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.

中文翻译：

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Kirchhoff型抛物积分微分方程的有限元分析

本文的目的是研究基尔霍夫型抛物线积分微分方程。我们通过Galerkin方法证明了该问题解的存在性和唯一性。使用相容有限元方法给出了该问题的半离散公式。作为Ritz-Volterra投影的结果，我们导出了半离散解及其时间导数的误差估计。为了找到此类方程的数值解，我们开发了两种不同类型的数值方案，它们基于后向Euler-Galerkin方法和Crank-Nicolson-Galerkin方法。建立了所提出方案在空间和时间方向上的先验界限和收敛估计。最后，我们通过进行一些数值实验来证实我们的理论结果，从而完成了这项工作。