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Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining so...

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Main Authors: Andang Sunarto, Praveen Agarwal, Jumat Sulaiman, Chew, Jackel Vui Lung, Shaher Momani
Format: Article
Language:English
English
Published: MDPI AG 2021
Subjects:
QA299.6-433 Analysis
Online Access:https://eprints.ums.edu.my/id/eprint/30876/1/Quarter-sweep%20preconditioned%20relaxation%20method%2C%20algorithm%20and%20efficiency%20analysis%20for%20fractional%20mathematical%20equation_ABSTRACT.pdf
https://eprints.ums.edu.my/id/eprint/30876/2/Quarter-sweep%20preconditioned%20relaxation%20method%2C%20algorithm%20and%20efficiency%20analysis%20for%20fractional%20mathematical%20equation.pdf
https://eprints.ums.edu.my/id/eprint/30876/
https://www.mdpi.com/2504-3110/5/3/98/htm
https://doi.org/10.3390/fractalfract5030098
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spelling my.ums.eprints.308762021-10-28T01:31:46Z https://eprints.ums.edu.my/id/eprint/30876/ Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation Andang Sunarto Praveen Agarwal Jumat Sulaiman Chew, Jackel Vui Lung Shaher Momani QA299.6-433 Analysis Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research. MDPI AG 2021 Article PeerReviewed text en https://eprints.ums.edu.my/id/eprint/30876/1/Quarter-sweep%20preconditioned%20relaxation%20method%2C%20algorithm%20and%20efficiency%20analysis%20for%20fractional%20mathematical%20equation_ABSTRACT.pdf text en https://eprints.ums.edu.my/id/eprint/30876/2/Quarter-sweep%20preconditioned%20relaxation%20method%2C%20algorithm%20and%20efficiency%20analysis%20for%20fractional%20mathematical%20equation.pdf Andang Sunarto and Praveen Agarwal and Jumat Sulaiman and Chew, Jackel Vui Lung and Shaher Momani (2021) Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation. Fractal and Fractional, 5. pp. 1-14. ISSN 2504-3110 https://www.mdpi.com/2504-3110/5/3/98/htm https://doi.org/10.3390/fractalfract5030098
institution Universiti Malaysia Sabah
building UMS Library
collection Institutional Repository
continent Asia
country Malaysia
content_provider Universiti Malaysia Sabah
content_source UMS Institutional Repository
url_provider http://eprints.ums.edu.my/
language English
English
topic QA299.6-433 Analysis
spellingShingle QA299.6-433 Analysis
Andang Sunarto
Praveen Agarwal
Jumat Sulaiman
Chew, Jackel Vui Lung
Shaher Momani
Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
description Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.
format Article
author Andang Sunarto
Praveen Agarwal
Jumat Sulaiman
Chew, Jackel Vui Lung
Shaher Momani
author_facet Andang Sunarto
Praveen Agarwal
Jumat Sulaiman
Chew, Jackel Vui Lung
Shaher Momani
author_sort Andang Sunarto
title Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
title_short Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
title_full Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
title_fullStr Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
title_full_unstemmed Quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
title_sort quarter-sweep preconditioned relaxation method, algorithm and efficiency analysis for fractional mathematical equation
publisher MDPI AG
publishDate 2021
url https://eprints.ums.edu.my/id/eprint/30876/1/Quarter-sweep%20preconditioned%20relaxation%20method%2C%20algorithm%20and%20efficiency%20analysis%20for%20fractional%20mathematical%20equation_ABSTRACT.pdf
https://eprints.ums.edu.my/id/eprint/30876/2/Quarter-sweep%20preconditioned%20relaxation%20method%2C%20algorithm%20and%20efficiency%20analysis%20for%20fractional%20mathematical%20equation.pdf
https://eprints.ums.edu.my/id/eprint/30876/
https://www.mdpi.com/2504-3110/5/3/98/htm
https://doi.org/10.3390/fractalfract5030098
_version_ 1760230819653222400
score 13.211869

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