Solitary wave solutions in time-fractional Korteweg-de Vries equations with power law kernel

Khalid Khan; Amir Ali; Muhammad Irfan; Zareen A. Khan; Khalid Khan; Amir Ali; Muhammad Irfan; Zareen A. Khan

doi:10.3934/math.2023039

AIMS Mathematics

2023, Volume 8, Issue 1: 792-814. doi: 10.3934/math.2023039

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Research article

Solitary wave solutions in time-fractional Korteweg-de Vries equations with power law kernel

1.
Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan
2.
Department of Physics, University of Malakand, Khyber Pakhtunkhwa, Pakistan
3.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 17 August 2022 Revised: 12 September 2022 Accepted: 18 September 2022 Published: 12 October 2022
MSC : 35Bxx, 35Qxx, 35Axx, 37Mxx, 65R10, 06A33, 42A38, 74-XX

The non-linear time-fractional Korteweg-de Vries and modified Korteweg-de Vries equations are studied with Caputo's fractional derivative. The general higher-order solitary wave solutions are derived using a novel technique called the Aboodh transform decomposition method. To validate the obtained results, two examples of Caputo's fractional derivative with appropriate subsidiary conditions are illustrated. The accuracy and efficiency are confirmed by using numerical simulations and error analysis, where good agreements are obtained. The numerical analysis shows that, in comparison to the time-fractional Korteweg-de Vries solution, the solitary wave solution for the time-fractional modified Korteweg-de Vries equation is less stable against the oscillations. The variations in the temporal variable $ t $ enhance the strength of the wave solutions. Moreover, the wave perturbations taper off as $ t $ attains large values. The parameter $ \alpha $ signifies the fractional derivative influence on the wave dispersion and nonlinearity effects. This affects the amplitude as well as the spatial extension of the solitary waves. With a relatively small value of $ t $, the obtained solutions admit pulse-shaped solitons. Moreover, the wave's solutions suffer from oscillations when the temporal variable attains large values. This effect cannot be noticed in the soliton solutions obtained in the integer order systems.
- non-linear time-fractional KdV,
- Caputo's fractional derivative,
- Aboodh transform,
- decomposition method,
- soliton solutions
Citation: Khalid Khan, Amir Ali, Muhammad Irfan, Zareen A. Khan. Solitary wave solutions in time-fractional Korteweg-de Vries equations with power law kernel[J]. AIMS Mathematics, 2023, 8(1): 792-814. doi: 10.3934/math.2023039

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Abstract

The non-linear time-fractional Korteweg-de Vries and modified Korteweg-de Vries equations are studied with Caputo's fractional derivative. The general higher-order solitary wave solutions are derived using a novel technique called the Aboodh transform decomposition method. To validate the obtained results, two examples of Caputo's fractional derivative with appropriate subsidiary conditions are illustrated. The accuracy and efficiency are confirmed by using numerical simulations and error analysis, where good agreements are obtained. The numerical analysis shows that, in comparison to the time-fractional Korteweg-de Vries solution, the solitary wave solution for the time-fractional modified Korteweg-de Vries equation is less stable against the oscillations. The variations in the temporal variable $ t $ enhance the strength of the wave solutions. Moreover, the wave perturbations taper off as $ t $ attains large values. The parameter $ \alpha $ signifies the fractional derivative influence on the wave dispersion and nonlinearity effects. This affects the amplitude as well as the spatial extension of the solitary waves. With a relatively small value of $ t $, the obtained solutions admit pulse-shaped solitons. Moreover, the wave's solutions suffer from oscillations when the temporal variable attains large values. This effect cannot be noticed in the soliton solutions obtained in the integer order systems.

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