Foreword ....................................................... xi
Preface to the Second Edition ................................ xiii
Preface to the First Edition ................................... xv
Chapter 1 Ordinary Differential Equations: Brief Review ........ 1
1.1 First-Order Equations ...................................... 1
1.2 Homogeneous Linear Equations with Constant Coefficients .... 3
1.3 Nonhomogeneous Linear Equations with Constant
Coefficients ............................................... 5
1.4 Cauchy-Euler Equations ..................................... 6
1.5 Functions and Operators .................................... 7
Exercises .................................................. 9
Chapter 2 Fourier Series ...................................... 11
2.1 The Full Fourier Series ................................... 11
2.2 Fourier Sine Series ....................................... 17
2.3 Fourier Cosine Series ..................................... 21
2.4 Convergence and Differentiation ........................... 23
Exercises ................................................. 24
Chapter 3 Sturm—Liouville Problems ............................ 27
3.1 Regular Sturm-Liouville Problems .......................... 27
3.2 Other Problems ............................................ 39
3.3 Bessel Functions .......................................... 41
3.4 Legendre Polynomials ...................................... 47
3.5 Spherical Harmonics ....................................... 50
Exercises ................................................. 54
Chapter 4 Some Fundamental Equations of Mathematical
Physics ............................................. 59
4.1 The Heat Equation ......................................... 59
4.2 The Laplace Equation ...................................... 67
4.3 The Wave Equation ......................................... 73
4.4 Other Equations ........................................... 78
Exercises ................................................. 81
Chapter 5 The Method of Separation of Variables ............... 83
5.1 The Heat Equation ......................................... 83
5.2 The Wave Equation ......................................... 95
5.3 The Laplace Equation ..................................... 101
5.4 Other Equations .......................................... 109
5.5 Equations with More than Two Variables ................... 113
Exercises ................................................ 124
Chapter 6 Linear Nonhomogeneous Problems ..................... 131
6.1 Equilibrium Solutions .................................... 131
6.2 Nonhomogeneous Problems .................................. 136
Exercises ................................................ 140
Chapter 7 The Method of Eigenfunction Expansion .............. 143
7.1 The Heat Equation ........................................ 143
7.2 The Wave Equation ........................................ 149
7.3 The Laplace Equation ..................................... 152
7.4 Other Equations .......................................... 155
Exercises ................................................ 159
Chapter 8 The Fourier Transformations ........................ 165
8.1 The Full Fourier Transformation .......................... 165
8.2 The Fourier Sine and Cosine Transformations .............. 172
8.3 Other Applications ....................................... 179
Exercises ................................................ 181
Chapter 9 The Laplace Transformation ......................... 187
9.1 Definition and Properties ................................ 187
9.2 Applications ............................................. 192
Exercises ................................................ 202
Chapter 10 The Method of Green's Functions .................... 205
10.1 The Heat Equation ........................................ 205
10.2 The Laplace Equation ..................................... 213
10.3 The Wave Equation ........................................ 217
Exercises ................................................ 223
Chapter 11 General Second-Order Linear Partial Differential
Equations with Two Independent Variables ........... 227
11.1 The Canonical Form ....................................... 227
11.2 Hyperbolic Equations ..................................... 231
11.3 Parabolic Equations ...................................... 235
11.4 Elliptic Equations ....................................... 238
Exercises ................................................ 239
Chapter 12 The Method of Characteristics ...................... 241
12.1 First-Order Linear Equations ............................. 241
12.2 First-Order Quasilinear Equations ........................ 248
12.3 The One-Dimensional Wave Equation ........................ 249
12.4 Other Hyperbolic Equations ............................... 256
Exercises ................................................ 260
Chapter 13 Perturbation and Asymptotic Methods ................ 263
13.1 Asymptotic Series ........................................ 263
13.2 Regular Perturbation Problems ............................ 266
13.3 Singular Perturbation Problems ........................... 274
Exercises ................................................ 280
Chapter 14 Complex Variable Methods ........................... 285
14.1 Elliptic Equations ....................................... 285
14.2 Systems of Equations ..................................... 291
Exercises ................................................ 294
Answers to Odd-Numbered Exercises ............................. 297
Appendix ...................................................... 313
Bibliography .................................................. 319
Index ......................................................... 321
| 
