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Fractional-Order Sine-Gordon Equation Involving Nonsingular Derivative

Author

Listed:
  • MUHAMMAD SHER

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

  • AZIZ KHAN

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

  • KAMAL SHAH

    (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)

  • THABET ABDELJAWAD

    (��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)

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.

Suggested Citation

  • Muhammad Sher & Aziz Khan & Kamal Shah & Thabet Abdeljawad, 2023. "Fractional-Order Sine-Gordon Equation Involving Nonsingular Derivative," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 31(10), pages 1-10.
    • Handle: RePEc:wsi:fracta:v:31:y:2023:i:10:n:s0218348x23400078
    DOI: 10.1142/S0218348X23400078
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