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Math Help - Non-homogeneous PDE with Non-homogeneous BC's - Eigenfunction Expansion

  1. #1
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    Non-homogeneous PDE with Non-homogeneous BC's - Eigenfunction Expansion

    I have an exam tomorrow, and I came across one question that I don't really know how to start and I was really hoping someone could help me out with it.

    Use the method of eigenfunction expansion to obtain a solution to

    u_t = u_{xx} + q(x,t)

    with initial condition:

    u(x,0) = f(x)

    BC's: u(\pi,t) = u_\pi, u(0,t) = u_0

    where u_\pi,  u_0 are given constants.

    So I need to start with a trial solution based on the homogeneous part of the problem i.e.

    u_t = u_{xx} which if the boundary conditions were something like u(\pi,t) = 0, u(0,t) = 0 I would start with a solution

    \displaystyle u(x,t) = \sum_1^{\infty} c_n(t) \sin (n \pi)

    but I don't know how to determine a trial solution if the BC's are non-homogeneous.
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  2. #2
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    You'll need to transform the problem such that the new problem has these kind of BC's. One usually tries

    u = v + ax + b

    and find a and b such that v(0,t) = 0, v(\pi,t) = 0.
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