Compact Finite Difference Scheme for One Dimensional Parabolic Inverse Problem

Inverse Problems is a research area dealing with inversion of models or data. An inverse problem is a mathematical model that is used to obtain information about a physical object or system from observed measurements. The solution to this problem is useful because it generally provides information about a physical parameter that we cannot directly observe or measure. These problems play a very important role in many areas of science and engineering. In this thesis, we formulated a compact finite difference scheme for a one dimensional parabolic inverse problem to determine the solution 𝜙�����(𝑥�����,𝑡�����) and control parameter 𝑐�����(𝑡�����). The global existence and uniqueness of the solution 𝜙�����(𝑥�����,𝑡�����) was proved using the method of retardation of the time variable along with an a priori estimation. Also, the existence of solution or solvability of the formulated compact scheme was proved by employing the homogeneous system of the tridiagonal system resulted from the formulated scheme. Some numerical results are presented to show that the accuracy of the space and time directions are improved, and computation time is shortened largely. Keywords: Parabolic Inverse Problems, Control Parameter, Existence and Uniqueness, Compact Difference Scheme.