ArticleOpen Access

FRACTIONAL-ORDER SINE-GORDON EQUATION INVOLVING NONSINGULAR DERIVATIVE

MUHAMMAD SHER

https://orcid.org/0009-0007-9632-1843

Department of Mathematics, University of Malakand, Chakdara Dir Lower 18000, Khyber Pakhtunkhwa, Pakistan

E-mail Address: msherpak55@gmail.com

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AZIZ KHAN

https://orcid.org/0000-0001-6185-9394

Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia

E-mail Address: akhan@psu.edu.sa

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KAMAL SHAH

https://orcid.org/0000-0002-8851-4844

Department of Mathematics, University of Malakand, Chakdara Dir Lower 18000, Khyber Pakhtunkhwa, Pakistan

Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia

Department of Computer Science and Mathematics, Lebanese American University, Byblos, Lebanon

E-mail Address: kamalshah408@gmail.com

E-mail Address: kshah@psu.edu.sa

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, and

THABET ABDELJAWAD

https://orcid.org/0000-0002-8889-3768

Department of Mathematics and Sciences, Prince Sultan University, P. O. Box 66833, Riyadh 11586, Saudi Arabia

Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea

E-mail Address: tabdeljawad@psu.edu.sa

Corresponding author.

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https://doi.org/10.1142/S0218348X23400078Cited by:7 (Source: Crossref)

This article is part of the issue:

Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems IV
Guest Editors: Yeliz Karaca, Dumitru Baleanu, Majaz Moonis, Yu-Dong Zhang and Osvaldo Gervasi

Abstract

The sine-Gordon equation has received attention since 1970s due to the existence of soliton solutions. The aforesaid equation has significant applications in the quantum field theory. The aforementioned problem has been treated by using various numerical and analytical techniques under the ordinary as well as fractional-order derivatives. The mentioned equation has been investigated under the usual Caputo fractional-order derivative. Since in some cases the nonsingular-type derivatives produce more significant results in the mathematical modelings of real-world nonlinear problems, therefore, the proposed problem is considered in this paper under the fractional-order case in the context of Atangana–Baleanu–Caputo (ABC) derivative for the analytical and approximate results. This fractional derivative has some useful properties involving Mittag-Leffler-type kernel that is nonlocal and nonsingular. Furthermore, Modified Homotopy Perturbation Method (MHPM) is utilized for the required approximate solution. We give appropriate examples depicting the sine-Gordon model. Also, we present our results for the approximate solution graphically to support all the results.

Keywords:

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Vol. 31, No. 10

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History

Received 20 January 2022

Accepted 20 September 2022

Published: September 2, 2023

Information

This is an Open Access article in the “Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems: Part IV”, edited by Yeliz Karaca https://orcid.org/0000-0001-8725-6719 (University of Massachusetts Chan Medical School, USA), Dumitru Baleanu https://orcid.org/0000-0002-0286-7244 (Cankaya University, Turkey & Institute of Space Sciences, Romania), Majaz Moonis https://orcid.org/0000-0002-9747-0003 (University of Massachusetts Chan Medical School, USA), Yu-Dong Zhang https://orcid.org/0000-0002-4870-1493 (University of Leicester, Leicester, UK) & Osvaldo Gervasi https://orcid.org/0000-0003-4327-520X (Perugia University, Perugia, Italy) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 (CC BY-NC) License which permits use, distribution and reproduction in any medium, provided that the original work is properly cited and is used for non-commercial purposes.

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FRACTIONAL-ORDER SINE-GORDON EQUATION INVOLVING NONSINGULAR DERIVATIVE

Abstract

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