. Equation of the Form ξ$- + a2ţŞ- =0 .1294 T8.4.2. Equation of the Form ţg- + a2\|^r = Φ(χ,ί) .1295 T8.4.3. Biharmonic Equation ΔΔ«; =0 .1297 T8.4.4. Nonhomogeneous Biharmonic Equation A Aw = Ф(х, у) . 1298 References for Chapter T8 .1299 T9. Nonlinear Mathematical Physics Equations .1301 T9.1. Parabolic Equations .1301 TO. 1.1. Nonlinear Heat Equations of the Form ^ = ^f + f(w) .1301 T9.1.2. Equations of the Form ^ = £[f(w)^] + g(w) .1303 T9. 1.3. Burgers Equation and Nonlinear Heat Equation in Radial Symmetric Cases . . 1307 T9.1.4. Nonlinear Schrödinger Equations .1309 T9.2. Hyperbolic Equations .1312 T9.2.1. Nonlinear Wave Equations of the Form &$- = af^ + f(w) .1312 T9.2.2. Other Nonlinear Wave Equations .1316 T9.3. Elliptic Equations .1318 T9.3.1. Nonlinear Heat Equations of the Form ţ^- + ţş- = f (w) .1318 T9.3.2. Equations of the Form £ [f(x)fé\ + % [g(y)%] = f(w) .1321 T9.3.3. Equations of the Form £ [f(w)fê] + £ [íK«)^] = Kw) .1322 T9.4. Other Second-Order Equations . ;_.1324 T9.4.1. Equations of Transonic Gas Flow .1324 T9.4.2. Monge-Ampère Equations .1326 T9.5. Higher-Order Equations .1327 T9.5.1. Third-Order Equations .1327 T9.5.2. Fourth-Order Equations .1332 References for Chapter T9 .1335 T10. Systems of Partial Differential Equations .1337 TlO.l. Nonlinear Systems of Two First-Order Equations ._.1337 T10.2. Linear Systems of Two Second-Order Equations .1341 Contents xxiii Т1О.З. Nonlinear Systems of Two Second-Order Equations .1343 T10.3.1. Systems of the Form ff = α0 + F(u,w), ^f = òf^ + G(u,w) .1343 TIO.3.2. Systems of the Form ^ = ^k-§^{xn^) + F(u,w), öt x71 ox \ ox / v * '. . T10.3.3. Systems of the Form Au = F(u, w), Aw = G(u, w) .1364 Т10.3.4. Systems of the Form ţg- - рг^(жп\|\|) + F(u,w), д2и> _±l/rnft»K η(η. „л 1 -ifrQ dt2 ~ xn дх У дх J "V"' w> . 1J"° Т10.3.5. Other Systems .1373 T10.4. Systems of General Form .1374 T10.4.1. Linear Systems .1374 T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t . . 1374 T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378 T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in í . 1381 References for Chapter T10 .1382 Til. Integral Equations .1385 Tl 1.1. Linear Equations of the First Kind with Variable Limit of Integration .1385 T11.2. Linear Equations of the Second Kind with Variable Limit of Integration .1391 Tl 1.3. Linear Equations of the First Kind with Constant Limits of Integration .1396 T 11.4. Linear Equations of the Second Kind with Constant Limits of Integration .1401 References for Chapter Til .1406 T12. Functional Equations .1409 Tl 2.1. Linear Functional Equations in One Independent Variable .1409 T 12.1.1. Linear Difference and Functional Equations Involving Unknown Function with Two Different Arguments .1409 T12.1.2. Other Linear Functional Equations .1421 T12.2. Nonlinear Functional Equations in One Independent Variable .1428 T12.2.1. Functional Equations with Quadratic Nonlinearity .1428 T12.2.2. Functional Equations with Power Nonlinearity .1433 T12.2.3. Nonlinear Functional Equation of General Form .1434 T12.3. Functional Equations in Several Independent Variables .1438 T12.3.1. Linear Functional Equations .1438 T12.3.2. Nonlinear Functional Equations .1443 References for Chapter Tl 2.1450 Supplement. Some Useful Electronic Mathematical Resources .1451 Index .1453
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id	DE-604.BV022388587
illustrated	Not Illustrated
index_date	2024-07-02T17:13:37Z
indexdate	2024-07-09T20:56:31Z
institution	BVB
isbn	1584885025 9781584885023
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-015597454
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open_access_boolean
owner	DE-29T DE-703
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physical	XXXII, 1509 S.
publishDate	2007
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spelling	Poljanin, Andrej D. 1951- Verfasser (DE-588)128391251 aut Handbook of mathematics for engineers and scientists Andrei D. Polyanin ; Alexander V. Manzhirov Boca Raton, Fla. [u.a.] Chapman & Hall/CRC 2007 XXXII, 1509 S. txt rdacontent n rdamedia nc rdacarrier Mathematik (DE-588)4037944-9 gnd rswk-swf Mathematik (DE-588)4037944-9 s DE-604 Manžirov, Aleksandr V. Verfasser (DE-588)120620758 aut Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015597454&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Poljanin, Andrej D. 1951- Manžirov, Aleksandr V. Handbook of mathematics for engineers and scientists Mathematik (DE-588)4037944-9 gnd
subject_GND	(DE-588)4037944-9
title	Handbook of mathematics for engineers and scientists
title_auth	Handbook of mathematics for engineers and scientists
title_exact_search	Handbook of mathematics for engineers and scientists
title_exact_search_txtP	Handbook of mathematics for engineers and scientists
title_full	Handbook of mathematics for engineers and scientists Andrei D. Polyanin ; Alexander V. Manzhirov
title_fullStr	Handbook of mathematics for engineers and scientists Andrei D. Polyanin ; Alexander V. Manzhirov
title_full_unstemmed	Handbook of mathematics for engineers and scientists Andrei D. Polyanin ; Alexander V. Manzhirov
title_short	Handbook of mathematics for engineers and scientists
title_sort	handbook of mathematics for engineers and scientists
topic	Mathematik (DE-588)4037944-9 gnd
topic_facet	Mathematik
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015597454&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT poljaninandrejd handbookofmathematicsforengineersandscientists AT manzirovaleksandrv handbookofmathematicsforengineersandscientists

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