Finite element analysis of time‐fractional integro‐differential equation of Kirchhoff type for non‐homogeneous materials

Abstract

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 bound as well as 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 by modifying 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 , where is the fractional derivative exponent and and 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 Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.

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