Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics
Abstract In applied research, fractional calculus plays an important role for comprehending a wide range of intricate physical phenomena. One of the Klein-Gordon model’s peculiar case yields the Phi-four equation. Additionally, throughout the past few decades it has been utilized to explain the kink...
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Nature Portfolio
2025-01-01
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| Series: | Scientific Reports |
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| Online Access: | https://doi.org/10.1038/s41598-025-86177-7 |
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| author | Ayesha Farooq Tooba Shafique Muhammad Abbas Asnake Birhanu Y. S. Hamed |
| author_facet | Ayesha Farooq Tooba Shafique Muhammad Abbas Asnake Birhanu Y. S. Hamed |
| author_sort | Ayesha Farooq |
| collection | DOAJ |
| description | Abstract In applied research, fractional calculus plays an important role for comprehending a wide range of intricate physical phenomena. One of the Klein-Gordon model’s peculiar case yields the Phi-four equation. Additionally, throughout the past few decades it has been utilized to explain the kink and anti-kink solitary waveform contacts that occur in biological systems and in the field of nuclear mechanics. In this current work, the key objective is to analyze the consequences of fractional variables on the soliton wave dynamic behavior in a nonlinear time-fractional Phi-four equation. Using the formulation of the conformable fractional derivative it illustrates some of the recovered solutions and analyze their dynamic behavior. The analytical solutions are drawn by using the extended direct algebraic and the Bernoulli Sub-ODE scheme. Various types of soliton solutions are proficiently expressed. Adjusting the specific values of fractional parameters allows to produce the periodic, kink, bell shape, anti-bell shape and W-shaped solitons. The impact of the conformable derivative on the precise solutions of the fractional Phi-four equation is demonstrated with a series of 2D, 3D and contour graphical representations. |
| format | Article |
| id | doaj-art-3c649e3c668e468bb7403540e5ccb55e |
| institution | Kabale University |
| issn | 2045-2322 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Nature Portfolio |
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| series | Scientific Reports |
| spelling | doaj-art-3c649e3c668e468bb7403540e5ccb55e2025-01-12T12:21:44ZengNature PortfolioScientific Reports2045-23222025-01-0115112610.1038/s41598-025-86177-7Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physicsAyesha Farooq0Tooba Shafique1Muhammad Abbas2Asnake Birhanu3Y. S. Hamed4Department of Mathematics, University of SargodhaDepartment of Mathematics, University of SargodhaDepartment of Mathematics, University of SargodhaDepartment of Mathematics, College of Science, Hawassa UniversityDepartment of Mathematics and Statistics, College of Science, Taif UniversityAbstract In applied research, fractional calculus plays an important role for comprehending a wide range of intricate physical phenomena. One of the Klein-Gordon model’s peculiar case yields the Phi-four equation. Additionally, throughout the past few decades it has been utilized to explain the kink and anti-kink solitary waveform contacts that occur in biological systems and in the field of nuclear mechanics. In this current work, the key objective is to analyze the consequences of fractional variables on the soliton wave dynamic behavior in a nonlinear time-fractional Phi-four equation. Using the formulation of the conformable fractional derivative it illustrates some of the recovered solutions and analyze their dynamic behavior. The analytical solutions are drawn by using the extended direct algebraic and the Bernoulli Sub-ODE scheme. Various types of soliton solutions are proficiently expressed. Adjusting the specific values of fractional parameters allows to produce the periodic, kink, bell shape, anti-bell shape and W-shaped solitons. The impact of the conformable derivative on the precise solutions of the fractional Phi-four equation is demonstrated with a series of 2D, 3D and contour graphical representations.https://doi.org/10.1038/s41598-025-86177-7Time-fractional Phi-Four equationBernoulli sub-ODE methodExtended direct algebraic method$$\beta$$ -DerivativeM-truncated derivativeConformable derivative |
| spellingShingle | Ayesha Farooq Tooba Shafique Muhammad Abbas Asnake Birhanu Y. S. Hamed Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics Scientific Reports Time-fractional Phi-Four equation Bernoulli sub-ODE method Extended direct algebraic method $$\beta$$ -Derivative M-truncated derivative Conformable derivative |
| title | Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics |
| title_full | Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics |
| title_fullStr | Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics |
| title_full_unstemmed | Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics |
| title_short | Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics |
| title_sort | explicit travelling wave solutions to the time fractional phi four equation and their applications in mathematical physics |
| topic | Time-fractional Phi-Four equation Bernoulli sub-ODE method Extended direct algebraic method $$\beta$$ -Derivative M-truncated derivative Conformable derivative |
| url | https://doi.org/10.1038/s41598-025-86177-7 |
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