Principles of Partial Differential Equations

Alexander Komech, Andrew Komech1441910956, 9781441910967, 9781441910950

This book is intended to give the reader an opportunity to master solving PDE problems. Our main goal was to have a concise text that would cover the classical tools of PDE theory that are used in today’s science and engineering, such as characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green’s functions. While introductory Fourier method – based PDE books do not give an adequate description of these areas, the more advanced PDE books are quite theoretical and require a high level of mathematical background from a reader. This book was written specifically to fill this gap, satisfying the demand of the wide range of end users who need the knowledge of how to solve the PDE problems and at the same time are not going to specialize in this area of mathematics. Arguably, this is the shortest PDE course, which stretches far beyond common, Fourier method – based PDE texts. For example, [Hab03], which is a common thorough textbook on partial differential equations, teaches a similar set of tools while being about five times longer.

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Table of contents :
Principles of Partial Differential Equations……Page 2
Principles of Partial Differential Equations……Page 3
Preface……Page 5
Acknowledgements……Page 6
Contents……Page 7
1 Derivation of the d’Alembert equation……Page 9
2 The d’Alembert method for infinite string……Page 15
3 Analysis of the d’Alembert formula……Page 20
4 Second-order hyperbolic equations in the plane……Page 27
5 Semi-infinite string……Page 38
6 Finite string……Page 52
7 Wave equation with many independent variables……Page 54
8 General hyperbolic equations……Page 64
9 Derivation of the heat equation……Page 72
10 Mixed problem for the heat equation……Page 74
11 The Sturm – Liouville problem……Page 75
12 Eigenfunction expansions……Page 81
13 The Fourier method for the heat equation……Page 85
14 Mixed problem for the d’Alembert equation……Page 90
15 The Fourier method for nonhomogeneous equations……Page 93
16 The Fourier method for nonhomogeneous boundary conditions……Page 100
17 The Fourier method for the Laplace equation……Page 102
18 Motivation……Page 111
19 Distributions……Page 115
20 Operations on distributions……Page 116
21 Differentiation of jumps and the product rule……Page 121
22 Fundamental solutions of ordinary differential equations……Page 124
23 Green’s function on an interval……Page 127
24 Solvability condition for the boundary value problems……Page 131
25 The Sobolev functional spaces……Page 134
26 Well-posedness of the wave equation in the Sobolev spaces……Page 136
27 Solutions to the wave equation in the sense of distributions……Page 137
28 Fundamental solutions of the Laplace operator in Rn……Page 138
29 Potentials and their properties……Page 142
30 Computing potentials via the Gauss theorem……Page 148
31 Method of reflections……Page 149
32 Green’s functions in 2D via conformal mappings……Page 154
Appendix A Classification of the second-order equations……Page 159
References……Page 163
Index……Page 164