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 88
... wave equation ) . Take the wave equation for u = u ( x1 , x2 , X3 ) . Let Z be the disc ( 5.7a ) Lu = Ux3x3 - Ux1x1 - Ux2x2 0 = x2 + x3 < 1 ; x1 = 0 ( 5.7b ) = in the " time - like ' plane x1 = 0. Show that every cone with base Z and ...
... wave equation ) . Take the wave equation for u = u ( x1 , x2 , X3 ) . Let Z be the disc ( 5.7a ) Lu = Ux3x3 - Ux1x1 - Ux2x2 0 = x2 + x3 < 1 ; x1 = 0 ( 5.7b ) = in the " time - like ' plane x1 = 0. Show that every cone with base Z and ...
Página 129
... wave equation ( 1.1 ) . * Formula ( 1.14 ) displays the relevant features of the solution u of the initial - value problem for the wave equation in the case n = 3 . First of all , writing our spherical means in the form ( 1.4 ) we can ...
... wave equation ( 1.1 ) . * Formula ( 1.14 ) displays the relevant features of the solution u of the initial - value problem for the wave equation in the case n = 3 . First of all , writing our spherical means in the form ( 1.4 ) we can ...
Página 135
... wave equation satisfied by Use formulas ( 1.24a , b ) . ] u ( x1 , x2 , t ) = cos ( x2 ) w ( x , t ) . 3. Solve the initial - value problem ( 1.1 ) , ( 1.2 ) for n = 4 by descent from the solution ... Wave Equation in n - Dimensional Space.
... wave equation satisfied by Use formulas ( 1.24a , b ) . ] u ( x1 , x2 , t ) = cos ( x2 ) w ( x , t ) . 3. Solve the initial - value problem ( 1.1 ) , ( 1.2 ) for n = 4 by descent from the solution ... Wave Equation in n - Dimensional Space.
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