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Finite Element Analysis of Time-Fractional Integro-differential Equation of Kirchhoff type for Non-homogeneous Materials
  • K Sreenadh,
  • Lalit Kumar,
  • Sivaji Ganesh Sista
K Sreenadh
Indian Institute of Technology Delhi

Corresponding Author:sreenadh@maths.iitd.ac.in

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Lalit Kumar
Indian Institute of Technology Bombay Department of Mathematics
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Sivaji Ganesh Sista
Indian Institute of Technology Bombay Department of Mathematics
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In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials ( P α ). As a consequence of energy argument, we derive L ∞ ( 0 , T ; H 0 1 ( Ω ) ) bound as well as L 2 ( 0 , T ; H 2 ( Ω ) ) bound on the solution of the problem ( P α ) by defining two new discrete Laplacian operators. Using these a priori bounds, existence and uniqueness of the weak solution to the considered problem is established. Further, we study semi discrete formulation of the problem ( P α ) by discretizing the space domain using a conforming FEM and keeping the time variable continuous. The semi discrete error analysis is carried out by modifying the standard Ritz-Volterra projection operator in such a way that it reduces the complexities arising from the Kichhoff type nonlinearity. Finally, we develop a new linearized L1 Galerkin FEM to obtain numerical solution of the problem ( P α ) with a convergence rate of O ( h + k 2 − α ) , where α (0 1) is the fractional derivative exponent, h and k are the discretization parameters in the space and time directions respectively. This convergence rate is improved to second order in the time direction by proposing a novel linearized L2-1 σ Galerkin FEM. We conduct a numerical experiment to validate our theoretical claims.