Dynamical structure of truncated M−fractional Klein–Gordon model via two integral schemes

Effect of the nonlinear dispersive coefficient on time-dependent variable coefficient soliton solutions of the Kolmogorov-Petrovsky-Piskunov model arising in biological and chemical science

“…Additionally, when dealing with nonlinear evolution equations it is important to have a thorough understanding of the characteristics of the equation itself, as well as the potential solutions, in order to ensure accuracy and avoid unnecessary computational effort. There are some techniques used to solve NLEEs successfully, such as N-fold Darboux transformation [ 16 ], simplified Hirota's direct method [ 17 ], modified exponential rational function scheme [ 18 ], Simplest equation and New form of modified Kudrusov procedure [ 19 ], new generalized exponential rational function technique [ 20 ], amplitude ansatz method [ 21 ], Hirota bilinear forms with Hirota direct method [ 22 , 23 ], unified scheme [ 24 ], New modified simple equation technique [ 25 ], backlund transformations [ 26 ], backlund transformation from the riccati form of an inverse method [ 26 ], simple symbolic computation approach [ 27 ], generalized projective riccati equations method [ 28 ] Sardar sub-equation and the MK approaches [ 29 ] sine-cosine scheme [ 30 ], reductive perturbation method [ 31 ], enhanced modified simple equation technique [ 32 ], and so on. The motivation of this work is to explore the variable coefficient solitary wave solution of classical Kolmogorov–Petrovsky–Piskunov (KPP) models.…”

A variety of soliton solutions of time M-fractional: Non-linear models via a unified technique

“…Recently, many researchers have developed different types of fractional derivatives and applied them to diverse nonlinear systems. M-fractional derivative is used to the Paraxial Wave model [27] and Klein-Gordon model [28], conformable fractional derivative is applied to the date-Jimbo-Kashiwara-Miwa equation [29] and variant Boussinesq equation [30], Caputo fractional derivative is applied to the prey-predator model [31] and Korteweg-de Vries (KdV) models [32], Riemann-Liouville fractional integral is used to the Kudryashov-Sinelshchikov equation [33] and relaxation-oscillation equation [34], fractional beta derivatives is used to Cubic Nonlinear Schro ¨dinger Equation [35] and heat equation [36], etc.…”

Abundant dynamical solitary waves solutions of M -fractional Oskolkov model

“…This literature review highlights the evolving landscape of numerical approaches for SMITHs, emphasizing the signifi cance of methodological comparisons to advance computational techniques in scientifi c research. This study presents an analytical investigation and discussion of the fractional Oskolkov model [26,27]. Therefore, it is crucial to look for wave solutions for NPDEs.…”