Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations

Mubashara Wali; Sadia Arshad; Sayed M Eldin; Imran Siddique; Mubashara Wali; Sadia Arshad; Sayed M Eldin; Imran Siddique

doi:10.3934/math.2023772

AIMS Mathematics

2023, Volume 8, Issue 7: 15129-15147. doi: 10.3934/math.2023772

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Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations

1.
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore 54000, Pakistan
2.
Center of Research, Faculty of Engineering, Future University in Egypt, New Cairo 11835, Egypt
3.
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan

Received: 09 December 2022 Revised: 19 February 2023 Accepted: 02 March 2023 Published: 23 April 2023
MSC : 34K28, 35-XX

In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps $ \delta t $ the recommended method converges at a rate of $ O(\delta t^2 + h^2) $. Based on graphic results and numerical examples, the application of the model is also examined.
- fractional diffusion equation,
- numerical approximation,
- Atangana-Baleanu Caputo derivative,
- non-singular kernel,
- stability-convergence
Citation: Mubashara Wali, Sadia Arshad, Sayed M Eldin, Imran Siddique. Numerical approximation of Atangana-Baleanu Caputo derivative for space-time fractional diffusion equations[J]. AIMS Mathematics, 2023, 8(7): 15129-15147. doi: 10.3934/math.2023772

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Abstract

In this study, we attempt to obtain the approximate solution for the time-space fractional linear and nonlinear diffusion equations. A finite difference approach is given for the solution of both linear and nonlinear fractional order diffusion problems. The Riesz fractional derivative in space is specifically approximated using the centered difference scheme. A system of Atangana-Baleanu Caputo equations that have been converted through spatial discretization is solved using a newly developed modified Simpson's 1/3 formula. A study of the proposed scheme is done to ascertain its stability and convergence. It has been shown that for mesh size h and time steps $ \delta t $ the recommended method converges at a rate of $ O(\delta t^2 + h^2) $. Based on graphic results and numerical examples, the application of the model is also examined.

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