Abstract
One of the types of the inverse problems is inverse Cauchy problem for modified Helmholtz equation. This type of problem arises in one of the real life application, which is the heat conduction in the fin. The goal of this thesis is the determination of temperature on the under-specified boundary benefiting from the accessible part of the boundary with Cauchy data which are the temperatures on the accessible boundary and the heat flux in this part.
This problem is solved using some numerical methods that is the meshless method by expressing the solution as a polynomial expansion and verifying our problem for this polynomial expansion which produce a linear system that is solved by two different numerical algorithms [Preconditioned Conjugate Gradient (PCG),Conjugate Gradient Least Square (CGLS)] are compared with the exact solution.
To investigate the accuracy of this proposed method several examples are studied with some polynomial and non-polynomial problems on regular domains by benefiting from the efficiency of the meshless method.
The inverse Cauchy problem is known to be an ill-posed problem and in addition our problem is highly ill-conditioned so the stability is confirmed by applying a noise for the Cauchy data. To reduce the effect of this highly ill-condition property, Tikhonov regularization and precondition are applied.
