Numerical Approximations of a One-Dimensional Time-Fractional Semilinear Parabolic Equation

Authors

  • Maan A. Rasheed Department of Mathematics, College of Basic Education, Mustansiriyah University, Baghdad, Iraq https://orcid.org/0000-0002-7955-1424
  • Maani A. Saeed Department of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, Iraq

DOI:

https://doi.org/10.24996/ijs.2023.64.12.28

Keywords:

Fractional order equation, Caputo fractional formula, Finite difference shames, Semilinear parabolic equation, Implicit Euler scheme, Crank-Nicolson method

Abstract

     The time fractional order differential equations are fundamental tools that are used for modeling neuronal dynamics. These equations are obtained by substituting the time derivative of order  where , in the standard equation with the Caputo fractional formula. In this paper, two implicit difference schemes: the linearly Euler implicit and the Crank-Nicolson (CN) finite difference schemes, are employed in solving a one-dimensional time-fractional semilinear equation with Dirichlet boundary conditions. Moreover, the consistency, stability and convergence of the proposed schemes are investigated. We prove that the IEM is unconditionally stable, while CNM is conditionally stable. Furthermore, a comparative study between these two schemes will be conducted via numerical experiments. The efficiency of the proposed schemes in terms of absolute errors, order of accuracy and computing time will be reported and discussed.

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Published

2023-12-30

Issue

Section

Mathematics

How to Cite

Numerical Approximations of a One-Dimensional Time-Fractional Semilinear Parabolic Equation. (2023). Iraqi Journal of Science, 64(12), 6445-6459. https://doi.org/10.24996/ijs.2023.64.12.28

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