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∞0,T;H01(Ω)$$ {L}^{\infty}\left(0,T;{H}_0^1\left(\Omega \right)\right) $$ bound as well as L2(0,T;H2(Ω))$$ {L}^2\left(0,T;{H}^2\left(\Omega \right)\right) $$ 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 O(h+k2−α)$$ O\left(h+{k}^{2-\alpha}\right) $$, where α(0<α<1)$$ \alpha \kern0.1em \left(0<\alpha <1\right) $$ is the fractional derivative exponent and h$$ h $$ and k$$ k $$ 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 σ$$ {}_{\sigma } $$ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims. [ABSTRACT FROM AUTHOR]