English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this article, we study the semi discrete and fully discrete formulationsfor a Kirchhoff type quasilinear integro-differential equation involvingtime-fractional derivative of order $\alpha \in (0,1) $. For the semi discreteformulation of the equation under consideration, we discretize the space domainusing a conforming FEM and keep the time variable continuous. We modify thestandard Ritz-Volterra projection operator to carry out error analysis for thesemi discrete formulation of the considered equation. In general, solutions ofthe time-fractional partial differential equations (PDEs) have a weaksingularity near time $t=0$. Taking this singularity into account, we develop anew linearized fully discrete numerical scheme for the considered equation on agraded mesh in time. We derive a priori bounds on the solution of this fullydiscrete numerical scheme using a new weighted $H^{1}(\Omega)$ norm. We provethat 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 freedomin the space and time directions respectively. The robustness and efficiency ofthe proposed numerical scheme are demonstrated by some numerical examples.

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