Partial Differential Equations: Theory and Completely Solved Problems

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Uniquely provides fully solved problems for linear partial differential equations and boundary value problems

Partial Differential Equations: Theory and Completely Solved Problems utilizes real-world physical models alongside essential theoretical concepts. With extensive examples, the book guides readers through the use of Partial Differential Equations (PDEs) for successfully solving and modeling phenomena in engineering, biology, and the applied sciences.

The book focuses exclusively on linear PDEs and how they can be solved using the separation of variables technique. The authors begin by describing functions and their partial derivatives while also defining the concepts of elliptic, parabolic, and hyperbolic PDEs. Following an introduction to basic theory, subsequent chapters explore key topics including:

• Classification of second-order linear PDEs

• Derivation of heat, wave, and Laplace’s equations

• Fourier series

• Separation of variables

• Sturm-Liouville theory

• Fourier transforms

Each chapter concludes with summaries that outline key concepts. Readers are provided the opportunity to test their comprehension of the presented material through numerous problems, ranked by their level of complexity, and a related website features supplemental data and resources.

Extensively class-tested to ensure an accessible presentation, Partial Differential Equations is an excellent book for engineering, mathematics, and applied science courses on the topic at the upper-undergraduate and graduate levels.

Preface v

PART I THEORY

1 Introduction 3

1.1 Partial Differential Equations 4

1.2 Classification of Second Order Linear PDEs 7

1.3 Side Conditions 10

1.4 Linear PDEs 12

1.5 Steady-State and Equilibrium Solutions 16

1.6 First Example for Separation of Variables 19

1.7 Derivation of the Diffusion Equation 24

1.8 Derivation of the Heat Equation 26

1.9 Derivation of the Wave Equation 29

1.10 Examples of Laplace's Equation 33

1.11 Summary 36

2 Fourier Series 39

2.1 Piecewise Continuous Functions 39

2.2 Even, Odd, and Periodic Functions 41

2.3 Orthogonal Functions 43

2.4 Fourier Series 49

2.5 Convergence of Fourier Series 56

2.6 Operations on Fourier Series 63

2.7 Mean Square Error 74

2.8 Complex Fourier Series 78

2.9 Summary 80

3 Separation of Variables 83

3.1 Homogeneous Equations 83

3.2 Nonhomogeneous Equations 95

3.3 Summary 111

4 Sturm-Liouville Theory 115

4.1 Formulation 115

4.2 Properties of Sturm-Liouville Problems 119

4.3 Eigenfunction Expansions 127

4.4 Rayleigh Quotient 135

4.5 Summary 141

5 Heat Eqn, Wave Eqn, and Laplace's Eqn 145

5.1 One-Dimensional Heat Equation 145

5.2 Two-Dimensional Heat Equation 150

5.3 One-DimensionalWave Equation 153

5.4 Laplace's Equation 163

5.5 Maximum Principle 168

5.6 Two-DimensionalWave Equation 169

5.7 Eigenfunctions in Two Dimensions 173

5.8 Summary 178

6 Polar Coordinates 181

6.1 Interior Dirichlet Problem for a Disk 181

6.2 Vibrating Circular Membrane 190

6.3 Bessel's Equation 193

6.4 Bessel Functions 198

6.5 Fourier-Bessel Series 213

6.6 Solution to the Vibrating Membrane Problem 217

6.7 Summary 221

7 Spherical Coordinates 225

7.1 Spherical Coordinates 225

7.2 Legendre's Equation 228

7.3 Legendre Functions 232

7.4 Spherical Bessel Functions 257

7.5 Interior Dirichlet Problem for a Sphere 258

7.6 Summary 262

8 Fourier Transforms 265

8.1 Fourier Integrals 265

8.2 Fourier Transforms 282

8.3 Summary 302

9 Fourier Transform Methods in PDEs 305

9.1 The Wave Equation 306

9.2 The Heat Equation 312

9.3 Laplace's Equation 326

9.4 Summary 335

10 Method of Characteristics 337

10.1 Introduction to the Method of Characteristics 337

10.2 Geometric Interpretation 342

10.3 d'Alembert's Solution 351

10.4 Extension to Quasilinear Equations 354

10.5 Summary 356

PART II EXPLICITLY SOLVED PROBLEMS

11 Fourier Series Problems 361

12 Sturm-Liouville Problems 393

13 Heat Equation Problems 433

14 Wave Equation Problems 491

15 Laplace's Equation Problems 543

16 Fourier Transform Problems 577

17 Method of Characteristics Problems 607

18 Four Sample Midterm Examinations 631

19 Four Sample Final Examinations 647

19.1 Final Exam 1 647

19.2 Final Exam 2 656

19.3 Final Exam 3 662

19.4 Final Exam 4 669

Appendices 677

Appendix A: Gamma Function 679

The Bibliography 683

Bibliography 685

The Index 687

Index 689

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