In this paper, we study a time-fractional initial-boundary value problem
of Kirchhoff type involving memory term for non-homogeneous materials (
P α ). As a consequence of energy argument, we derive L ∞ ( 0 , T ; H 0
1 ( Ω ) ) bound as well as L 2 ( 0 , T ; H 2 ( Ω ) ) bound on the
solution of the problem ( P α ) by defining two new discrete Laplacian
operators. Using these a priori bounds, existence and uniqueness of the
weak solution to the considered problem is established. Further, we
study semi discrete formulation of the problem ( P α ) by discretizing
the space domain using a conforming 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 Kichhoff type nonlinearity.
Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical
solution of the problem ( P α ) with a convergence rate of O ( h + k 2 −
α ) , where