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    Steady State Solution

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    Suppose that we have the heat equation with the boundary-initial data

    where T_0 and T_1 are positive constants. Find a steady state solution of this equation. Use this knowledge to rewrite the solution u(x,t) of the initial-boundary value problem in the form u = v + w where w(x,t) has homogenous boundary conditions. Write down the initial-boundary value data problems that w and v satisfy.

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    Solution Preview

    To find the steady state solution without solving, we must put all the terms including the derivative of t in them equal to zero (i.e. consider u only a function of x). Therefore, we have:

    u_xx=0 ---> u= Ax+ B.

    Now we use the conditions to find A and B:
    u(0)=T0 and 0= T1- u(a) (or u(a)= T1)

    From these two conditions we get:
    B= T0 and Aa+ ...

    Solution Summary

    A steady state solution is found in the following problem with explanation along with steps for solving each part of the equation.