Partial Differential Equations by Hillen, Thomas -- Read -- Imperial Library of Trantor

Index

Cover Page Table of Contents Title Copyright Preface Part I Theory

1 Introduction

1.1 Partial Differential Equations 1.2 Classification of Second-Order Linear PDEs 1.3 Side Conditions 1.4 Linear PDEs 1.5 Steady-State and Equilibrium Solutions 1.6 First Example for Separation of Variables 1.7 Derivation of the Diffusion Equation 1.8 Derivation of the Heat Equation 1.9 Derivation of the Wave Equation 1.10 Examples of Laplace's Equation 1.11 Summary

2 Fourier Series

2.1 Piecewise Continuous Functions 2.2 Even, Odd, and Periodic Functions 2.3 Orthogonal Functions 2.4 Fourier Series 2.5 Convergence of Fourier Series 2.6 Operations on Fourier Series 2.7 Mean Square Error 2.8 Complex Fourier Series 2.9 Summary

3 Separation of Variables

3.1 Homogeneous Equations 3.2 Nonhomogeneous Equations 3.3 Summary

4 Sturm-Liouville Theory

4.1 Formulation 4.2 Properties of Sturm-Liouville Problems 4.3 Eigenfunction Expansions 4.4 Rayleigh Quotient 4.5 Summary

5 Heat, Wave, and Laplace Equations

5.1 One-Dimensional Heat Equation 5.2 Two-Dimensional Heat Equation 5.3 One-Dimensional Wave Equation 5.4 Laplace's Equation 5.5 Maximum Principle 5.6 Two-Dimensional Wave Equation 5.7 Eigenfunctions in Two Dimensions 5.8 Summary

6 Polar Coordinates

6.1 Interior Dirichlet Problem for a Disk 6.2 Vibrating Circular Membrane 6.3 Bessel's Equation 6.4 Bessel Functions 6.5 Fourier-Bessel Series 6.6 Solution to the Vibrating Membrane Problem 6.7 Summary

7 Spherical Coordinates

7.1 Spherical Coordinates 7.2 Legendre's Equation 7.3 Legendre Functions 7.4 Spherical Bessel Functions 7.5 Interior Dirichlet Problem for a Sphere 7.6 Summary

8 Fourier Transforms

8.1 Fourier Integrals 8.2 Fourier Transforms 8.3 Summary

9 Fourier Transform Methods in PDEs

9.1 The Wave Equation 9.2 The Heat Equation 9.3 Laplace's Equation 9.4 Summary

10 Method of Characteristics

10.1 Introduction to the Method of Characteristics 10.2 Geometric Interpretation 10.3 d'Alembert's Solution 10.4 Extension to Quasilinear Equations 10.5 Summary

Part II Explicitly Solved Problems

11 Fourier Series Problems 12 Sturm-Liouville Problems 13 Heat Equation Problems 14 Wave Equation Problems 15 Laplace Equation Problems 16 Fourier Transform Problems 17 Method of Characteristics Problems 18 Four Sample Midterm Examinations

18.1 Midterm Exam 1 18.2 Midterm Exam 2 18.3 Midterm Exam 3 18.4 Midterm Exam 4

19 Four Sample Final Examinations

19.1 Final Exam 1 19.2 Final Exam 2 19.3 Final Exam 3 19.4 Final Exam 4

Appendix A: Gamma Function Bibliography Index