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- Introduction to Applied Partial Differential Equations

# Introduction to Applied Partial Differential Equations

## First Edition| ©2013 John M. Davis

Drawing on his decade of experience teaching the differential equations course

**,**John Davis offers a refreshing and effective new approach to partial differential equations that is equal parts computational proficiency, visualization, and physical interpretation of the problem at hand.Drawing on his decade of experience teaching the differential equations course

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**,**John Davis offers a refreshing and effective new approach to partial differential equations that is equal parts computational proficiency, visualization, and physical interpretation of the problem at hand.
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**, **John Davis offers a refreshing and effective new approach to partial differential equations that is equal parts computational proficiency, visualization, and physical interpretation of the problem at hand.

Features

**Focus on Deeper Concepts Not Mundane Computation**

The text requires hand calculation selectively, and encourages the use of appropriate computing technology throughoutÑfor example, in computing the integrals arising in Fourier series coefficients or plotting 3D animations of solution surfaces. By utilizing technology appropriately, the focus can be placed on more complex, realistic problems.

**Emphasis on Geometric Iinsight and Physical Interpretation**

Davis' text gives the computational aspect of the course a vivid context by continually reflecting on what the combination of calculation, visualization, and physical interpretation reveals about a problem. This helps students gain some understanding of the qualitative properties of solutions and what they tell us about the real world physical system.

**Embraces the Tools and Language of Vector Calculus**

The book takes full advantage of the opportunity of using the partial differential equation course to solidify students' understanding of vector calculus. For example, deriving the multidimensional heat and wave equations from a variational viewpoint are occasions to invoke the Divergence Theorem and Stokes' Theorem.

**Sets the Stage for Future Topics in Analysis**

To make sure students are prepared for upper level mathematics courses, the text introduces a number of topics (tackle vector spaces, inner product spaces, eigenvalue problems, orthogonality, various modes of convergence, and basic L2 theory) at an appropriate level.

New to This Edition

**
Introduction to Applied Partial Differential Equations**

First Edition| ©2013

John M. Davis

# Digital Options

**Introduction to Applied Partial Differential Equations**

First Edition| 2013

John M. Davis

## 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

**Introduction to Applied Partial Differential Equations**

First Edition| 2013

John M. Davis

## Authors

### John M. Davis

**John Davis**received his Ph.D. in Mathematics from Auburn University in 1998 and joined the faculty at Baylor University in 1999. His interdisciplinary research in ordinary and partial differential equations, hybrid dynamical systems, and applications to control theory and signal processing has been funded by the National Science Foundation, resulting in more than 50 peer reviewed publications. He won the Mathematical Association of America’s Distinguished University Teaching Award in 2009.

**Introduction to Applied Partial Differential Equations**

First Edition| 2013

John M. Davis

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