lec18 - Math 124A December 02, 2010 Viktor Grigoryan 18...

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Math 124A – December 02, 2010 ± Viktor Grigoryan 18 Separation of variables: Neumann conditions The same method of separation of variables that we discussed last time for boundary problems with Dirichlet conditions can be applied to problems with Neumann, and more generally, Robin boundary conditions. We illustrate this in the case of Neumann conditions for the wave and heat equations on the finite interval. Substituting the separated solution u ( x,t ) = X ( x ) T ( t ) into the wave Neumann problem ( u tt - c 2 u xx = 0 , 0 < x < l, u ( x, 0) = φ ( x ) , u t ( x, 0) = ψ ( x ) , u x (0 ,t ) = u x ( l,t ) = 0 , (1) gives the same equations for X and T as in the Dirichlet case, - X 00 = λX, and - T 00 = 2 T. However, the boundary conditions now imply X 0 (0) T ( t ) = X 0 ( l ) T ( t ) = 0 , t X 0 (0) = X 0 ( l ) = 0 . To find all the separated solutions, we need to find all the eigenvalues and eigenfunctions satisfying these boundary conditions. To do this, we need to consider the cases λ = 0, λ < 0 and λ > 0 separately. Assume λ = 0, then the equation for X is X 00 = 0, which has the solution X ( x ) = C + Dx . The derivative is then X 0 ( x ) = D , and the boundary conditions imply that D = 0. So every constant function, X ( x ) = C , is an eigenfunction for the eigenvalue λ 0 = 0. Next, we assume that λ = - γ 2 < 0, in which case the equation for X takes the form X 00 = γ 2 X. The solution to this equation is X ( x ) = Ce γx + De γx , so X 0 ( x ) = Cγe γx - Dγe - γx . Checking the boundary conditions gives ± - = 0 Cγe γl - Dγe - γl = 0 ± C = D ( e 2 γl - 1) = 0 C = D = 0 , since γ 6 = 0, and hence, also e 2 γl - 1 6 = 0. This leads to the identically zero solution
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This note was uploaded on 01/10/2011 for the course MATH 124A taught by Professor Ponce during the Fall '08 term at UCSB.

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lec18 - Math 124A December 02, 2010 Viktor Grigoryan 18...

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