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Mathematics
Differential Equations
Boundary conditions in the time evolution of Spectral Method in PDE
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[QUOTE="Leonardo Machado, post: 6353055, member: 599520"] [B]TL;DR Summary:[/B] I need some help to implement a boundary condition to a 1D heat equation when dealing with it using spectral expansion. Hi everyone! I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example, $$ u_t=k u_{xx}, $$ $$ u(t,-1)=\alpha, $$ $$ u(t,1)=\beta, $$ $$ u(0,x)=f(x), $$ $$ u(t,x)=\sum^{N-1}_{n=0} a_n(t) T_{n}(x). $$ T(x) is for Chebyshev polynomials. I can easily create through MMT or FFT the spectral expansion for u(0,x), including the boundaries, to initiate the time-stepping via the first equation as, $$ \sum^{N-1}_{n=0} \frac{da_n(t)}{dt} T_n(x)=\sum^{N-1}_{n=0} a^{(2)}_n(t) T_n(x). $$ which implies in $$ a_n(t+ \delta t)=k a_n(t)+a^{(2)}_n \delta t. $$ But the point is.. how do i use the boundary conditions in the time-step? I can't find it in the bibliography. I think that the usual thing to do would be use the boundary conditions as $$ \sum^{N-1}_{n=0} a_n(t+\delta t) T_n(-1)=\sum^{N-1}_{n=0} a_n(t+\delta t)=\alpha, $$ $$ \sum^{N-1}_{n=0} a_n(t+\delta t) T_n(1)=\sum^{N-1}_{n=0} a_n(t+\delta t)=\beta, $$ but I'm not sure how to implement it in the time-stepping... Any suggestions? I would love to know a book that deals with it! PS: I had problems using the caption in the LaTeX, the idex runs for n instead of N and t_N should be T_n! I don't know why it became like this. [/QUOTE]
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Boundary conditions in the time evolution of Spectral Method in PDE
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