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Unformatted text preview: Math 124A November 30, 2010 Viktor Grigoryan 17 Separation of variables: Dirichlet conditions Earlier in the course we solved the Dirichlet problem for the wave equation on the finite interval 0 < x < l using the reflection method. This required separating the domain ( x,t ) (0 ,l ) (0 , ) into different regions according to the number of reflections that the backward characteristic originating in the regions undergo before reaching the x axis. In each of these regions the solution was given by a different ex- pression, which is impractical in applications, and the method does not generalize to higher dimensions or other equations. We now study a different method of solving the boundary value problems on the finite interval, called separation of variables . Let us start by considering the wave equation on the finite interval with homogeneous Dirichlet con- ditions. ( u tt- c 2 u xx = 0 , < x < l, u ( x, 0) = ( x ) , u t ( x, 0) = ( x ) , u (0 ,t ) = u ( l,t ) = 0 . (1) The idea of the separation of variables method is to find the solution of the boundary value problem as a linear combination of simpler solutions (compare this to finding the simpler solution S ( x,t ) of the heat equation, and then expressing any other solution in terms of the heat kernel). The building blocks in this case will be the separated solutions , which are the solutions that can be written as a product of two functions, one of which depends only on x , and the other only on t , i.e. u ( x,t ) = X ( x ) T ( t ) . (2) Let us try to find all the separated solutions of the wave equation. Substituting (2) into the equation gives X ( x ) T 00 ( t ) = c 2 X 00 ( x ) T ( t ) . Dividing both sides of these identity by- c 2 X ( x ) T ( t ), we get- X 00 ( x ) X ( x ) =- T 00 ( t ) c 2 T ( t ) = . (3) Clearly is a constant, since it is independent of x from =- T 00 / ( c 2 T ), and is independent of t from =- X 00 /X . We will shortly see that the boundary conditions force....
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