Advanced Transform Techniques for the One-Dimensional Non-Homogeneous Heat Equation with Non-Homogeneous BCs and IC

This study addresses the one-dimensional non-homogeneous heat equation with non-homogeneous boundary conditions using a transformation method. We introduce a new dependent variable V(x,t) and a function ψ(x) to simplify the PDE into a homogeneous form, solving it analytically. The solution involves separating variables and applying Fourier series, leading to: Numerical simulations confirm the theoretical results, illustrating the method’s robustness for modeling heat conduction problems.