A first course in partial differential equations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore
World Scientific
[2018]
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xviii, 606 Seiten Illustrationen, Diagramme |
| ISBN: | 9789811211317 9789813226432 |
Internformat
MARC
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| 245 | 1 | 0 | |a A first course in partial differential equations |c J. Robert Buchanan, Zhoude Shao (Millersville University, USA) |
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Datensatz im Suchindex
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| adam_text | Contents Preface vii About the Authors xi Acknowledgments xiii 1. Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2. 3. 1 Preliminaries: Notation, Definitions, and the Principle of Superposition....................................................................... Classification of Second-Order Partial Differential Equations Heat Conduction and the Heat Equation........................... Vibrating Strings and the Wave Equation........................ Laplace’s Equation.............................................................. Separation of Variables........................................................ Exercises................................................................................ 1 7 8 18 25 27 35 First-Order Partial Differential Equations 41 2.1 2.2 2.3 2.4 41 49 57 67 First-Order Linear Equations............................................. First-Order Quasilinear Equations .................................... Applications.......................................................................... Exercises................................................................................ Fourier Series 71 3.1 3.2 3.3 3.4 3.5 73 75 76 81 86 Periodic Functions.............................................................. The Trigonometric System and Orthogonality.................. Euler-Fourier Formulas and Fourier Series........................ Even and Odd Functions..................................................... Even or Odd Extension of Functions................................. XV
A First Course in Partial Differential Equations xvi 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 4. The Heat Equation 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5. 6. Convergence Theorem........................................................... The Gibbs Phenomenon and Uniform Convergence .... Differentiation and Integration of Fourier Series................ Mean Square Approximation and Parseval’s Identity . . . Complex Form of the Fourier Series..................................... Proofs of Two Theoretical Results ..................................... Historical and Supplemental Remarks ............................... Exercises.................................................................................... Homogeneous Boundary Value Problems on Bounded Intervals.................................................................................... Nonhomogeneous Boundary Value Problems...................... A Maximum Principle and Uniqueness of Solutions .... The Heat Equation on Unbounded Intervals...................... The Heat Equation on a Rectangular Domain................... Supplemental Remarks and Suggestions for Further Study Exercises.................................................................................... 92 97 102 106 110 112 117 118 123 124 134 143 149 165 173 174 The Wave Equation 181 5.1 5.2 5.3 5.4 5.5 5.6 181 190 199 207 214 216 Wave Equation with Homogeneous Boundary Conditions . d’Alembert’s Approach........................................................... Solving the Wave Equation ֊ Revisited............................... Nonhomogeneous Cases
........................................................ The Energy Integral and Uniqueness of Solutions............ Exercises.................................................................................... Laplace’s Equation 223 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 224 226 233 240 249 253 255 260 265 268 Boundary Value Problems of Laplace’s Equation............ Dirichlet Problems on Rectangles........................................ Dirichlet Problems on Disks.................................................. Dirichlet Problems on Domains Related to Disks............ Neumann Problems on Rectangles............... Neumann Problems on Disks ............................................... Mixed Boundary Conditions on Rectangles......................... Poisson’s Formula and Mean Value Property...................... Maximum Principle and Uniqueness .................................. Exercises....................................................................................
Contents 7. Sturm-Liouville Theory 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 8. 9. xvii 275 Two-Point Boundary Value Problems of Second-Order Differential Equations ........................................................... 276 Properties of Eigenvalues and Eigenfunctions.......................280 Zeros of Eigenfunctions........................................................... 286 Generalized Fourier Series..................................................... 289 Estimating Eigenvalues and the RayleighQuotient .... 291 Existence of Eigenfunctions and Eigenvalues...................... 295 Readings and Further Results.............................................. 304 Exercises.................................................................................... 305 Special Functions 311 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 311 316 338 355 358 370 377 385 The Gamma Function........................................................... Bessel’s Equation of Order p................................................. The Legendre Equation ........................................................ Spherical Harmonics.............................................................. The Laguerre Equation........................................................... The Hermite Polynomials and Their Properties................ The Chebyshev Polynomials and Their Properties............ Exercises................................................................................... Applications of PDEs in the PhysicalSciences 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Swinging
Chain....................................................................... 393 Vibration of a Circular Drum Head..................................... 397 Steady-State Temperature in a Sphere............................... 402 Vibrating Sphere.................................................................... 405 Heat Equation in a Sphere..................................................... 412 Quantum Harmonic Oscillator.............................................. 415 Model of the Hydrogen Atom.............................................. 424 Exercises................................................................................... 428 10. Nonhomogeneous Initial Boundary ValueProblems 10.1 10.2 10.3 10.4 393 Duhamel’s Principle and Its Motivation .......................... Nonhomogeneous Heat Equation........................................ Nonhomogeneous Wave Equation........................................ Exercises................................................................................... 435 437 439 451 458
A First Course in Partial Differential Equations xviii 11. Nonlinear Partial Differential Equations 11.1 11.2 11.3 11.4 11.5 11.6 Waves and Traveling Waves.................................................. Burgers’ Equation ................................................................. The Korteweg-de Vries Equation........................................ The Nonlinear Schrödinger Equation.................................. Further Comments................................................................. Exercises.................................................................................... 12. Numerical Solutions to PDEs Using Finite Differences 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 Complex Arithmetic and Calculus Complex Numbers.................................................................. Complex Functions.................................................................. Complex Differentiation......................................................... Complex Integration............................................................... Appendix В B.l B.2 B.3 B.4 B.5 B.6 462 466 483 488 492 493 497 Discretization of Derivatives.................................................. 498 Discretization of Initial and Boundary Conditions.................502 The Heat Equation................................................................. 503 The Wave Equation .............................................................. 512 Laplace’s and Poisson’s Equations ..................................... 518 Iterative
Methods.................................................................... 525 Convergence and Stability..................................................... 533 Exercises.................................................................................... 549 Appendix A A.l A.2 A.3 A.4 461 Linear Algebra Primer Matrices and Their Properties.............................................. Determinant of a Square Matrix........................................... Vector and Matrix Norms.................................................... Eigenvalues and Eigenvectors.............................................. Diagonally Dominant Matrices ........................................... Positive Definite Matrices.................................................... 559 559 561 565 568 573 573 578 580 582 585 590 Bibliography 593 Index 599
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| isbn | 9789811211317 9789813226432 |
| language | English |
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| physical | xviii, 606 Seiten Illustrationen, Diagramme |
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| spelling | Buchanan, J. Robert (DE-588)102711332X aut A first course in partial differential equations J. Robert Buchanan, Zhoude Shao (Millersville University, USA) Singapore World Scientific [2018] © 2018 xviii, 606 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Differential equations, Partial Textbooks Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Shao, Zhoude (DE-588)1159999740 aut Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030144848&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Buchanan, J. Robert Shao, Zhoude A first course in partial differential equations Differential equations, Partial Textbooks Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4123623-3 |
| title | A first course in partial differential equations |
| title_auth | A first course in partial differential equations |
| title_exact_search | A first course in partial differential equations |
| title_full | A first course in partial differential equations J. Robert Buchanan, Zhoude Shao (Millersville University, USA) |
| title_fullStr | A first course in partial differential equations J. Robert Buchanan, Zhoude Shao (Millersville University, USA) |
| title_full_unstemmed | A first course in partial differential equations J. Robert Buchanan, Zhoude Shao (Millersville University, USA) |
| title_short | A first course in partial differential equations |
| title_sort | a first course in partial differential equations |
| topic | Differential equations, Partial Textbooks Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Differential equations, Partial Textbooks Partielle Differentialgleichung Lehrbuch |
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