Finite element analysis of parabolic integro‐differential equations of Kirchhoff type

Abstract

The aim of this paper is to study parabolic integro‐differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.

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

“…From Tables 1, 2, and 3, we conclude that the convergence rate is first order in the space direction and (2 − 𝛼) in the time direction as predicted in Theorem 3.4. We also observe that as 𝛼 → 1, then convergence rate is approaching to first order in the time direction that coincides with the results established in [55] for classical diffusion case. Remark 3.…”