# Solving linear and nonlinear transient diffusion problems with the Laplace Transform Dual Reciprocity Method

1996

## Degree Name

Doctor of Philosophy

## Department

Department of Mathematics

## Abstract

In this thesis, a new numerical method, with the Laplace Transform and the Dual Reciprocity Method (DRM) combined into the so called Laplace Transform Dual Reciprocity Method (LTDRM), is proposed and applied to solve linear and nonlinear transient diffusion problems. The method comprises of three crucial steps. Firstly, the Laplace transform is applied to the partial differential equation and boundary conditions in a given differential system. Secondly, the dual reciprocity method is employed to solve the transformed differential system. Thirdly, a numerical inversion is utilised to retrieve the solution in the time domain.

The LTDRM is first applied for the solution of the linear transient diffusion equation. A time-free and boundary-only integral formulation is produced due to the first and second steps of the method. In this work, only the fundamental solution of the Laplace's equation is utilised in the dual reciprocity method. That is, the Laplacian operator is treated as the main operator and the nonhomogeneous terms, such as those obtained from the Laplace transform of the temporal derivative, sources or sinks, or other terms, are left to a domain integral. The DRM technique then requires all these terms be approximated by a finite sum of interpolation functions that will allow the domain integral to be taken onto the boundary.

Several problems are then analysed to demonstrate the efficiency and accuracy of the L T D R M . A numerical inversion due to the Stehfest's algorithm is examined and found to be satisfactory in terms of the numerical accuracy, efficiency and ease of implementation.

Next, the LTDRM is extended to the solution of the diffusion problems with nonlinear source terms. A linearisation of the nonlinear governing equation is required before the LTDRM can be applied. Two linearisation techniques are adopted. The convergence of solution of the linearised differential system to the true solution of the original nonlinear system is studied and found to be quite satisfactory. Then, the LTDRM is applied to solve some practical nonhnear problems of microwave heating process and spontaneous ignition

Finally, the diffusion problems with nonlinear material properties and nonlinear boundary conditions are solved by the LTDRM. Three integral formulations are presented; one of them is based on the use of the Kirchhoff transform to simplify the governing differential system before the LTDRM is applied while other two are based on the direct approach with the LTDRM being applied directly to the governing differential system. Due to the presence of spatial derivatives in two of these formulations, another set of interpolation functions, which is from that used to cast the domain integral into the boundary integrals, is employed to approximate these derivative terms. These formulations are applied to solve a variety of problems, and their advantages and disadvantages are discussed.

It may be noteworthy that for all the cases, a time-free and boundary-only integral formulation is produced. As a result of both step-by-step calculation in the time domain and computation of domain integrals being eliminated, the dimension of the problem is virtually reduced by two. The results of numerical examples presented throughout these research projects demonstrate the efficiency and accuracy of the LTDRM. For linear problems, the LTDRM is shown to be very efficient when a solution at large time is required. In addition, solutions at small time and large time can be obtained with the same level of accuracy. Similar conclusions can be drawn for nonlinear cases. As stated before, the LTDRM is shown to possess good convergence properties for nonlinear problems presented herein.

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Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.