Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions
In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional...
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| Veröffentlicht in: | Mathematical problems in engineering 2021-10, Vol.2021, p.1-15 |
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| creator | Saifullah, Sayed Ali, Amir Irfan, Muhammad Shah, Kamal |
| description | In this article, we study the time-fractional nonlinear Klein–Gordon equation in Caputo–Fabrizio’s sense and Atangana–Baleanu–Caputo’s sense. The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient α significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations. |
| doi_str_mv | 10.1155/2021/6858592 |
| format | Article |
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The modified double Laplace transform decomposition method is used to attain solutions in the form of series of the proposed model under aforesaid fractional operators. The suggested method is the composition of the double Laplace transform and decomposition method. The convergence of the considered method is demonstrated for the considered model. It is observed that the obtained solutions converge to the exact solution of the proposed model. For validity, we consider two particular examples with appropriate initial conditions and derived the series solution in the sense of both operators for the considered model. From numerical solutions, it is observed that the considered model admits pulse-shaped solitons. It is also observed that the wave amplitude enhances with variations in time, which infers the coefficient α significantly increases the wave amplitude and affects the nonlinearity/dispersion effects, therefore may admit monotonic shocks. The physical behavior of the considered numerical examples is illustrated explicitly which reveals the evolution of localized shock excitations.</description><identifier>ISSN: 1024-123X</identifier><identifier>EISSN: 1563-5147</identifier><identifier>DOI: 10.1155/2021/6858592</identifier><language>eng</language><publisher>New York: Hindawi</publisher><subject>Amplitudes ; Calculus ; Control theory ; Convergence ; Decomposition ; Engineering ; Exact solutions ; Hypotheses ; Initial conditions ; Klein-Gordon equation ; Laplace transforms ; Mathematical models ; Nonlinearity ; Operators (mathematics) ; Ordinary differential equations ; Partial differential equations ; Physics ; Quantum field theory ; Shock waves ; Solitary waves</subject><ispartof>Mathematical problems in engineering, 2021-10, Vol.2021, p.1-15</ispartof><rights>Copyright © 2021 Sayed Saifullah et al.</rights><rights>Copyright © 2021 Sayed Saifullah et al. This is an open access article distributed under the Creative Commons Attribution License (the “License”), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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| subjects | Amplitudes Calculus Control theory Convergence Decomposition Engineering Exact solutions Hypotheses Initial conditions Klein-Gordon equation Laplace transforms Mathematical models Nonlinearity Operators (mathematics) Ordinary differential equations Partial differential equations Physics Quantum field theory Shock waves Solitary waves |
| title | Time-Fractional Klein–Gordon Equation with Solitary/Shock Waves Solutions |
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