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Based on two algorithm integrations, such as the exp (- Φ (ξ)) -expansion method and the hyperbolic function method, we build dark, bright and trigonometric function solution to the Klein–Gordon equations with M-fractional derivative of order α . By adopting the travelling-wave transformation, the constraint condition between the model coefficients and the travelling-wave frequency coefficient for the existence of soliton solutions is also obtained. Moreover, miscellaneous soliton solutions obtained is depicted in 3D and 2D. [ABSTRACT FROM AUTHOR]

This work addresses the analytical investigation of the prey–predator behavior modeled by nonlinear evolution equation systems with fractional derivative order. Through the New Extended Algebraic Method (NEAM), we unearthed diverse types of soliton solutions including bright, dark solitons, combined trigonometric function solutions, and singular solutions. Besides the results obtained in the work of Khater, some new complex soliton solutions are also unearthed. The NEAM can also be used like the synthesis of the two mathematical tools. [ABSTRACT FROM AUTHOR]