Partial Differential EquationsSpringer, 1982 - 249 páginas This 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. |
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Página 57
... condition det ( A + ) # 0 ( 2.9 ) does not depend on the Cauchy data on S ; in the quasi - linear case , however , where the A , depend on the Du with | B | < m −1 and on x , one has to know the 1⁄2 in order to decide if S is ...
... condition det ( A + ) # 0 ( 2.9 ) does not depend on the Cauchy data on S ; in the quasi - linear case , however , where the A , depend on the Du with | B | < m −1 and on x , one has to know the 1⁄2 in order to decide if S is ...
Página 149
... condition . Equation ( 2.14a ) is hyperbolic if there exists a real number c such that P ( i§ , iλ ) # 0 for all §ER " and all complex λ with Imλ < -c . ( 2.28 ) Condition ( 2.28 ) is equivalent to the statement that all of the m roots ...
... condition . Equation ( 2.14a ) is hyperbolic if there exists a real number c such that P ( i§ , iλ ) # 0 for all §ER " and all complex λ with Imλ < -c . ( 2.28 ) Condition ( 2.28 ) is equivalent to the statement that all of the m roots ...
Página 151
... condition . Then by ( 2.36 ) sin ( cst ) Z ( $ , t ) = cl ] ( 2.37d ) Thus the standard problem for the wave equation has the solution u ( x , t ) = ( 2π ) ̄n / 2 Jeive si sin ( cst ) 8 ( ξ ) αξ c | $ | for gЄ Cn + 3 ( R " ) . ( 2.37e ) ...
... condition . Then by ( 2.36 ) sin ( cst ) Z ( $ , t ) = cl ] ( 2.37d ) Thus the standard problem for the wave equation has the solution u ( x , t ) = ( 2π ) ̄n / 2 Jeive si sin ( cst ) 8 ( ξ ) αξ c | $ | for gЄ Cn + 3 ( R " ) . ( 2.37e ) ...
Contenido
Chapter | 1 |
Examples | 2 |
Analytic Solution and Approximation Methods in a Simple Example Problems 4 Quasilinear Equations | 4 |
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analytic functions assume ball boundary bounded uniformly Cauchy data Cauchy problem Cauchy sequence Chapter characteristic curves class C² coefficients compact support complex constant continuous converge defined denote derivatives of orders difference quotients Dirichlet problem domain of dependence elliptic exists follows formula Fourier function f fundamental solution Gårding Gårding's inequality given harmonic function heat equation hence Hint Ho(N 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 real numbers satisfies scalar Show solution u(x,t solved space square integrable sufficiently small test functions theorem u₁ vanish variables vector wave equation x₁ ΘΩ