A Hybrid Laplace Transform/Finite Difference Method for Diffusion Problems

The thesis carries out a series of investigations to improve the understanding and effciency of the Laplace Transform Finite Difference Method (LTFDM). In chapter two, I begin by investigating the noise handling properties of the Fourier series and the Stehfest and Talbot algorithms for inverting the Laplace transform. Here noise is added to various test functions, and the results are compared to the exact solutions. I find that the Talbot algorithm successfully reconstructs the function while both the Stehfest and the Fourier series methods fail to invert these functions accurately. Chapter three extends the investigation by examining the performance of five of the main algorithms for inverting the Laplace transform in standard 16 digits precision and multi-precision. The results show that Talbot generally outper- forms the other algorithms in standard precision while the Stehfest is the best in multi-precision. The LTFDM is then used to solve the Fisher KPP (Kolmogorov, Petrovsky and Piskunov) equation. This equation has a travelling wave solution, and Fourier and Laplace transform numerical methods have difficulty reconstruct- ing travelling waves. Using the knowledge gained in chapters two and three and understanding the nature of the perturbations in this equation, accurate representations of several solutions to this equation were produced. The LTDFM is then successfully applied to a series of linear and non-linear diffusion equations. Comparisons are then made with the Froward Time Cen- tral Difference and the Crank Nicholson methods, with the LTDFM showing advantages over these schemes in both time and accuracy.