Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics

Ayesha Farooq; Tooba Shafique; Muhammad Abbas; Asnake Birhanu; Y S Hamed

doi:10.1038/s41598-025-86177-7

Explicit travelling wave solutions to the time fractional Phi-four equation and their applications in mathematical physics

Sci Rep. 2025 Jan 11;15(1):1683. doi: 10.1038/s41598-025-86177-7.

Authors

Ayesha Farooq¹, Tooba Shafique¹, Muhammad Abbas¹, Asnake Birhanu², Y S Hamed³

Affiliations

¹ Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan.
² Department of Mathematics, College of Science, Hawassa University, Hawassa, Ethiopia. asnakeb@hu.edu.et.
³ Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia.

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.

Keywords: β-Derivative; Bernoulli sub-ODE method; Conformable derivative; Extended direct algebraic method; M-truncated derivative; Solitons; Time-fractional Phi-Four equation.