Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels

Amir Ali; Abid Ullah Khan; Obaid Algahtani; Sayed Saifullah; Amir Ali; Abid Ullah Khan; Obaid Algahtani; Sayed Saifullah

doi:10.3934/math.2022820

AIMS Mathematics

2022, Volume 7, Issue 8: 14975-14990. doi: 10.3934/math.2022820

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Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels

1.
Department of Mathematics, University of Malakand, Chakdara, Dir(L), Khyber Pakhtunkhwa, Pakistan
2.
Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 27 April 2022 Revised: 28 May 2022 Accepted: 06 June 2022 Published: 14 June 2022
MSC : 26A33, 35Cxx, 35Qxx, 35R11, 41Axx

In this article, we study the nonlinear sine-Gordon equation (sGE) under Mittag-Leffler and exponential decay type kernels in a fractal fractional sense. The Laplace Adomian decomposition method (LADM) is applied to investigate the sGE under the above-mentioned operators. The convergence analysis is provided for the proposed method. The results are validated by considering numerical examples with different initial conditions for both kernels and confirm the competence of the proposed technique. It is revealed that the obtained series solutions of the model with fractal fractional operators converge to the exact solutions. The numerical results converge to the particular exact solutions, proving the significance of LADM for nonlinear systems under fractal fractional derivatives. The absolute error analysis between the exact and obtained series solutions with both operators is shown in the tabulated form. The physical interpretations of the attained results with different fractal and fractional parameters are discussed in detail.
- sine-Gordon Equation,
- Laplace transform,
- ABC and C-F fractal-fractional derivatives
Citation: Amir Ali, Abid Ullah Khan, Obaid Algahtani, Sayed Saifullah. Semi-analytical and numerical computation of fractal-fractional sine-Gordon equation with non-singular kernels[J]. AIMS Mathematics, 2022, 7(8): 14975-14990. doi: 10.3934/math.2022820

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Abstract

In this article, we study the nonlinear sine-Gordon equation (sGE) under Mittag-Leffler and exponential decay type kernels in a fractal fractional sense. The Laplace Adomian decomposition method (LADM) is applied to investigate the sGE under the above-mentioned operators. The convergence analysis is provided for the proposed method. The results are validated by considering numerical examples with different initial conditions for both kernels and confirm the competence of the proposed technique. It is revealed that the obtained series solutions of the model with fractal fractional operators converge to the exact solutions. The numerical results converge to the particular exact solutions, proving the significance of LADM for nonlinear systems under fractal fractional derivatives. The absolute error analysis between the exact and obtained series solutions with both operators is shown in the tabulated form. The physical interpretations of the attained results with different fractal and fractional parameters are discussed in detail.

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