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Numerical Solution of Two-Dimensional Time Fractional Mobile/Immobile Equation Using Explicit Group Methods.

Fouad Mohammad Salama1, Umair Ali2, Ajmal Ali3

  • 1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Penang Malaysia.

International Journal of Applied and Computational Mathematics
|July 21, 2022
PubMed
Summary

New fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods efficiently solve time fractional mobile/immobile equations. These numerical methods reduce computation time and iterations while maintaining accuracy.

Keywords:
ConvergenceCrank-Nicolson finite differenceFractional mobile/immobile equationGrouping strategyNumerical experimentsStability

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Time fractional mobile/immobile equations are crucial in modeling complex physical phenomena.
  • Existing numerical methods may face challenges in computational efficiency and accuracy for these equations.
  • The Crank-Nicolson (C-N) finite difference method is a common approach, but advancements are sought.

Purpose of the Study:

  • To develop and present two novel explicit group schemes: fractional explicit group (FEG) and modified fractional explicit group (MFEG).
  • To solve the time fractional mobile/immobile equation in two spatial dimensions using these new methods.
  • To rigorously analyze the stability and convergence properties of the proposed FEG and MFEG algorithms.

Main Methods:

  • Formulation of FEG and MFEG methods based on two Crank-Nicolson (C-N) finite difference schemes.
  • Application of Fourier analysis to rigorously prove the stability and convergence of the developed schemes.
  • Implementation of numerical experiments to validate the efficiency and accuracy of the proposed methods.

Main Results:

  • The stability and convergence of the FEG and MFEG methods are rigorously proven.
  • Numerical experiments demonstrate the effectiveness of the proposed algorithms.
  • FEG and MFEG significantly reduce computational times and iterations compared to the standard C-N finite difference method.

Conclusions:

  • The developed fractional explicit group (FEG) and modified fractional explicit group (MFEG) methods are efficient and accurate for solving time fractional mobile/immobile equations.
  • These novel methods offer a significant improvement in computational performance over traditional approaches.
  • The rigorous mathematical analysis confirms the reliability of FEG and MFEG for practical applications.