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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