Partial Differential EquationsThis book is a very well-accepted introduction to the subject. In it, the author identifies the significant aspects of the theory and explores them with a limited amount of machinery from mathematical analysis. Now, in this fourth edition, the book has again been updated with an additional chapter on Lewy’s example of a linear equation without solutions. |
Contents
Chapter | 1 |
Quasilinear Equations | 9 |
Examples | 15 |
The Cauchy Problem | 24 |
Chapter 2 | 33 |
The OneDimensional Wave Equation | 40 |
Systems of FirstOrder Equations | 46 |
A Quasilinear System and Simple Waves | 52 |
Proof of Existence of Solutions for the Dirichlet Problem Using | 111 |
Solution of the Dirichlet Problem by HilbertSpace Methods | 117 |
Chapter 5 | 126 |
HigherOrder Hyperbolic Equations with Constant Coefficients | 143 |
Symmetric Hyperbolic Systems | 163 |
Chapter 6 | 185 |
More on the Hilbert Space Ho and the Assumption of Boundary Values | 198 |
Chapter 7 | 206 |
Chapter 8 | 53 |
The LagrangeGreen Identity | 79 |
Distribution Solutions | 89 |
The Maximum Principle | 103 |
The InitialValue Problem for General SecondOrder Linear | 227 |
H Lewys Example of a Linear Equation | 235 |
Bibliography | 241 |
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Common terms and phrases
Apply assume ball Cauchy data Cauchy problem Cauchy sequence Cauchy-Kowalevski Chapter characteristic curves class C¹ coefficients compact subset compact support complex constant continuous converge defined denote derivatives of orders determined uniquely Dirichlet problem domain of dependence elliptic exists follows formula Fourier fundamental solution Gårding given harmonic function heat equation hence Hilbert space Hint holds identity implies inequality initial data initial values initial-value problem integral surface Laplace equation Lemma linear matrix maximum principle neighborhood non-characteristic norm obtained open set partial differential equation plane polynomial power series prescribed proof real analytic functions real numbers replaced S₁ satisfies scalar Show solved square integrable sufficiently small test functions theorem u₁ vanish variables vector wave equation x₁ ίξ ξΕΩ Ω ΘΩ Ω Ω