Finite element analysis of time-fractional integro-differential equation 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. As a consequence of energy argument, we derive L-infinity (0, T; H-0(1)(Omega)) bound as well as L-2(0, T; H-2(Omega)) bound on the solution of the considered problem by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem are established. Further, we study semi discrete formulation of the problem by discretizing the space domain using a conforming finite element method (FEM) and keeping the time variable continuous. The semi discrete error analysis is carried out bymodifying the standard Ritz-Volterra projection operator in such a way that it reduces the complexities arising from the Kirchhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem under consideration. This method has a convergence rate of O(h + k(2-alpha)), where alpha (0 < alpha < 1) is the fractional derivative exponent and h and k are the discretization parameters in the space and time directions, respectively. This convergence rate is further improved to second order in the time direction by proposing a novel linearized L2-1(sigma). Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.