Numerical Solution of linear Second Order Partial Differential Equation

This study explores the analytical and numerical solutions of partial differential equations (PDEs), focusing on parabolic (heat). The first part presents their analytical solutions using initial and boundary conditions and delves into the finite difference method (FDM), discussing forward, backward, and central difference schemes. These methods are applied to numerically solve one- and two-dimensional heat. The Crank-Nicolson method, recognized for its unconditional stability, is employed to improve the accuracy of heat equation solutions, overcoming limitations of explicit and implicit schemes. We then analyze the performance, strengths, and weaknesses of FDM through numerical simulations of one-dimensional heat. Due to computational constraints, Crank-Nicolson for 1D simulation, was not executed. Results indicate that the implicit backward difference method demonstrates superior stability by allowing unrestricted step sizes compared to the explicit forward difference method. These findings contribute to a deeper understanding of numerical PDE solutions and stability considerations in computational mathematics.