Comparative Study of Explicit and Implicit Schemes for the 1D Heat Equation

Abstract

This thesis presents a comprehensive and in-depth comparative study between explicit and implicit numerical computational schemes for solving the one-dimensional (1D) unsteady heat equation, which is mathematically classified as a parabolic partial differential equation (PDE). The primary objective of this research is to evaluate the mathematical performance, computational efficiency, and numerical stability of both the explicit Forward-Time Central- Space (FTCS) scheme and the implicit Backward-Time Central-Space (BTCS) scheme under Dirichlet boundary conditions. The theoretical framework of the thesis covers the formulation of the Finite Difference Method (FDM) based on Taylor series expansion, while focusing on the three fundamental pillars of numerical analysis: consistency, stability, and convergence. An integrated simulation program was developed within the MATLAB environment to evaluate five distinct case studies by varying the parametric stability parameter r (ranging from 0.40 to 0.65) and comparing the results against the exact analytical solution derived via Fourier series expansion.