Boundary Element Method for Solving Inverse Heat Source Problems

In this thesis, the boundary element method (BEM) is applied
for solving inverse source problems for the heat equation. Through the employment of the Green’s formula and fundamental solution, the BEM naturally reduces the dimensionality of the problem by one although domain integrals are still present due to the initial condition
and the heat source. We mainly consider the identification of time-dependent source for heat equation with several types of conditions such as non-local, non-classical, periodic, fixed point, time-average and integral which are considered as boundary or overdetermination conditions. Moreover, the more challenging cases of finding the space- and time-dependent heat source functions for additive and multiplicative cases are also considered.
Under the above additional conditions a unique solution is known to exist, however, the inverse problems are still ill-posed since small errors in the input measurements result in large errors in the output heat source solution. Then some type of regularisation method is required to stabilise the solution. We utilise regularisation methods such as
the Tikhonov regularisation with order zero, one, two, or the truncated singular value decomposition (TSVD) together with various choices of the regularisation parameter.

The numerical results obtained from several benchmark test
examples are presented in order to verify the efficiency of adopted computational methodology. The retrieved numerical solutions are compared with their analytical solutions, if available, or with the corresponding direct numerical solution, otherwise. Accurate and stable numerical solutions have been obtained throughout for all the inverse heat source problems considered.