TY - JOUR

T1 - FRACTIONAL-ORDER SINE-GORDON EQUATION INVOLVING NONSINGULAR DERIVATIVE

AU - Sher, Muhammad

AU - Khan, Aziz

AU - Shah, Kamal

AU - Abdeljawad, Thabet

N1 - Publisher Copyright:
© 2023 World Scientific Publishing Company.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - ABC Fractional-Order Derivative

KW - Modified Homotopy Perturbation Method

KW - Sine-Gordon Equations

UR - http://www.scopus.com/inward/record.url?scp=85170225572&partnerID=8YFLogxK

U2 - 10.1142/S0218348X23400078

DO - 10.1142/S0218348X23400078

M3 - Article

AN - SCOPUS:85170225572

SN - 0218-348X

JO - Fractals

JF - Fractals

M1 - 2340007

ER -