VARIANTS OF METHOD OF FUNDAMENTAL SOLUTIONS FOR INVERSE HEAT CONDUCTION PROBLEM

Abstract

In this study, we propose and investigate an application of the variants of the method of fundamental solutions (MFS) to the parabolic inverse heat conduction problems (IHCP). The focus lies on three schemes, in particular, application of heat polynomial based variant of MFS introduced by Malihe Rostamian and Alimardan Shahrezaee (2017); the application of energy MFS, introduced by Chein-Shan Liu and Fajie Wang (2018); and a combination study of the heat polynomial based MFS and energy MFS newly developed in this thesis. The MFS being a collocation based meshless scheme offers a great deal of flexibility in the building of novel meshfree schemes. For starters, we extend the idea of Malihe Rostamian and Alimardan Shahrezaee for two-dimensional IHCP in an anisotropic medium and for the steady state inverse heat conduction in an anisotropic medium. Next, we extend the algorithm of Chein-Shan Liu and Fajie Wang for a two dimensional inverse Cauchy problem of heat conduction. Lastly, we culminate these two schemes and introduce a novel meshfree scheme, which is tested for its efficiency and accuracy for a two dimensional IHCP set in an isotropic medium. The study also includes theoretical properties and results, along with discussions around the optimal placement of source points and numerical investigations on the efficiency and accuracy of these schemes. Owing to the illposed nature of an inverse problem, truncated singular value decomposition (TSVD) regularization in conjunction with the L-Curve criteria, for parameter determination, is used to solve the obtained ill conditioned system of equations. A list of references related to the study is also incorporated at the end of this thesis.