Modern mathematical methods for scientists and engineers: a street-smart introduction
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World Scientific
[2023]
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| Physical Description: | xxii, 545 Seiten Illustrationen, Diagramme |
| ISBN: | 9781800611832 9781800611801 |
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| 245 | 1 | 0 | |a Modern mathematical methods for scientists and engineers |b a street-smart introduction |c Athanassios Fokas, University of Cambridge, UK & University of Southern California, USA, Efthimios Kaxiras, Harvard University, USA |
| 264 | 1 | |a New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo |b World Scientific |c [2023] | |
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| adam_text | Contents Preface vii About the Authors xiii Acknowledgments xv Part I Functions of Real Variables i Chapter i Functions of a Single Variable 3 1.1 Introduction: The various types of numbers................................. 3 1.2 Elementary functions....................................................................... 7 1.2.1 Polynomials........................................................................... 7 1.2.2 The inverse function............................................................. 9 1.2.3 Geometric shapes: Circle, ellipse, hyperbola.................... 10 1.2.4 Trigonometric functions...................................................... 12 1.2.5 Exponential, logarithm, hyperbolic functions.................... 13 1.3 Continuity and derivatives.............................................................. 15 1.3.1 Continuity.............................................................................. 15 1.3.2 Definition of derivatives....................................................... 16 1.3.3 Geometric interpretation of derivatives............................. 17 1.3.4 Finding roots by using derivatives...................................... 19 1.3.5 Chain rule and implicit differentiation ............................. 20 1.4 Integrals........................................................................................... 21 1.4.1 The definite and indefinite integrals................................... 21 1.4.2 Integration by change of variables...................................... 23 1.4.3 Integration by
parts............................................................. 24 1.4.4 The principal value integral .............................................. 25 1.5 Norms and moments of a function................................................ 27 1.5.1 The norms of a function....................................................... 27 1.5.2 The normalized Gaussian function ................................... 28 1.5.3 Moments of a function.......................................................... 30 1.6 Generalized functions: ժ-function and ¿-function ...................... 31 1.6.1 The 0-function or step-function............................................ 32 1.6.2 The ¿-function........................................................................ 33 1.7 Application: Signal filtering, convolution, correlation.................... 36
xviii Modern Mathematical Methods Chapter 2 Functions of Many Variables 4і 2.1 General considerations......................................................................... 4і շ.շ Coordinate systems and change of variables.................................... 43 2.2.1 Coordinate systems in two dimensions................................. 43 2.2.2 Coordinate systems in three dimensions.............................. 46 2.2.3 Change of variables ............................................................. 49 2.3 Vector operations............................................................................. 52 2.3.1 The dot or scalar product........................................................ 53 2.3.2 The cross or vector product................................................. 54 2.3.3 The wedge product................................................................ 54 2.4 Differential operators......................................................................... 55 2.5 The vector calculus integrals ........................................................... 60 2.5.1 Simple curves and simply-connected domains...................... 60 2.5.2 Green s theorem................................................................... 61 2.5.3 The divergence and curl theorems...................................... 62 2.5.4 Geometric interpretation of the divergence and curl theorems................................................................ 65 2.6 Function optimization..................................................................... 67 2.6.1 Finding the extrema of a
function.......................................... 67 2.6.2 Constrained optimization: Lagrange multipliers............... 69 2.6.3 Calculus of variations.......................................................... 70 2.7 Application: Feedforward neural network......................................... 73 2.7.1 Definition of feedforward neural network ........................ 74 2.7.2 Training of the network....................................................... 78 Chapter 3 Series Expansions 85 3.1 Infinite sequences and series ......................................................... 85 3.1.1 Series convergence tests....................................................... 87 3.1.2 Some important number series............................................ 88 3.2 Series expansions of functions......................................................... 91 3.2.1 Power series expansion: The Taylor series ........................... 92 3.2.2 Inversion of power series.................................................... 96 3.2.3 Trigonometric series expansions......................................... 97 3.2.4 Polynomial series expansion............................................... 99 3.3 Convergence of series expansion of functions.................................. 101 3.3.1 Uniform convergence............................................................ 102 3.3.2 Uniform convergence criteria.................................................106 3.4 Truncation error in series expansions ............................................. 107 3.5 Application: Wavelet
analysis............................................................109 3.5.1 Haar wavelet expansion......................................................... 110 3.5.2 The wavelet transform............................................................ 113 Part II Complex Analysis and the Fourier Transform 117 Chapter 4 Functions of Complex Variables 119 4.1 Complex numbers ............................................................................ 119 4.2 Complex variables ............................................................................ 122 4.2.1 The complex exponential.......................................................123
Contents xix 4.3 4.4 4.5 4.6 4.7 4.8 4.2.2 Multi-valuedness in polar representation............................ 123 4.2.3 The complex logarithm......................................................... 124 Continuity and derivatives................................................................. 125 Analyticity.......................................................................................... 127 4.4.1 The d-bar derivative...............................................................130 4.4.2 Cauchy-Riemann relations and harmonic functions . . . 130 4.4.3 Consequences of analyticity................................................... 132 4.4.4 Explicit formulas for computing an analytic function in terms of its real or imaginaryparts....................... 134 Complex integration: Cauchy s theorem.......................................... 136 4.5.1 Integration over a contour...................................................... 136 4.5.2 Cauchy s theorem ..................................................................137 4.5.3 Contour deformation ......................... 139 4.5.4 Cauchy s Integral Formula................................................... 141 Extensions of Cauchy s theorem..................................................... 141 4.6.1 General form of Cauchy s theorem........................................ 142 4.6.2 Pompeiu s formula................................................................. 142 4.6.3 The ¿-function of complex argument .................................. 143 Complex power series
expansions.................................................. 144 4.7.1 Taylor series............................................................................. 146 4.7.2 Laurent series.......................................................................... 147 Application: The 2D Fourier transform ..........................................150 Chapter 5 Singularities, Residues, Contour Integration 159 5.1 Types of singularities......................................................................... 159 5.2 Residue theorem ............................................................................... 161 5.3 Integration by residues...................................................................... 162 5.3.1 Integrands with simple poles................................................ 162 5.3.2 Integrands that are ratios of polynomials............................ 165 5.3.3 Integrands with trigonometric or hyperbolic functions...................................................................... 170 5.3.4 Skipping a simple pole .........................................................172 5.4 Jordan s lemma................................................................................. 173 5.5 Scalar Riemann-Hilbert problems.................................................. 176 5.5.1 Derivation of the inverse Fourier transform ...................... 180 5.6 Branch points and branch cuts........................................................ 181 5.7 The source-sink function and related problems ........................... 184 5.8 Application: Medical
imaging...........................................................188 Chapter 6 Mappings Produced by Complex Functions 195 6.1 Mappings produced by positive powers.......................................... 196 6.2 Mappings produced by 1/z.............................................................. 199 6.3 Mappings produced by the exponential function........................... 203 6.4 Conformal mapping......................................................................... 209 6.5 Application: Fluid flow around an obstacle.................................... 211 Chapter 7 The Fourier Transform 219 7.1 Fourier series expansions................................................................. 219 7.1.1 Real Fourier expansion ......................................................... 219
xx Modern Mathematical Methods 7.1.2 Odd and even function expansions ..................................... 222 7.1.3 Handling of discontinuities in Fourier expansions............ 223 7.1.4 Fourier expansion of arbitrary range .................................. 223 7.1.5 Complex Fourier expansion....................................................224 7.1.6 Fourier expansion of convolution and correlation............225 7.1.7 Conditions for the validity of a Fourier expansion........... 226 7.2 The Fourier transform........................................................................226 7.2.1 Definition of the Fourier transform..................................... 226 7.2.2 Properties of the Fourier transform..................................... 227 7.2.3 Symmetries of the Fourier transform..................................234 7.2.4 The sine- and cosine-transforms.......................................... 235 7.3 Fourier transforms of special functions ........................................... 236 7.3.1 The FT of the normalized Gaussian function...................... 236 7.3.2 FT of the 0-function and of the ¿-function............................ 238 7.4 Application: Signal analysis............................................................... 242 7.4.1 Aliasing: The Nyquist frequency........................................... 243 Part III Applications to Partial Differential Equations 253 Chapter 8 Partial Differential Equations: Introduction 255 8.1 General remarks.................................................................................. 255 8.2 Overview of
traditional approaches................................................ 256 8.2.1 Examples of ODEs.................................................................. 257 8.2.2 Eigenvalue equations ............................................................. 259 8.2.3 The Green s function method .............................................. 260 8.2.4 The Fourier series method.......................................................263 8.3 Evolution equations............................................................................ 263 8.3.1 Motivation of evolution equations........................................ 263 8.3.2 Separation of variables and initial value problems............ 265 8.3.3 Traditional integral transforms for problems on the half-line....................................................................... 270 8.3.4 Traditional infinite series for problems on a finite interval..................................................................272 8.4 The wave equation............................................................................ 273 8.4.1 Derivation oí the wave equation...........................................273 8.4.2 The solution of d Alembert................................................... 275 8.4.3 Traditional transforms ......................................................... 277 8.5 The Laplace and Poisson equations................................................... 280 8.5.1 Motivation of the Laplace and Poisson equations............ 282 8.5.2 Integral representations through the fundamental
solution.................................................................................. 283 8.6 The Helmholtz and modified Helmholtz equations...................... 285 8.7 Disadvantages of traditional integral transforms............................ 288 8.8 Application: The Black-Scholes equation....................................... 290 Chapter 9 Unified Transform. I: Evolution PDEs on the Half-line 297 9.1 The unified transform is based on analyticity................................. 297 9.2 The heat equation............................................................................... 300
Contents xxi The general methodology of the unified transform ..................... 304 9.3.1 Advantages of the unified transform .................................. 306 9.4 A PDE with a third-order spatial derivative....................................314 9.5 Inhomogeneous PDEs and other considerations........................... 320 9.5.1 Robin boundary conditions.................................................... 322 9.5.2 From PDEs to ODEs............................................................... 328 9.5.3 Green s functions..................................................................... 329 9.6 Application: Heat flow along a solid rod ...................................... 331 9.6.1 The conventional solution....................................................... 332 9.6.2 The solution through the unified transform ....................... 333 9.3 Chapter 10 Unified Transform II: Evolution PDEs on a Finite Interval 341 10.1 The heat equation.............................................................................. 341 10.2 Advantages of the unified transform............................................... 347 10.3 A PDE with a third-order spatial derivative................................... 349 10.4 Inhomogeneous PDEs and other considerations.......................... 351 10.4.1 Robin boundary conditions................................................... 351 10.4.2 From PDEs to ODEs.............................................................. 354 10.4.3 Green s functions.................................................................... 355 10.5
Application: Detection of colloid concentration............................. 356 Chapter 11 Unified Transform III: The Wave Equation 365 11.1 An alternative derivation of the global relation............................. 365 11.2 The wave equation on the half-line .............................................. 366 11.2.1 The Dirichlet problem........................................................... 368 11.3 The wave equation on a finite interval........................................... 371 11.3.1 The Dirichlet-Dirichlet problem.......................................... 372 11.4 The forced problem...........................................................................381 Unified Transform IV: Laplace, Poisson, and Helmholtz Equations 385 Introduction...................................................................................... 385 12.1.1 Integral representations in the fc-complex plane...............387 The Laplace and Poisson equations ................................................ 389 12.2.1 Global relations for a convex polygon.................................. 390 12.2.2 An integral representation in the fc-complex plane............ 391 12.2.3 The approximate global relation for a convex polygon ............................................... 399 12.2.4 The general case.................................................................... 403 The Helmholtz and modified Helmholtz equations...................... 403 12.3.1 Global relations for a convex polygon ............................... 404 12.3.2 Novel integral representations for the modified Helmholtz
equation.................................................. 405 12.3.3 The general case.................................................................... 411 12.3.4 Computing the solution in the interior of a convex polygon..................................................412 Generalizations.................................................................................. 413 Application: Water waves................................................................. 413 Chapter 12 12.1 12.2 12.3 12.4 12.5
xxii Modern Mathematical Methods Part IV Probabilities, Numerical, and Stochastic Methods 423 Chapter 13 Probability Theory 425 13.1 Probability distributions..................................................................... 425 13.1.1 General concepts..................................................................... 425 13.1.2 Probability distributions andtheir features......................... 428 13.2 Common probability distributions....................................................43° 13.2.1 Uniform distribution .............................................................43° 13.2.2 Binomial distribution .............................................................431 13.2.3 Poisson distribution................................................................433 13.2.4 Normal distribution................................................................ 43$ 13.2.5 Arbitrary probability distributions........................................ 445 13.3 Probabilities, information and entropy.............................................. 447 13.4 Multivariate probabilities .................................................................. 450 13.5 Composite random variables............................................................ 454 13.5.1 Central Limit Theorem............................................................ 458 13.6 Conditional probabilities..................................................................... 460 13.6.1 Bayes theorem...................................................................... 462 13.6.2 The Fokker-Planck
equation................................................ 465 13.7 Application: Hypothesis testing......................................................... 468 Chapter 14 Numerical Methods 479 14.1 Calculating with digital computers................................................... 479 14.2 Numerical representation of functions ................ 483 14.2.1 Grid and spectral methods................................................... 483 14.2.2 Accuracy and stability............................................................ 484 14.3 Numerical evaluation of derivatives................................................ 485 14.4 Numerical evaluation of integrals ................................................... 488 14.4.1 The trapezoid formula........................................................... 489 14.4.2 Simpson s 1/3-rule formula.................................................. 490 14.5 Numerical solution of ODEs............................................................... 491 14.5.1 First-order ODEs................................................................... 491 14.5.2 Second-order ODEs............................................................. 494 14.6 Numerical solution of PDEs............................................................... 498 14.7 Solving differential equations with neural networks...................... 503 14.8 Computer-generated random numbers ...........................................507 14.9 Application: Monte Carlo integration............................................. 508 Chapter 15 Stochastic Methods 515 15.1 Stochastic
simulation: Random walks and diffusion...................... 515 15.2 Stochastic optimization....................................................................... 520 15.2.1 The method of importance sampling ............................... 522 15.2.2 The Metropolis algorithm...................................................... 526 15.3 The simulated annealing method ................................................ 529 15.4 Application: The traveling salesman problem.................................. 533 Appendix A Solution of the Black-Scholes Equation 537 Appendix В Gaussian Integral Table 539 Index 541
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| adam_txt |
Contents Preface vii About the Authors xiii Acknowledgments xv Part I Functions of Real Variables i Chapter i Functions of a Single Variable 3 1.1 Introduction: The various types of numbers. 3 1.2 Elementary functions. 7 1.2.1 Polynomials. 7 1.2.2 The inverse function. 9 1.2.3 Geometric shapes: Circle, ellipse, hyperbola. 10 1.2.4 Trigonometric functions. 12 1.2.5 Exponential, logarithm, hyperbolic functions. 13 1.3 Continuity and derivatives. 15 1.3.1 Continuity. 15 1.3.2 Definition of derivatives. 16 1.3.3 Geometric interpretation of derivatives. 17 1.3.4 Finding roots by using derivatives. 19 1.3.5 Chain rule and implicit differentiation . 20 1.4 Integrals. 21 1.4.1 The definite and indefinite integrals. 21 1.4.2 Integration by change of variables. 23 1.4.3 Integration by
parts. 24 1.4.4 The principal value integral . 25 1.5 Norms and moments of a function. 27 1.5.1 The norms of a function. 27 1.5.2 The normalized Gaussian function . 28 1.5.3 Moments of a function. 30 1.6 Generalized functions: ժ-function and ¿-function . 31 1.6.1 The 0-function or step-function. 32 1.6.2 The ¿-function. 33 1.7 Application: Signal filtering, convolution, correlation. 36
xviii Modern Mathematical Methods Chapter 2 Functions of Many Variables 4і 2.1 General considerations. 4і շ.շ Coordinate systems and change of variables. 43 2.2.1 Coordinate systems in two dimensions. 43 2.2.2 Coordinate systems in three dimensions. 46 2.2.3 Change of variables . 49 2.3 Vector operations. 52 2.3.1 The dot or scalar product. 53 2.3.2 The cross or vector product. 54 2.3.3 The wedge product. 54 2.4 Differential operators. 55 2.5 The vector calculus integrals . 60 2.5.1 Simple curves and simply-connected domains. 60 2.5.2 Green's theorem. 61 2.5.3 The divergence and curl theorems. 62 2.5.4 Geometric interpretation of the divergence and curl theorems. 65 2.6 Function optimization. 67 2.6.1 Finding the extrema of a
function. 67 2.6.2 Constrained optimization: Lagrange multipliers. 69 2.6.3 Calculus of variations. 70 2.7 Application: Feedforward neural network. 73 2.7.1 Definition of feedforward neural network . 74 2.7.2 Training of the network. 78 Chapter 3 Series Expansions 85 3.1 Infinite sequences and series . 85 3.1.1 Series convergence tests. 87 3.1.2 Some important number series. 88 3.2 Series expansions of functions. 91 3.2.1 Power series expansion: The Taylor series . 92 3.2.2 Inversion of power series. 96 3.2.3 Trigonometric series expansions. 97 3.2.4 Polynomial series expansion. 99 3.3 Convergence of series expansion of functions. 101 3.3.1 Uniform convergence. 102 3.3.2 Uniform convergence criteria.106 3.4 Truncation error in series expansions . 107 3.5 Application: Wavelet
analysis.109 3.5.1 Haar wavelet expansion. 110 3.5.2 The wavelet transform. 113 Part II Complex Analysis and the Fourier Transform 117 Chapter 4 Functions of Complex Variables 119 4.1 Complex numbers . 119 4.2 Complex variables . 122 4.2.1 The complex exponential.123
Contents xix 4.3 4.4 4.5 4.6 4.7 4.8 4.2.2 Multi-valuedness in polar representation. 123 4.2.3 The complex logarithm. 124 Continuity and derivatives. 125 Analyticity. 127 4.4.1 The d-bar derivative.130 4.4.2 Cauchy-Riemann relations and harmonic functions . . . 130 4.4.3 Consequences of analyticity. 132 4.4.4 Explicit formulas for computing an analytic function in terms of its real or imaginaryparts. 134 Complex integration: Cauchy's theorem. 136 4.5.1 Integration over a contour. 136 4.5.2 Cauchy's theorem .137 4.5.3 Contour deformation . 139 4.5.4 Cauchy's Integral Formula. 141 Extensions of Cauchy's theorem. 141 4.6.1 General form of Cauchy's theorem. 142 4.6.2 Pompeiu's formula. 142 4.6.3 The ¿-function of complex argument . 143 Complex power series
expansions. 144 4.7.1 Taylor series. 146 4.7.2 Laurent series. 147 Application: The 2D Fourier transform .150 Chapter 5 Singularities, Residues, Contour Integration 159 5.1 Types of singularities. 159 5.2 Residue theorem . 161 5.3 Integration by residues. 162 5.3.1 Integrands with simple poles. 162 5.3.2 Integrands that are ratios of polynomials. 165 5.3.3 Integrands with trigonometric or hyperbolic functions. 170 5.3.4 Skipping a simple pole .172 5.4 Jordan's lemma. 173 5.5 Scalar Riemann-Hilbert problems. 176 5.5.1 Derivation of the inverse Fourier transform . 180 5.6 Branch points and branch cuts. 181 5.7 The source-sink function and related problems . 184 5.8 Application: Medical
imaging.188 Chapter 6 Mappings Produced by Complex Functions 195 6.1 Mappings produced by positive powers. 196 6.2 Mappings produced by 1/z. 199 6.3 Mappings produced by the exponential function. 203 6.4 Conformal mapping. 209 6.5 Application: Fluid flow around an obstacle. 211 Chapter 7 The Fourier Transform 219 7.1 Fourier series expansions. 219 7.1.1 Real Fourier expansion . 219
xx Modern Mathematical Methods 7.1.2 Odd and even function expansions . 222 7.1.3 Handling of discontinuities in Fourier expansions. 223 7.1.4 Fourier expansion of arbitrary range . 223 7.1.5 Complex Fourier expansion.224 7.1.6 Fourier expansion of convolution and correlation.225 7.1.7 Conditions for the validity of a Fourier expansion. 226 7.2 The Fourier transform.226 7.2.1 Definition of the Fourier transform. 226 7.2.2 Properties of the Fourier transform. 227 7.2.3 Symmetries of the Fourier transform.234 7.2.4 The sine- and cosine-transforms. 235 7.3 Fourier transforms of special functions . 236 7.3.1 The FT of the normalized Gaussian function. 236 7.3.2 FT of the 0-function and of the ¿-function. 238 7.4 Application: Signal analysis. 242 7.4.1 Aliasing: The Nyquist frequency. 243 Part III Applications to Partial Differential Equations 253 Chapter 8 Partial Differential Equations: Introduction 255 8.1 General remarks. 255 8.2 Overview of
traditional approaches. 256 8.2.1 Examples of ODEs. 257 8.2.2 Eigenvalue equations . 259 8.2.3 The Green's function method . 260 8.2.4 The Fourier series method.263 8.3 Evolution equations. 263 8.3.1 Motivation of evolution equations. 263 8.3.2 Separation of variables and initial value problems. 265 8.3.3 Traditional integral transforms for problems on the half-line. 270 8.3.4 Traditional infinite series for problems on a finite interval.272 8.4 The wave equation. 273 8.4.1 Derivation oí the wave equation.273 8.4.2 The solution of d'Alembert. 275 8.4.3 Traditional transforms . 277 8.5 The Laplace and Poisson equations. 280 8.5.1 Motivation of the Laplace and Poisson equations. 282 8.5.2 Integral representations through the fundamental
solution. 283 8.6 The Helmholtz and modified Helmholtz equations. 285 8.7 Disadvantages of traditional integral transforms. 288 8.8 Application: The Black-Scholes equation. 290 Chapter 9 Unified Transform. I: Evolution PDEs on the Half-line 297 9.1 The unified transform is based on analyticity. 297 9.2 The heat equation. 300
Contents xxi The general methodology of the unified transform . 304 9.3.1 Advantages of the unified transform . 306 9.4 A PDE with a third-order spatial derivative.314 9.5 Inhomogeneous PDEs and other considerations. 320 9.5.1 Robin boundary conditions. 322 9.5.2 From PDEs to ODEs. 328 9.5.3 Green's functions. 329 9.6 Application: Heat flow along a solid rod . 331 9.6.1 The conventional solution. 332 9.6.2 The solution through the unified transform . 333 9.3 Chapter 10 Unified Transform II: Evolution PDEs on a Finite Interval 341 10.1 The heat equation. 341 10.2 Advantages of the unified transform. 347 10.3 A PDE with a third-order spatial derivative. 349 10.4 Inhomogeneous PDEs and other considerations. 351 10.4.1 Robin boundary conditions. 351 10.4.2 From PDEs to ODEs. 354 10.4.3 Green's functions. 355 10.5
Application: Detection of colloid concentration. 356 Chapter 11 Unified Transform III: The Wave Equation 365 11.1 An alternative derivation of the global relation. 365 11.2 The wave equation on the half-line . 366 11.2.1 The Dirichlet problem. 368 11.3 The wave equation on a finite interval. 371 11.3.1 The Dirichlet-Dirichlet problem. 372 11.4 The forced problem.381 Unified Transform IV: Laplace, Poisson, and Helmholtz Equations 385 Introduction. 385 12.1.1 Integral representations in the fc-complex plane.387 The Laplace and Poisson equations . 389 12.2.1 Global relations for a convex polygon. 390 12.2.2 An integral representation in the fc-complex plane. 391 12.2.3 The approximate global relation for a convex polygon . 399 12.2.4 The general case. 403 The Helmholtz and modified Helmholtz equations. 403 12.3.1 Global relations for a convex polygon . 404 12.3.2 Novel integral representations for the modified Helmholtz
equation. 405 12.3.3 The general case. 411 12.3.4 Computing the solution in the interior of a convex polygon.412 Generalizations. 413 Application: Water waves. 413 Chapter 12 12.1 12.2 12.3 12.4 12.5
xxii Modern Mathematical Methods Part IV Probabilities, Numerical, and Stochastic Methods 423 Chapter 13 Probability Theory 425 13.1 Probability distributions. 425 13.1.1 General concepts. 425 13.1.2 Probability distributions andtheir features. 428 13.2 Common probability distributions.43° 13.2.1 Uniform distribution .43° 13.2.2 Binomial distribution .431 13.2.3 Poisson distribution.433 13.2.4 Normal distribution. 43$ 13.2.5 Arbitrary probability distributions. 445 13.3 Probabilities, information and entropy. 447 13.4 Multivariate probabilities . 450 13.5 Composite random variables. 454 13.5.1 Central Limit Theorem. 458 13.6 Conditional probabilities. 460 13.6.1 Bayes' theorem. 462 13.6.2 The Fokker-Planck
equation. 465 13.7 Application: Hypothesis testing. 468 Chapter 14 Numerical Methods 479 14.1 Calculating with digital computers. 479 14.2 Numerical representation of functions . 483 14.2.1 Grid and spectral methods. 483 14.2.2 Accuracy and stability. 484 14.3 Numerical evaluation of derivatives. 485 14.4 Numerical evaluation of integrals . 488 14.4.1 The trapezoid formula. 489 14.4.2 Simpson's 1/3-rule formula. 490 14.5 Numerical solution of ODEs. 491 14.5.1 First-order ODEs. 491 14.5.2 Second-order ODEs. 494 14.6 Numerical solution of PDEs. 498 14.7 Solving differential equations with neural networks. 503 14.8 Computer-generated random numbers .507 14.9 Application: Monte Carlo integration. 508 Chapter 15 Stochastic Methods 515 15.1 Stochastic
simulation: Random walks and diffusion. 515 15.2 Stochastic optimization. 520 15.2.1 The method of "importance sampling". 522 15.2.2 The Metropolis algorithm. 526 15.3 The "simulated annealing" method . 529 15.4 Application: The traveling salesman problem. 533 Appendix A Solution of the Black-Scholes Equation 537 Appendix В Gaussian Integral Table 539 Index 541 |
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| author | Fokas, Athanassios S. 1952- Kaxiras, Efthimios |
| author_GND | (DE-588)113194471 (DE-588)1170424910 |
| author_facet | Fokas, Athanassios S. 1952- Kaxiras, Efthimios |
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| author_sort | Fokas, Athanassios S. 1952- |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV048877389 |
| illustrated | Illustrated |
| index_date | 2024-07-03T21:45:15Z |
| indexdate | 2024-07-10T09:48:35Z |
| institution | BVB |
| isbn | 9781800611832 9781800611801 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-034142179 |
| oclc_num | 1381302050 |
| open_access_boolean | |
| owner | DE-703 DE-83 |
| owner_facet | DE-703 DE-83 |
| physical | xxii, 545 Seiten Illustrationen, Diagramme |
| publishDate | 2023 |
| publishDateSearch | 2023 |
| publishDateSort | 2023 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Fokas, Athanassios S. 1952- Verfasser (DE-588)113194471 aut Modern mathematical methods for scientists and engineers a street-smart introduction Athanassios Fokas, University of Cambridge, UK & University of Southern California, USA, Efthimios Kaxiras, Harvard University, USA New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2023] xxii, 545 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Fokas-Methode (DE-588)1309266816 gnd rswk-swf Fourier-Transformation (DE-588)4018014-1 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Vektoranalysis (DE-588)4191992-0 gnd rswk-swf Mathematische Methode (DE-588)4155620-3 s Vektoranalysis (DE-588)4191992-0 s Funktionentheorie (DE-588)4018935-1 s Fourier-Transformation (DE-588)4018014-1 s Partielle Differentialgleichung (DE-588)4044779-0 s Fokas-Methode (DE-588)1309266816 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Kaxiras, Efthimios Verfasser (DE-588)1170424910 aut Erscheint auch als Online-Ausgabe 978-1-80061-181-8 Erscheint auch als Online-Ausgabe 978-1-80061-182-5 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034142179&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fokas, Athanassios S. 1952- Kaxiras, Efthimios Modern mathematical methods for scientists and engineers a street-smart introduction Partielle Differentialgleichung (DE-588)4044779-0 gnd Funktionentheorie (DE-588)4018935-1 gnd Fokas-Methode (DE-588)1309266816 gnd Fourier-Transformation (DE-588)4018014-1 gnd Mathematische Methode (DE-588)4155620-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Vektoranalysis (DE-588)4191992-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4018935-1 (DE-588)1309266816 (DE-588)4018014-1 (DE-588)4155620-3 (DE-588)4128130-5 (DE-588)4079013-7 (DE-588)4191992-0 |
| title | Modern mathematical methods for scientists and engineers a street-smart introduction |
| title_auth | Modern mathematical methods for scientists and engineers a street-smart introduction |
| title_exact_search | Modern mathematical methods for scientists and engineers a street-smart introduction |
| title_exact_search_txtP | Modern mathematical methods for scientists and engineers a street-smart introduction |
| title_full | Modern mathematical methods for scientists and engineers a street-smart introduction Athanassios Fokas, University of Cambridge, UK & University of Southern California, USA, Efthimios Kaxiras, Harvard University, USA |
| title_fullStr | Modern mathematical methods for scientists and engineers a street-smart introduction Athanassios Fokas, University of Cambridge, UK & University of Southern California, USA, Efthimios Kaxiras, Harvard University, USA |
| title_full_unstemmed | Modern mathematical methods for scientists and engineers a street-smart introduction Athanassios Fokas, University of Cambridge, UK & University of Southern California, USA, Efthimios Kaxiras, Harvard University, USA |
| title_short | Modern mathematical methods for scientists and engineers |
| title_sort | modern mathematical methods for scientists and engineers a street smart introduction |
| title_sub | a street-smart introduction |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Funktionentheorie (DE-588)4018935-1 gnd Fokas-Methode (DE-588)1309266816 gnd Fourier-Transformation (DE-588)4018014-1 gnd Mathematische Methode (DE-588)4155620-3 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd Vektoranalysis (DE-588)4191992-0 gnd |
| topic_facet | Partielle Differentialgleichung Funktionentheorie Fokas-Methode Fourier-Transformation Mathematische Methode Numerisches Verfahren Wahrscheinlichkeitstheorie Vektoranalysis |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=034142179&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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