# Introduction to Applied Partial Differential Equations

## First EditionJohn M. Davis

©2013
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*Introduction to Applied Partial Differential **Equ**ations* takes a new approach to partial differential equations to enhance comprehension.

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Learn More## Table of Contents

**Preface**

**1 Introduction to PDEs**

1.1 ODEs vs. PDEs

1.2 How PDEs Are Born: Conservation Laws, Fluids, and Waves

1.3 Boundary Conditions in One Space Dimension

1.4 ODE Solution Methods

**2 Fourier's Method: Separation of Variables**

2.1 Linear Algebra Concepts

2.2 The General Solution via Eigenfunctions

2.3 The Coefficients via Orthogonality

2.4 Consequences of Orthogonality

2.5 Robin Boundary Conditions

2.6 Nonzero Boundary Conditions: Steady-States and Transients*

**3 Fourier Series Theory**

3.1 Fourier Series: Sine, Cosine, and Full

3.2 Fourier Series vs. Taylor Series: Global vs. Local Approximations*

3.3 Error Analysis and Modes of Convergence

3.4 Convergence Theorems

3.5 Basic L2 Theory

3.6 The Gibbs Phenomenon*

**4 General Orthogonal Series Expansions**

4.1 Regular and Periodic Sturm-Liouville Theory

4.2 Singular Sturm-Liouville Theory

4.3 Orthogonal Expansions: Special Functions

4.4 Computing Bessel Functions: The Method of Frobenius

4.5 The Gram-Schmidt Procedure*

**5 PDEs in Higher Dimensions**

5.1 Nuggets from Vector Calculus

5.2 Deriving PDEs in Higher Dimensions

5.3 Boundary Conditions in Higher Dimensions

5.4 Well-Posed Problems: Good Models

5.5 Laplace's Equation in 2D

5.6 The 2D Heat and Wave Equations

**6 PDEs in Other Coordinate Systems**

6.1 Laplace's Equation in Polar Coordinates

6.2 Poisson's Formula and Its Consequences*

6.3 The Wave Equation and Heat Equation in Polar Coordinates

6.4 Laplace's Equation in Cylindrical Coordinates

6.5 Laplace's Equation in Spherical Coordinates

**7 PDEs on Unbounded Domains**

7.1 The Infinite String: d'Alembert's Solution

7.2 Characteristic Lines

7.3 The Semi-infinite String: The Method of Reflections

7.4 The Infinite Rod: The Method of Fourier Transforms

**Appendix**

Selected Answers

Credits

Index

Selected Answers

Credits

Index