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