Solving Elliptic Boundary Value Problems using Finite Difference Method in Steady-State Heat Conduction

Abstract

This work investigates the application of the finite difference method (FDM) to solve the boundary value problem of an elliptic equation in one-dimensional steady-state conduction. The objectives of this work are two-fold: first, to formulate the boundary value problems into a numerical model. Secondly, to validate the accuracy of the proposed model by solving it with the aid of MATLAB programming. The proposed work addresses the challenge of predicting the temperature distributions in problems where the analytical solution is not readily possible. The approach to solving the steady-state conduction equation involves first converting the steady-state conduction problems into one-dimensional Poisson’s equation, and secondly, dividing the solution domain into grid points with the central difference method, and finally, rewriting the resulting equation into a system of linear algebraic equations. The solutions of these problems can be obtained manually with coarse grids. Additionally, the solution can be obtained with fine grids manually in MATLAB. The work has clearly demonstrated that the FDM yields exact solutions. Additionally, errors can be as low as machine precision with fine grids. The later result theoretically proves that this method is second-order accurate because of error norms. Finally, the paper has confirmed the validity of the finite difference method for the solution of one-dimensional steady-state heat conduction equations. It has made clear that it is a simple, consistent, and efficient method for the numerical solution of the equations, thereby providing a firm basis for the use of thermal analysis.