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Title:  The radial integration boundary integral and integrodifferential equation methods for numerical solution of problems with variable coefficients 
Authors:  AlJawary, Majeed 
Advisors:  Wrobel, LC 
Keywords:  Boundary element method Boundarydomain integral equation Heat conduction with variable coefficient Helmholtz equation Diffusion equation 
Publication Date:  2012 
Publisher:  Brunel University School of Engineering and Design PhD Theses 
Abstract:  The boundary element method (BEM) has become a powerful method for the numerical
solution of boundaryvalue problems (BVPs), due to its ability (at least for problems with
constant coefficients) of reducing a BVP for a linear partial differential equation (PDE)
defined in a domain to an integral equation defined on the boundary, leading to a simplified discretisation process with boundary elements only. On the other hand, the coefficients in the mathematical model of a physical problem typically correspond to the material parameters of the problem. In many physical problems, the governing equation is likely to involve variable coefficients. The application of the BEM to these equations is hampered by the difficulty of finding a fundamental solution.
The first part of this thesis will focus on the derivation of the boundary integral equation (BIE) for the Laplace equation, and numerical results are presented for some examples using constant elements. Then, the formulations of the boundarydomain integral or integrodifferential equation (BDIE or BDIDE) for heat conduction problems with variable coefficients are presented using a parametrix (Levi function), which is usually available. The second part of this thesis deals with the extension of the BDIE and BDIDE formulations to the treatment of the twodimensional Helmholtz equation with variable coefficients. Four possible cases are investigated, first of all when both material parameters and wave number are constant, in which case the zeroorder Bessel function of the second kind is used as fundamental solution. Moreover, when the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz
equation to a BDIE or a BDIDE. Finally, when material parameters are constant (with
variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. In the third part, the radial integration method (RIM) is introduced and discussed in detail. Modifications are introduced to the RIM, particularly the fact that the radial integral is calculated by using a pure boundaryonly integral which relaxes the “starshaped” requirement of the RIM. Then, the RIM is used to convert the domain integrals appearing in both BDIE and BDIDE for heat conduction and Helmholtz equations to equivalent boundary integrals. For domain integrals consisting of known functions the transformation is straightforward, while for domain integrals that include unknown variables the transformation is accomplished with the use of augmented radial basis functions (RBFs). The most attractive feature of the method is that the transformations are very simple and
have similar forms for both 2D and 3D problems. Finally, the application of the RIM is discussed for the diffusion equation, in which the parabolic PDE is initially reformulated as a BDIE or a BDIDE and the RIM is used to convert the resulting domain integrals to equivalent boundary integrals. Three cases have been investigated, for homogenous, nonhomogeneous and variable coefficient diffusion problems. 
Description:  This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University. 
URI:  http://bura.brunel.ac.uk/handle/2438/6449 
Appears in Collections:  School of Engineering and Design Theses Mechanical and Aerospace Engineering

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