Partial Differential Equations and Boundary Value Problems with Maple (eBook)

Front Cover 1
Partial Differential Equations and Boundary Value Problems with Maple 4
Copyright Page 5
Contents 6
Preface 10
Chapter 0. Basic Review 14
0.1 Preparation for Maple Worksheets 14
0.2 Preparation for Linear Algebra 17
0.3 Preparation for Ordinary Differential Equations 21
0.4 Preparation for Partial Differential Equations 23
Chapter 1. Ordinary Linear Differential Equations 26
1.1 Introduction 26
1.2 First-Order Linear Differential Equations 27
1.3 First-Order Initial-Value Problem 32
1.4 Second-Order Linear Differential Equations with Constant Coefficients 36
1.5 Second-Order Linear Differential Equations with Variable Coefficients 41
1.6 Finding a Second Basis Vector by the Method of Reduction of Order 45
1.7 The Method of Variation of Parameters—Second-Order Green’s Function 49
1.8 Initial-Value Problem for Second-Order Differential Equations 58
1.9 Frobenius Method of Series Solutions to Ordinary Differential Equations 62
1.10 Series Sine and Cosine Solutions to the Euler Differential Equation 64
1.11 Frobenius Series Solution to the Bessel Differential Equation 69
Chapter Summary 76
Exercises 78
Chapter 2. Sturm-Liouville Eigenvalue Problems and Generalized Fourier Series 86
2.1 Introduction 86
2.2 The Regular Sturm-Liouville Eigenvalue Problem 86
2.3 Green’s Formula and the Statement of Orthonormality 88
2.4 The Generalized Fourier Series Expansion 94
2.5 Examples of Regular Sturm-Liouville Eigenvalue Problems 99
2.6 Nonregular or Singular Sturm-Liouville Eigenvalue Problems 142
Chapter Summary 159
Exercises 160
Chapter 3. The Diffusion or Heat Partial Differential Equation 174
3.1 Introduction 174
3.2 One-Dimensional Diffusion Operator in Rectangular Coordinates 174
3.3 Method of Separation of Variables for the Diffusion Equation 176
3.4 Sturm-Liouville Problem for the Diffusion Equation 178
3.5 Initial Conditions for the Diffusion Equation in Rectangular Coordinates 181
3.6 Example Diffusion Problems in Rectangular Coordinates 183
3.7 Verification of Solutions—Three-Step Verification Procedure 199
3.8 Diffusion Equation in the Cylindrical Coordinate System 203
3.9 Initial Conditions for the Diffusion Equation in Cylindrical Coordinates 207
3.10 Example Diffusion Problems in Cylindrical Coordinates 209
Chapter Summary 218
Exercises 219
Chapter 4. The Wave Partial Differential Equation 230
4.1 Introduction 230
4.2 One-Dimensional Wave Operator in Rectangular Coordinates 230
4.3 Method of Separation of Variables for the Wave Equation 232
4.4 Sturm-Liouville Problem for the Wave Equation 234
4.5 Initial Conditions for the Wave Equation in Rectangular Coordinates 237
4.6 Example Wave Equation Problems in Rectangular Coordinates 241
4.7 Wave Equation in the Cylindrical Coordinate System 257
4.8 Initial Conditions for the Wave Equation in Cylindrical Coordinates 262
4.9 Example Wave Equation Problems in Cylindrical Coordinates 264
Chapter Summary 274
Exercises 275
Chapter 5. The Laplace Partial Differential Equation 288
5.1 Introduction 288
5.2 Laplace Equation in the Rectangular Coordinate System 289
5.3 Sturm-Liouville Problem for the Laplace Equation in Rectangular Coordinates 291
5.4 Example Laplace Problems in the Rectangular Coordinate System 297
5.5 Laplace Equation in Cylindrical Coordinates 312
5.6 Sturm-Liouville Problem for the Laplace Equation in Cylindrical Coordinates 314
5.7 Example Laplace Problems in the Cylindrical Coordinate System 320
Chapter Summary 338
Exercises 340
Chapter 6. The Diffusion Equation in Two Spatial Dimensions 352
6.1 Introduction 352
6.2 Two-Dimensional Diffusion Operator in Rectangular Coordinates 352
6.3 Method of Separation of Variables for the Diffusion Equation in Two Dimensions 354
6.4 Sturm-Liouville Problem for the Diffusion Equation in Two Dimensions 355
6.5 Initial Conditions for the Diffusion Equation in Rectangular Coordinates 360
6.6 Example Diffusion Problems in Rectangular Coordinates 364
6.7 Diffusion Equation in the Cylindrical Coordinate System 378
6.8 Initial Conditions for the Diffusion Equation in Cylindrical Coordinates 384
6.9 Example Diffusion Problems in Cylindrical Coordinates 387
Chapter Summary 407
Exercises 408
Chapter 7. The Wave Equation in Two Spatial Dimensions 422
7.1 Introduction 422
7.2 Two-Dimensional Wave Operator in Rectangular Coordinates 422
7.3 Method of Separation of Variables for the Wave Equation 424
7.4 Sturm-Liouville Problem for the Wave Equation in Two Dimensions 425
7.5 Initial Conditions for the Wave Equation in Rectangular Coordinates 430
7.6 Example Wave Equation Problems in Rectangular Coordinates 433
7.7 Wave Equation in the Cylindrical Coordinate System 450
7.8 Initial Conditions for the Wave Equation in Cylindrical Coordinates 456
7.9 Example Wave Equation Problems in Cylindrical Coordinates 460
Chapter Summary 479
Exercises 480
Chapter 8. Nonhomogeneous Partial Differential Equations 490
8.1 Introduction 490
8.2 Nonhomogeneous Diffusion or Heat Equation 490
8.3 Initial Condition Considerations for the Nonhomogeneous Heat Equation 501
8.4 Example Nonhomogeneous Problems for the Diffusion Equation 503
8.5 Nonhomogeneous Wave Equation 523
8.6 Initial Condition Considerations for the Nonhomogeneous Wave Equation 533
8.7 Example Nonhomogeneous Problems for the Wave Equation 536
Chapter Summary 559
Exercises 560
Chapter 9. Infinite and Semi-infinite Spatial Domains 570
9.1 Introduction 570
9.2 Fourier Integral 570
9.3 Fourier Sine and Cosine Integrals 574
9.4 Nonhomogeneous Diffusion Equation over Infinite Domains 577
9.5 Convolution Integral Solution for the Diffusion Equation 581
9.6 Nonhomogeneous Diffusion Equation over Semi-infinite Domains 583
9.7 Example Diffusion Problems over Infinite and Semi-infinite Domains 586
9.8 Nonhomogeneous Wave Equation over Infinite Domains 599
9.9 Wave Equation over Semi-infinite Domains 601
9.10 Example Wave Equation Problems over Infinite and Semi-infinite Domains 607
9.11 Laplace Equation over Infinite and Semi-infinite Domains 619
9.12 Example Laplace Equation over Infinite and Semi-infinite Domains 625
Chapter Summary 632
Exercises 634
Chapter 10. Laplace Transform Methods for Partial Differential Equations 652
10.1 Introduction 652
10.2 Laplace Transform Operator 652
10.3 Inverse Transform and Convolution Integral 654
10.4 Laplace Transform Procedures on the Diffusion Equation 655
10.5 Example Laplace Transform Problems for the Diffusion Equation 659
10.6 Laplace Transform Procedures on the Wave Equation 679
10.7 Example Laplace Transform Problems for the Wave Equation 684
Chapter Summary 706
Exercises 707
References 722
Index 724