Accurate analytical/numerical solution of the heat conduction equation

Abstract

Purpose – This paper addresses a novel method for solving parabolic partial differential equations in general, wherein the heat conduction equation constitutes an important particular case. The new method, appropriately named the Improved Transversal Method Of Lines (ITMOL), is inspired in the Transversal Method Of Lines (TMOL), with strong insight from the method of separation of variables. Methodology – The essence of ITMOL revolves around an exponential variation of the dependent variable in the parabolic partial differential equation for the evaluation of the time derivative. As will be demostrated later, this key step is responsible for improving the accuracy of ITMOL over its predecesor TMOL. Throughout the paper, the theoretical properties of ITMOL, such as consistency, stability, convergence and accuracy are analyzed in depth. In addition, ITMOL has proven to be unconditionally stable in the Fourier sense. Findings – In a case study, the 1-D heat conduction equation for a large plate with symmetric Dirichlet boundary conditions is transformed into a nonlinear ordinary differential equation by means of ITMOL. The numerical solution of the resulting differential equation is straightforward and brings forth a nearly zero truncation error over the entire time domain, which is practically nonexistent. Originality – Accurate levels of the analytical/numerical solution of the 1-D heat conduction equation by ITMOL are easily established in the entire time domain.