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    1D heat equation

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    Consider the following problem

    ut = 4uxx 0<x<Pi, t>0

    u(0,t)=a(t), u(Pi,t)=b(t) t>0

    u(x,0)=f(x) 0<x<Pi

    (a) Show that the solution (which exists and is unique for reasonably nice functions f,a,b) u(x,t) is of the form

    U(x,t) = v(x,t)+(1-x/Pi) a(t)+x/Pi b(t)
    where v solves a heat equation of the form vt = 4vxx + q(x, t) with homogeneous boundary conditions:
    v(0,t) = v(π,t) = 0 for t > 0. Determine q(x,t).
    . (b) Assume a(t) ≡ a0,b(t) ≡ b0 are constant. Determine the steady state solution uE. How does this 
solution depend on the initial value f(x)? 

    . (c) Show that for large t one has u(x, t) ≈ uE (t) + C e−4t sin x, for some constant C . Determine C . 

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    Solution Preview

    The equation is:
    The boundary conditions are
    And the initial condition is
    If we set
    We see that
    Plugging (1.4) into (1.1) we obtain:


    Then we define
    Is a solution to the original equation if

    Now the boundary functions are constants
    For the steady state we require and the boundary conditions still hold

    Solution Summary

    The solution shows how to convert an equation with time-dependent boundary values to a simple non-homogeneous equation, and in a special case how to get the steady state and asymptotic behavior