1D heat equation with variable diffusivity
Provide a detailed solution to the following problem:
Solve this problem by first showing that there exists a set of appropriate eigenfunctions for this PDE given by βn(x)=(1/√x)sin (πnln x/ln 2)
where n is an integer. Develop a series solution for the initial boundary value problem using these eigenfunctions.
Consider the solution of the heat equation for the temperature in a rod given by φ(x, t) but with a variable diffusivity:
φt=A2(∂/∂x)[x2(∂φ/∂x)]
where A is a constant. Suppose the rod occupies the interval 1 ≤ x ≤ 2 and the boundary conditions are given by φ(1, t) = 0 φ(2, t) = 0, and the initial condition is φ(x, 0) = f(x).