Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel

Sayed Saifullah; Amir Ali; Zareen A. Khan; Sayed Saifullah; Amir Ali; Zareen A. Khan

doi:10.3934/math.2022293

AIMS Mathematics

2022, Volume 7, Issue 4: 5275-5290. doi: 10.3934/math.2022293

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Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel

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

Received: 21 October 2021 Revised: 27 December 2021 Accepted: 30 December 2021 Published: 05 January 2022
MSC : 35Bxx, 35Qxx, 37Mxx, 65Mxx, 41Axx

We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor $ \omega $ considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.
- Klein-Gordon equation,
- Caputo operator,
- double Laplace transform,
- decomposition method,
- localized modes excitations
Citation: Sayed Saifullah, Amir Ali, Zareen A. Khan. Analysis of nonlinear time-fractional Klein-Gordon equation with power law kernel[J]. AIMS Mathematics, 2022, 7(4): 5275-5290. doi: 10.3934/math.2022293

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Abstract

We investigate the nonlinear Klein-Gordon equation with Caputo fractional derivative. The general series solution of the system is derived by using the composition of the double Laplace transform with the decomposition method. It is noted that the obtained solution converges to the exact solution of the model. The existence of the model in the presence of Caputo fractional derivative is performed. The validity and precision of the presented method are exhibited with particular examples with suitable subsidiary conditions, where good agreements are obtained. The error analysis and its corresponding surface plots are presented for each example. From the numerical solutions, we observe that the proposed system admits soliton solutions. It is noticed that the amplitude of the wave solution increases with deviations in time, that concludes the factor $ \omega $ considerably increases the amplitude and disrupts the dispersion/nonlinearity properties, as a result, may admit the excitation in the dynamical system. We have also depicted the physical behavior that states the advancement of localized mode excitations in the system.

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