Partial Differential Equations In Fluid

This book is concerned with partial differential equations applied to fluids problems in science and engineering and is designed for two potential audiences. First, this book can function as a text for a course in mathematical methods in fluid mechanics in non-mathematics departments or in mathematics service courses. The authors have taught both. Second, this book is designed to help provide serious readers of journals (professionals, researchers, and graduate students) in analytical science and engineering with tools to explore and extend the missing steps in an analysis. The topics chosen for the book are those that the authors have found to be of considerable use in their own research careers. These topics are applicable in many areas, such as aeronautics and astronautics; biomechanics; chemical, civil, and mechanical engineering; fluid mechanics; and geophysical flows. Continuum ideas arise in other contexts, and the techniques included have applications there as well.

Preface	p. xi
Acknowledgments	p. xiii
Review of Analytic Function Theory	p. 1
Preamble	p. 1
Fundamentals of Complex Numbers	p. 1
Analytic Functions	p. 4
Integration and Cauchy's Theorem	p. 8
Application to Real Integrals	p. 19
References	p. 30
Exercises	p. 30
Special Functions	p. 37
Preamble	p. 37
The Gamma Function	p. 39
Functions Defined by Differential Equations	p. 43
Integral Representations	p. 51
References	p. 55
Exercises	p. 55
Eigenvalue Problems and Eigenfunction Expansions	p. 62
Preamble	p. 62
Synge's Setup for Rayleigh's Criterion	p. 62
Sturm-Liouville Problems	p. 65
Expansions in Eigenfunctions	p. 71
Worked Examples	p. 74
Nonstandard Eigenvalue Problems	p. 78
Fourier-Bessel Series	p. 85
Continuous versus Discrete Spectra	p. 91
References	p. 94
Exercises	p. 95
Green's Functions for Boundary-Value Problems	p. 106
Preamble	p. 106
Green's Function	p. 110
Connections with Distributions	p. 119
First-Order System: Green's Matrices	p. 121
Generalized Green's Functions	p. 122
Expansions in Eigenfunctions	p. 125
References	p. 132
Exercises	p. 133
Appendix: Linear Ordinary Differential Equations	p. 142
Laplace Transform Methods	p. 148
Preamble	p. 148
The Laplace Transform and Its Inverse	p. 148
Worked Examples	p. 156
Bilateral Laplace Transform	p. 171
References	p. 175
Exercises	p. 176
Fourier Transform Methods	p. 183
Preamble	p. 183
The Fourier Transform and Its Inverse	p. 183
Worked Examples	p. 188
Mellin Transforms	p. 200
Worked Examples	p. 205
References	p. 207
Exercises	p. 208
Particular Physical Problems	p. 217
Preamble	p. 217
Lee Waves	p. 217
The Far Momentum Wake	p. 220
Kelvin-Helmholtz Instability	p. 223
The Boundary-Layer Signal Problem	p. 225
Stability of Plane Couette Flow	p. 228
Generalized Transform Techniques	p. 231
References	p. 235
Exercises	p. 236
Asymptotic Expansions of Integrals	p. 244
Preamble	p. 244
Asymptotic Expansions	p. 244
Integration by Parts	p. 245
Laplace-Type Integrals; Watson's Lemma	p. 246
Generalized Laplace Integrals: Laplace's Method	p. 253
Method of Steepest Descent	p. 256
Method of Stationary Phase; Kelvin's Results	p. 266
References	p. 268
Exercises	p. 268
Index	p. 279
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Partial Differential Equations In Fluid Dynamics

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