Solving the heat equation with variable thermal conductivity

We consider the heat equation with spatially variable thermal conductivity and homogeneous Dirichlet boundary conditions. Using Method of Fokas or Unified Transform Method, we derive solution representations as limit solutions constant-coefficient interface problems where number subdomains interfaces becomes unbounded. This produces an explicit representation solution, from which can compute determine its properties. this expression, find eigenvalues corresponding variable-coefficient eigenvalue problem roots a transcendental function. write eigenfunctions explicitly in terms eigenvalues. The is first example more general second-order initial–boundary value that be solved using approach.