A note on the solution to the diffusion equation

Three methods (Gauss-Legendre method, Stehfest method and Laplace transform method) are used to evaluate a solution of a coupled heat-fluid linear diffusion equation. Comparing with the results by Jaeger, the accuracy and efficiency of the Stehfest and Gauss-Legendre methods and the limitations of the truncated solutions obtained by Laplace transformation are discussed. It is concluded that the Stehfest method gives accurate results and is numerically more efficient than the other two methods, particularly for the solutions in early time. Two transformations with u = -ln(x) and u = arctan(x pi/2), where u is the original integral variable, are considered in the Gauss-Legendre method.