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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 Summary

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

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The eqaution is:
(1.1)
With boundary condition
(1.2)
And initial conditions
(1.3)

The steady state is, by definition:
(1.4)
The function is now only x-dependent so the partial derivatives become full derivatives and the ordinary differential equation is
(1.5)
And the boundary conditions must still hold:
(1.6)
Integrating (1.5) twice we obtain:

(1.7)

Applying boundary conditions we get the steady state solution.

(1.8)
We would like to turn the system into a homogenous system.
So we write:
(1.9)
When we apply it to the original equation we get:
(1.10)
If we set
(1.11)
the equation becomes homogenous.
For the boundary conditions we ...

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