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Partial Differential Equations : Separation Solutions, Initial Conditions, Boundary Conditions and Boundary Value Problems

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Partial Differential Equations, Separation Solutions, Initial Conditions, Boundary Conditions and Boundary Value Problems are investigated.

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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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