# 1D heat equation with variable diffusivity

Please provide a detailed solution to the attached problem.

Consider the solution of the heat equation for the temperature in a rod of length L=1 with variable diffusivity:

u_t = A^2 d/dx (x^2 du/dx)

The derivatives are partial derivatives.

The boundary conditions are:

u(1,t)=u(2,t)=0

And

u(x,0) = f(x)

solve this problem by first showing that there exists set of appropriate eigenfunctions for this PDE given by:

b_n(x) = 1/sqrt(x) * sin (n*Pi*ln(x)/ln(2) )

Where n is an integer. develop a series solution for teh initial boundary value problem using these eigenfunctions.

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#### Solution Summary

The solution is 11 pages long. It includes full derivation of the solution to the PDE. It shows how the eigenfunctions were obtained (instead of just showing that they satisfy the equation). It then continues to derive the sereis expansion of the equation's solution.