Finite-Difference Approximations to the Heat Equation

Abstract

Partial differential equations (PDEs) are useful tools for mathematical modelling in the field of physics, engineering and Applied Mathematics. Useful as these equations are, only a few of them can be solved analytically. Numerical methods have been proven to perform exceedingly well in solving difficult partial differential equations. A popularly known numerical method known as finite difference method has been applied expansively for solving partial differential equations successfully. In this study, explicit finite difference scheme is established and applied to a simple problem of one-dimensional heat equation by means of C. These sample calculations show that the accuracy of the predictions depends on mesh spacing and time step. The result of the study reveals that the solutions of the heat equation decay from an initial state to a non-varying fixed state circumstance, the temporary performance of these solutions are smooth and bounded, the solution does not improve local or global utmost that are outside the range of the initial data.