The method of fundamental solutions for some direct and inverse problems

Reeve, Thomas Henry (2013). The method of fundamental solutions for some direct and inverse problems. University of Birmingham. Ph.D.

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Abstract

We propose and investigate applications of the method of fundamental solutions (MFS) to several parabolic time-dependent direct and inverse heat conduction problems (IHCP). In particular, the two-dimensional heat conduction problem, the backward heat conduction problem (BHCP), the two-dimensional Cauchy problem, radially symmetric and axisymmetric BHCPs, the radially symmetric IHCP, inverse one and two-phase linear Stefan problems, the inverse Cauchy-Stefan problem, and the inverse two-phase one-dimensional nonlinear Stefan problem. The MFS is a collocation method therefore it does not require mesh generation or integration over the solution boundary, making it suitable for solving inverse problems, like the BHCP, an ill-posed problem. We extend the MFS proposed in Johansson and Lesnic (2008) for the direct one-dimensional heat equation, and Johansson and Lesnic (2009) for the direct one-phase one-dimensional Stefan problem, with source points placed outside the space domain of interest and in time. Theoretical properties, including linear independence and denseness, the placement of source points, and numerical investigations are included showing that accurate results can be efficiently obtained with small computational cost. Regularization techniques, in particular, Tikhonov regularization, in conjunction with the L-curve criterion, are used to solve the illconditioned systems generated by this method. In Chapters 6 and 8, investigating the linear and nonlinear Stefan problems, the MATLAB toolbox lsqnonlin, which is designed to minimize a sum of squares, is used.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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