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 151
... roots are distinct . One finds that then m eiλkt Z ( § , t ) = Σ k = 1 P1 ( i§ , iλ ) * ( 2.36 ) As an example consider the n - dimensional wave equation corresponding to the operator P ( D , 7 ) = □ ] = 72 — c2 Σ D2 . ( 2.37a ) Here k ...
... roots are distinct . One finds that then m eiλkt Z ( § , t ) = Σ k = 1 P1 ( i§ , iλ ) * ( 2.36 ) As an example consider the n - dimensional wave equation corresponding to the operator P ( D , 7 ) = □ ] = 72 — c2 Σ D2 . ( 2.37a ) Here k ...
Página 152
... roots of a polynomial with highest coefficient 1 depend continu- ously on the coefficients , we see that for p → ∞o the roots μ of ( 2.42 ) will tend * to the roots of η Pm ( n , μ ) = 0 . ( 2.43 ) Let there exist for a certain ʼn a root ...
... roots of a polynomial with highest coefficient 1 depend continu- ously on the coefficients , we see that for p → ∞o the roots μ of ( 2.42 ) will tend * to the roots of η Pm ( n , μ ) = 0 . ( 2.43 ) Let there exist for a certain ʼn a root ...
Página 154
... roots μ of ( 2.53 ) are real and distinct for / p0 , it follows for the roots μ of ( 2.42 ) that = 0 ( 1 + 151 ) Imμ = C ρ as long as / p is bounded away from zero , and hence as long as ¿ \ / p < . Thus the roots λ of ( 2.49 ) satisfy ...
... roots μ of ( 2.53 ) are real and distinct for / p0 , it follows for the roots μ of ( 2.42 ) that = 0 ( 1 + 151 ) Imμ = C ρ as long as / p is bounded away from zero , and hence as long as ¿ \ / p < . Thus the roots λ of ( 2.49 ) satisfy ...
Contenido
Chapter | 1 |
Quasilinear Equations | 9 |
The Cauchy Problem | 24 |
Derechos de autor | |
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assume ball boundary bounded uniformly Cauchy data Cauchy problem Cauchy sequence Cauchy-Kowalevski Chapter characteristic curves class C² coefficients compact support complex constant continuous converge defined denote derivatives of orders determined uniquely difference quotients Dirichlet problem domain of dependence elliptic exists follows formula Fourier function f(x fundamental solution Gårding given Green's identity heat equation hence Hint holds identity implies inequality initial data initial values initial-value problem integral surface Laplace equation Lemma linear matrix neighborhood non-characteristic norm obtained open set partial differential equation plane polynomial power series prescribed proof quasi-linear real numbers satisfies scalar second derivatives Show solved space square integrable sufficiently small test functions theorem u₁ vanish variables vector wave equation x₁ ΘΩ