Solving heat equation using finite difference method

This paper presents a numerical solution of the one-dimensional heat equation using the Crank-Nicolson finite difference method. The heat equation is a second order partial differential equation that models the distribution of heat over time ina given medium. Due to the limitations of analytical methods in solving problems with complex boundary conditions, numerical approaches are essential. In this study, we focus on a test problem involving Dirichlet boundary conditions at both ends of the domain. The continuous heat equation is discretized in both space and time to form a system of linear algebraic equations. The Crank-Nicolson scheme is selected for its stability and accuracy. Numerical implementation is carried out using Wolfram Mathematica, which also enables the generation of 3D surface plots for better visualization. Error analysis is performed by comparing the numerical solution to the exact analytical solution using the L₂ norm and maximum absolute error. Results indicate that the method yields accurate and stable approximations for the heat distribution over time.