A Linearized L1-Galerkin FEM for Non-smooth Solutions of Kirchhoff type Quasilinear Time-fractional Integro-differential Equation

TL;DR

This paper develops a new linearized finite element method for solving non-smooth solutions of Kirchhoff type time-fractional integro-differential equations, providing error analysis and demonstrating accuracy and efficiency through numerical examples.

Contribution

It introduces a novel linearized fully discrete scheme on graded meshes for Kirchhoff type equations with fractional derivatives, including error bounds and stability analysis.

Findings

Achieves an accuracy rate of O(P^{-1}+N^{-(2-eta)}) in relevant norms.

Provides a priori bounds using a new weighted H^1 norm.

Demonstrates robustness and efficiency through numerical experiments.

Abstract

In this article, we study the semi discrete and fully discrete formulations for a Kirchhoff type quasilinear integro-differential equation involving time-fractional derivative of order $\alpha \in (0,1) $. For the semi discrete formulation of the equation under consideration, we discretize the space domain using a conforming FEM and keep the time variable continuous. We modify the standard Ritz-Volterra projection operator to carry out error analysis for the semi discrete formulation of the considered equation. In general, solutions of the time-fractional partial differential equations (PDEs) have a weak singularity near time $t=0$. Taking this singularity into account, we develop a new linearized fully discrete numerical scheme for the considered equation on a graded mesh in time. We derive a priori bounds on the solution of this fully discrete numerical scheme using a new weighted $H^{1}(\Omega)$ norm. We prove that the developed numerical scheme has an accuracy rate of $O(P^{-1}+N^{-(2-\alpha)})$ in $L^{\infty}(0,T;L^{2}(\Omega))$ as well as in $L^{\infty}(0,T;H^{1}_{0}(\Omega))$, where $P$ and $N$ are degrees of freedom in the space and time directions respectively. The robustness and efficiency of the proposed numerical scheme are demonstrated by some numerical examples.