Two New Modifications of the Exp-Function Method for Solving the Fractional-Order Hirota-Satsuma Coupled KdV

Abstract

Two novel modifications of the exp-function method have been suggested to solve the nonlinear system of the partial differential equation of the fractional order for the first time. The fractional-order Hirota-Satsuma coupled KdV system has been solved precisely; as a result, some exact solutions, which include soliton-type and rational solutions, will be derived. Eventually, the graphs of the obtained results have been illustrated at the end of the article. The newly used methods are highly accurate, flexible, effective, and programmable to solve fractional-order devices.

Non‐singular complexiton, singular complexiton and complex multiple soliton solutions to the (3 + 1)‐dimensional nonlinear evolution equation

“…Soliton theory is an important part of nonlinear scientific content, and finding efficacious approaches to develop soliton solutions is the hot spot for scholars. As yet, a good many reliable and efficacious techniques have been put forward to solve the NPDEs, for example, the Bäcklund transformation approach [13–15], direct algebraic approach [16], Darboux transformation technique [17–19], subequation approach [20–22], trial‐equation technique [23–25], extended F‐Expansion approach [26, 27], exp‐function approach [28, 29], and many others [30–36]. In the presented study, we will inquire into the (3 + 1)‐dimensional nonlinear evolution equation (NEE) that reads as [37]: where the inverse operator is defined as: …”

Complexiton, complex multiple kink soliton and the rational wave solutions to the generalized (3 + 1)-dimensional kadomtsev-petviashvili equation

“…In order to better understand and explain the nonlinear phenomena, finding exact solutions to the NPDEs has become an important focus of scholars' attention and research. In the past half century, mathematicians and physicists have been dedicated to studying exact solutions to NPDEs, including the Jacobi elliptic function expansion approach [12], modified generalized exponential rational function method [13], direct algebraic approach [14], (G'/G 2 )-expansion method [15], variational approach [16], trial-equation technique [17,18], Bäcklund transformation approach [19,20], subequation approach [21,22], Darboux transformation technique [23,24], exp-function approach [25], modified Kudryashov method [26], extended F-Expansion approach [27], sinh-Gordon equation expansion method [28] and so on. Although mathematical physicists have developed a large number of methods, it has been found that, due to the diversity and complexity of NPDEs, there is currently no unified method to solve them, and often only the corresponding methods can be selected based on specific equations.…”

Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform

“…Approximative or numerical techniques are typically used because some fractional-order partial differential equations do not have accurate analytic solutions. In the literature, there are numerous analytical and numerical techniques for handling FPDEs, such as the multiple exp-function algorithm for solving the nonlinear fractional Sharma-Tasso-Olver equation [18] and the fractional-order Hirota-Satsuma coupled KdV [19]; the homotopy perturbation transform method for solving time-fractional Belousov-Zhabotinsky reactions [20] and diffusion equations of fractional-order in a plasma and fluids [21]; the fractional complex transformation for fractional nonlinear partial differential equations in mathematical physics [22] and the time-fractional heat conduction equation [23]; the variational iteration transform method for the investigation of the Newell-Whitehead-Segel equations having fractional order [24]; the homotopy analysis method for solving fractional Lorenz systems [25] and time-fractional Schrödinger equations [26]; the finite element method for parabolic equations of fractional order [27] and the time-fractional Fokker-Planck equation [28]; the fractional sub-equation method for generalised fractional KdV-Zakharov-Kuznetsov equations [29]; and so on.…”