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    Non homogeneous 1D heat equation

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    ut = 3uxx + 2, 0 < x < 4, t > 0,
    u(0,t) = 0, u(4,t) = 0, t = 0

    u(x,0) = 5sin2πx,0 < x < 4.

    (a) Find the steady state solution uE(x)

    (b) Find an expression for the solution.

    (c) Verify, from the expression of the solution, that limt→∞ u(x, t) = uE (x)
    for all x, 0 < x < 4.

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    Solution Preview

    The eqaution is:
    With boundary condition
    And initial conditions

    The steady state is, by definition:
    The function is now only x-dependent so the partial derivatives become full derivatives and the ordinary differential equation is
    And the boundary conditions must still hold:
    Integrating (1.5) twice we obtain:


    Applying boundary conditions we get the steady state solution.

    We would like to turn the system into a homogenous system.
    So we write:
    When we apply it to the original equation we get:
    If we set
    the equation becomes homogenous.
    For the boundary conditions we ...

    Solution Summary

    The solution contains 10 pages of step by step explanation how to solve the heat equation u_t = u_xx + 2.