Numerical Study of Inverse Problem with Computing the Functions and Parameters of Fractional Heat Equation Using the Transform of α-Fractional Derivative

Abstract

The inverse problem (IP) of the differential equations focusses for determining an unknown parameter(s) or a function. However, the IP of the fractional partial differential equations (FPDEs) problem plays a big role in engineering and applied science. Accordingly, the fractional differential equations (FDEs) have the significant rule in the mathematical modeling of science and engineering. As well as, finding the solutions of the FPDEs is a significant subject and a wide field. The objectives of this article is to study the method of solutions of the IP for determining unknown parameters or functions of the problem of FPDEs by using the definition of α-fractional derivative transform which is converted a FPDEs to a partial differential equation (PDE) and then, we can use the method of lines (MOL) with finite difference method for solving a quasi-linear PDE by converted it to an ordinary-differential equations (ODEs) system. The characteristic of α-fractional derivative transform is very appropriate, significant, and powerful for solving the problems of FPDEs. Additionally, the useful properties of the definition of α-fractional derivative transform are used in converting the quasi-linear FPDEs to a quasi-linear PDE. Hence, it is converted to a system of ODEs by using the MOL and finite difference formulas. Some implementations of inverse problems of the FPDEs are solved using the proposed method and then, they have compared with the numerical solutions. The test implementations showed that the two approximated solutions using the proposed method are identical. Hence, the algorithm of the proposed method proved to be efficient and accurate.