In this study, the compatible extensions of the $\left(G^{\prime }/G\right)$-expansion approach and the generalized $\left(G^{\prime }/G\right)$-expansion scheme are proposed to generate scores of radical closed-form solutions of nonlinear fractional evolution equations. The originality and improvements of the extensions are confirmed by their application to the fractional space-time paired Burgers equations. The application of the proposed extensions highlights their effectiveness by providing dissimilar solutions for assorted physical forms in nonlinear science. In order to explain some of the wave solutions geometrically, we represent them as two- and three-dimensional graphs. The results demonstrate that the techniques presented in this study are effective and straightforward ways to address a variety of equations in mathematical physics with conformable derivative.

In current study, modified (G’/G)-expansion method is applied to develop the travelling wave solutions for time-space-fractional Boussinesq (TSFB) equation. In this solution, modified Riemann-Liouville derivative will be used together with nonlinear fractional complex transformation. The transformation expression is applied to transform the nonlinear fractional equation into nonlinear ordinary differential equation (NODE). A variety of travelling wave solutions can be produced from this solution which is found to be applicable in physical sciences.

In this paper, we propose and execute the Multistep Modified Reduced Differential Transform Method (MMRDTM) to obtain the solution of fractional nonlinear Schrodinger equations (FNLSEs). Using this proposed technique, the nonlinear term is substituted by the related Adomian polynomials followed by application of a multistep approach. In addition, the fractional NLSE solutions can be obtained with less computational effort than other approximate analytical solution such as Adomian decomposition method, homotopy pertubation method, variational iteration method and differential transformation method. Besides that, it offers precise estimated solutions over a longer time period. We considered a few FNLSEs and represent the features of these solutions in the form of graphs to demonstrate the power and precision of the MMRDTM.

Recently, Kaplan and Bekir have used the exp (−ϕ(ξ)) method to extract the exact solutions of fractional Hirota-Satsuma coupled KdV (FHScKdV) system of equations. In this study, we apply a different technique called a (G′/G) -expansion method proposed by Wang et al in the sense of conformable fractional derivative for finding other novel travelling wave solutions for the same model mentioned above. The obtained results from these two different methods will be compared. The obtained solutions have shown that the proposed method by Wang et al with conformable fractional derivative is efficient and reliable for FHScKdV system.

The study of solitons and compactons is important in nonlinear physics. In this paper we combined the Adomian polynomials with the multi-step approach to present a new technique called Multi-step Modified Reduced Differential Transform Method (MMRDTM). The proposed technique has the advantage of producing an analytical approximation in a fast converging sequence with a reduced number of calculated terms. The MMRDTM is presented with some modification of the Reduced Differential Transformation Method (RDTM) with multi-step approach and its nonlinear term is replaced by the Adomian polynomials. Therefore, the nonlinear initial value problem can easily be solved with less computational effort. Besides that, the multi-step approach produces a solution in fast converging series that converges the solution in a wide time area. Two examples are provided to demonstrate the capability and benefits of the proposed method for approximating the solution of NKdVEs with compactons. Graphical inputs are used to represent the solution and to demonstrate the precision and validity of the MMRDTM in graphic illustration. From the results, it was found that it is possible to obtain highly accurate results or exact solutions by using the MMRDTM.

The cubic B-spline and cubic trigonometric B-spline functions are used to set up the collocation in finding solutions for the Buckmaster equation. These splines are applied as interpolating functions in the spatial dimension while the finite difference method (FDM) is used to discretize the time derivative. The Buckmaster equation is linearized using Taylor’s expansion and solved using two schemes, namely Crank-Nicolson and fully implicit. The von Neumann stability analysis is carried out on the two schemes and they are shown to be conditionally stable. In order to demonstrate the capability of the schemes, some problems are solved and compared with analytical and FDM solutions. The proposed methods are found to generate more accurate results than the FDM.