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Finite element analysis of parabolic integro‐differential equations of Kirchhoff type
Author(s) -
Kumar Lalit,
Sista Sivaji Ganesh,
Sreenadh Konijeti
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6607
Subject(s) - mathematics , galerkin method , finite element method , backward euler method , uniqueness , mathematical analysis , superconvergence , convergence (economics) , numerical analysis , projection (relational algebra) , type (biology) , partial differential equation , a priori and a posteriori , euler equations , algorithm , ecology , philosophy , physics , epistemology , biology , economics , thermodynamics , economic growth
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.