Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level.The proposed book has a very comprehensive coverage on partial differential equations in a variety of coordinate systems and geometry, and their solutions using the method of separation of variables. The treatment includes complete details on going from the basic theory (including separability conditions not presented in introductory texts) to full implementation for applications. A very good choice of examples is inspired by the authors? research on semiconductor nanostructures and metamaterials and include modern applications like quantum dots.The fluency of the text and the high quality of graphics make the topic easy accessible. The organization of the content by coordinate systems rather than by equation types is unique and offers an easy access.The authors consider recent research results which have led to a much increased pedagogical understanding of not just this topic but of many other related topics in mathematical physics, and which like the explicit discussion on differential geometry shows - yet have not been treated in the older texts. To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Homework problems are included as well as numerical algorithms for computing special functions. Thus this book can serve as a reference text for advanced undergraduate students, as a textbook for graduate level courses, and as a self-study book and reference manual for physicists, theoretically oriented engineers and traditional mathematicians.MA4300, PH2300 suitable for graduate level course; could serve as one of two main texts of a partial differential equations course
Author(s): Morten Willatzen, Lok C. Lew Yan Voon
Edition: 1
Publisher: Wiley-VCH
Year: 2011
Language: English
Pages: 399
Tags: Математика;Математическая физика;
Contents......Page 6
Preface......Page 22
Part One Preliminaries......Page 23
1 Introduction......Page 25
2.2 Canonical Partial Differential Equations......Page 29
2.3 Differential Operators in Curvilinear Coordinates......Page 30
2.4 Separation of Variables......Page 32
2.5 Series Solutions......Page 42
2.6 Boundary-Value Problems......Page 48
2.7 Physical Applications......Page 52
2.8 Problems......Page 58
Part Two Two-Dimensional Coordinate Systems......Page 61
3.2 Coordinate System......Page 63
3.3 Differential Operators......Page 64
3.4 Separable Equations......Page 65
3.5 Applications......Page 68
3.6 Problems......Page 71
4.2 Coordinate System......Page 73
4.3 Differential Operators......Page 74
4.4 Separable Equations......Page 75
4.5 Applications......Page 78
4.6 Problems......Page 81
5.2 Coordinate System......Page 83
5.3 Differential Operators......Page 85
5.4 Separable Equations......Page 86
5.5 Applications......Page 88
5.6 Problems......Page 90
6.2 Coordinate System......Page 93
6.3 Differential Operators......Page 94
6.4 Separable Equations......Page 95
6.5 Applications......Page 97
6.6 Problems......Page 98
Part Three Three-Dimensional Coordinate Systems......Page 101
7.2 Coordinate System......Page 103
7.3 Differential Operators......Page 104
7.4 Separable Equations......Page 105
7.5 Applications......Page 109
7.6 Problems......Page 111
8.2 Coordinate System......Page 113
8.3 Differential Operators......Page 114
8.4 Separable Equations......Page 116
8.5 Applications......Page 118
8.6 Problems......Page 119
9.2 Coordinate System......Page 121
9.3 Differential Operators......Page 123
9.4 Separable Equations......Page 124
9.5 Applications......Page 127
9.6 Problems......Page 129
10.2 Coordinate System......Page 131
10.3 Differential Operators......Page 134
10.4 Separable Equations......Page 135
10.5 Applications......Page 137
10.6 Problems......Page 146
11.2 Coordinate System......Page 147
11.3 Differential Operators......Page 148
11.4 Separable Equations......Page 149
11.5 Applications......Page 152
11.6 Problems......Page 159
12.2 Coordinate System......Page 161
12.3 Differential Operators......Page 163
12.4 Separable Equations......Page 164
12.5 Applications......Page 166
12.6 Problems......Page 176
13.2 Coordinate System......Page 177
13.3 Differential Operators......Page 179
13.4 Separable Equations......Page 181
13.5 Applications......Page 183
13.6 Problems......Page 185
14.2 Coordinate System......Page 187
14.3 Differential Operators......Page 189
14.4 Separable Equations......Page 190
14.5 Applications......Page 193
14.6 Problems......Page 201
15.2 Coordinate System......Page 203
15.3 Differential Operators......Page 205
15.4 Separable Equations......Page 206
15.5 Applications......Page 209
15.6 Problems......Page 211
16.1 Introduction......Page 213
16.2 Coordinate System......Page 214
16.3 Differential Operators......Page 217
16.4 Separable Equations......Page 219
16.5 Applications......Page 222
16.6 Problems......Page 237
17.2 Coordinate System......Page 239
17.3 Differential Operators......Page 241
17.4 Separable Equations......Page 243
17.5 Applications......Page 249
17.6 Problems......Page 251
Part Four Advanced Formulations......Page 253
18.2 Review of Differential Geometry......Page 255
18.3 Problems......Page 261
19.2 Laplacian in a Tubular Neighborhood of a Curve – Arc-Length Parameterization......Page 263
19.3 Application to the Schrödinger Equation......Page 270
19.4 Schrödinger Equation in a Tubular Neighborhood of a Curve – General Parameterization......Page 272
19.5 Applications......Page 273
19.6 Perturbation Theory Applied to the Curved-Structure Problem......Page 281
19.7 Problems......Page 291
20.2 Laplacian in Curved Coordinates......Page 293
20.4 Applications......Page 296
20.5 Problems......Page 303
21.1 Nondegenerate States......Page 305
21.2 Degenerate States......Page 307
21.3 Applications......Page 308
21.4 Problems......Page 315
Appendix A Hypergeometric Functions......Page 317
Appendix B Baer Functions......Page 327
Appendix C Bessel Functions......Page 331
Appendix D Lamé Functions......Page 343
Appendix E Legendre Functions......Page 351
Appendix F Mathieu Functions......Page 361
Appendix G Spheroidal Wave Functions......Page 373
Appendix H Weber Functions......Page 379
Appendix I Elliptic Integrals and Functions......Page 383
References......Page 391
Index......Page 397
