PARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS

Transcription

1 PARTIAL DIFFERENTIAL EQUATIONS and BOUNDARY VALUE PROBLEMS NAKHLE H. ASMAR University of Missouri PRENTICE HALL, Upper Saddle River, New Jersey 07458

2 Contents Preface vii A Preview of Applications and Techniques What Is a Partial Differential Equation? Solving and Interpreting a Partial Differential Equation 7 Fourier Series Periodic Functions Fourier Series Fourier Series of Functions with Arbitrary Periods Half-Range Expansions: The Cosine, and Sine Series Mean Square Approximation and Parseval's Identity Complex Form of Fourier Series 56 Supplement on Convergence 2.7 Uniform Convergence of Sequences and Series of Functions Dirichlet Test and Convergence of Fourier Series 72 Partial Differential Equations in Rectangular Coordinates Partial Differential Equations in Physics and Engineering Modeling: Vibrating Strings and the Wave Equation Solution of the One Dimensional Wave Equation: The Method of Separation of Variables D'Alembert's Method The One Dimensional Heat Equation Heat Conduction in Bars: Varying the Boundary Conditions The Two Dimensional Wave and Heat Equations Laplace's Equation in Rectangular Coordinates 138

3 iv Contents 3.9 Poisson's Equation: The Method of Eigenfunction Expansions The Maximum Principle 154 Partial Differential Equations in Polar and Cylindrical Coordinates 160 J 4.1 The Laplacian in Various Coordinate Systems Vibrations of a Circular Membrane: Symmetric Case Vibrations of a Circular Membrane: General Case Steady-State Temperature in a Disk Steady-State Temperature in a Cylinder The Helmholtz and Poisson Equations 195 Supplement on Bessel Functions 4.7 Bessel's Equation and Bessel Functions Bessel Series Expansions 212 Partial Differential Equations in Spherical Coordinates Preview of Problems and Methods Dirichlet Problems with Symmetry Spherical Harmonics and the General Dirichlet Problem The Helmholtz Equation with Applications to the Poisson, Heat, and Wave Equations 248 Supplement on Legendre Functions 5.5 Legendre's Differential Equation Legendre Polynomials and Legendre Series Expansions Associated Legendre Functions and Series Expansions 275 Sturm Liouville Theory with Engineering Applications Orthogonal Functions Sturm-Liouville Theory The Hanging Chain Fourth Order Sturm-Liouville Theory Elastic Vibrations and Buckling of Beams 313

4 Contents V The Fourier Transform and its Applications 325 / 7.1 The Fourier Integral Representation The Fourier Transform The Fourier Transform Method The Heat Equation and Gauss's Kernel A Dirichlet Problem and the Poisson Integral Formula The Fourier Cosine and Sine Transforms Problems Involving Semi-Infinite Intervals 372 The Laplace and Hankel Transforms with Applications The Laplace Transform Further Properties of the Laplace Transform The Laplace Transform Method The Hankel Transform with Applications 404 Finite Difference Numerical Methods The Finite Difference Method for the Heat Equation The Finite Difference Method for the Wave Equation The Finite Difference Method for Laplace's Equation Iteration Methods for Laplace's Equation 437 Sampling and Discrete Fourier Analysis with Applications to Partial Differential Equations The Sampling Theorem Partial Differential Equations and the Sampling Theorem The Discrete and Fast Fourier Transforms The Fourier and Discrete Fourier Transforms ~\_ An Introduction to Quantum Mechanics Schrodinger's Equation The Hydrogen Atom Heisenberg's Uncertainty Principle 487 Supplement on Orthogonal Polynomials 11.4 Hermite and Laguerre Polynomials 494

5 vi Contents APPENDIXES Ordinary Differential Equations: Review of Concepts and Methods 507 A.I Linear Ordinary Differential Equations 508 A.2 Linear Ordinary Differential Equations with Constant Coefficients 516 A.3 Methods for Solving Ordinary Differential Equations 525 A.4 The Method of Power Series 532 A.5 The Method of Frobenius 540 Tables of Transforms B.I Fourier Transforms 555 B.2 Fourier Cosine Transforms 557 B.3 Fourier Sine Transforms 558 B.4 Laplace Transforms 559 References 562 Answers to Selected Exercises 566 Index 587