Beginning Partial Differential Equations

1.1 Two Partial Differential Equations 1

1.2 Fourier Series 10

1.3 Two Eigenvalue Problems 28

1.4 A Proof of the Fourier Convergence Theorem 30

2. Solutions of the Heat Equation 39

2.1 Solutions on an Interval (0, L) 39

2.2 A Nonhomogeneous Problem 64

2.3 The Heat Equation in Two space Variables 71

2.4 The Weak Maximum Principle 75

3. Solutions of the Wave Equation 81

3.1 Solutions on Bounded Intervals 81

3.2 The Cauchy Problem 109

3.3 The Wave Equation in Higher Dimensions 137

4. Dirichlet and Neumann Problems 147

4.1 Laplace’s Equation and Harmonic Functions 147

4.2 The Dirichlet Problem for a Rectangle 153

4.3 The Dirichlet Problem for a Disk 158

4.4 Properties of Harmonic Functions 165

4.5 The Neumann Problem 187

4.6 Poisson’s Equation 197

4.7 Existence Theorem for a Dirichlet Problem 200

5. Fourier Integral Methods of Solution 213

5.1 The Fourier Integral of a Function 213

5.2 The Heat Equation on a Real Line 220

5.3 The Debate over the Age of the Earth 230

5.4 Burger’s Equation 233

5.5 The Cauchy Problem for a Wave Equation 239

5.6 Laplace’s Equation on Unbounded Domains 244

6. Solutions Using Eigenfunction Expansions 253

6.1 A Theory of Eigenfunction Expansions 253

6.2 Bessel Functions 266

6.3 Applications of Bessel Functions 279

6.4 Legendre Polynomials and Applications 288

7. Integral Transform Methods of Solution 307

7.1 The Fourier Transform 307

7.2 Heat and Wave Equations 318

7.3 The Telegraph Equation 332

7.4 The Laplace Transform 334

8 First-Order Equations 341

8.1 Linear First-Order Equations 342

8.2 The Significance of Characteristics 349

8.3 The Quasi-Linear Equation 354

9 End Materials 361

9.1 Notation 361

9.2 Use of MAPLE 363

9.3 Answers to Selected Problems 370

Index 434