Finite Element Analysis of Time Fractional Integro-differential Equations of Kirchhoff type for Non-homogeneous Materials

In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials. The energy argument is applied to derive the a priori bounds on the solution of the considered problem. Consequently, we prove the existence and uniqueness of the weak solution to the problem under consideration. We keep the time variable continuous and discretize the space domain using a conforming FEM to obtain the semi discrete formulation of the problem. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra projection operator. To obtain the numerical solution to the problem efficiently, we develop a new linearized L1 Galerkin FEM. This numerical scheme is shown to have a convergence rate of $O(h+k^{2-\alpha})$, where $\alpha~ (0<\alpha<1)$ is the fractional derivative exponent, $h$ and $k$ are the discretization parameters in the space and time directions respectively. Further, this convergence rate is improved in the time direction by proposing a novel linearized L2-1$_{\sigma}$ Galerkin FEM. We prove that this numerical scheme has an accuracy rate of $O(h+k^{2})$. Finally, a numerical experiment is conducted to validate our theoretical claims.