On the inverse problem of time fractional heat equation using Crank-Nicholson type methods and genetic algorithm

Abstract

In this paper, the time-fractional heat equation with the Caputo derivative of order $\alpha$ where $0<\alpha \le 1$ is considered. The parametric Crank-Nicholson type method for direct problems is used. But for the inverse problem, for finding the best conduction parameter $c$ and the best order of fractional derivative $\alpha$, we use genetic algorithm ($GA$) for minimizing fitting error such that the numerical solution obtained for direct problem at the final time be fitted by the final conditions.
Several examples are carried out to describe the method and to support the theoretical claims. Finally, we conclude that fractional partial equations ($FPDE$) are more flexible than partial equations ($PDE$) and have a higher ability to model physical phenomena.