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# Partial Differential Equatons : Homogeneous Heat Conduction

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Use separation of variables to find the solution, in the form of an infinite series, of the homogeneous heat conduction problem with mixed boundary conditions:
Ehi 02u
PDE: ....
BCs: ....
ICs: ....
Proceed as follows:

(a) Assume u(x, t) = (x)G(t) and derive the ODEs satisfied by q(x) and G(t).
(b) Solve the ODEs for q(x) and G(t), arid determine the allowed values for the separation constant ..
(c) Show that the eigenfunctions of the spatial eigenvalue-eigenfunction problem are mutually orthogonal.
(d) Write the solution in terms of an infinite series with coefficients B7, and derive a formula for the B in terms of an integral involving the intial condition u(x, 0) = f(x).

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##### Solution Summary

A homogeneous heat conduction problem is investigated. The solution is detailed and well presented.

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