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In this research, Genocchi wavelets method, a quite new type of wavelet-like basis, is adopted to obtain a numerical solution for the classical and time-fractional Rosenau-Hyman or K(n, n) equation arising in the formation of patterns in liquid drops. The considered partial differential equation can be transformed into a system of non-linear algebraic equations by utilizing the wavelets method including an integral operational matrix and then discretizing the equation at the collocation points. The system can be simply solved by several traditional methods. Finally, the algorithm is implemented for some numerical examples and the numerical solutions are compared with the exact solutions using MAPLE. The obtained results are demonstrated using figures and tables. When the results are compared, it is evinced that the algorithm is quite effective and advantageous due to its easily computable algorithm, high accuracy, and less process time. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

Purpose: In this paper, optical solitons of higher-order nonlinear Schrödinger equation with Kudryashov's sextic power-law of nonlinear refractive index are investigated via the direct mapping method. The considered model identifies the optical soliton pulse propagation in the optical fibers. Deriving the optical solutions of investigated model such as sextic power is critical but difficult work. The primary aim of this paper is to graphically examine the impact of power-law nonlinearity (pLawNL) and chromatic dispersion (CD) parameters clarifying self-phase modulation (SPM) in the equation on soliton behavior as well as obtaining optical soliton solutions. Methodology: To according to the used technique, we first used the complex wave transform to generate the nonlinear ordinary differential equation (NLODE) form of the nonlinear Schrödinger equation (NLSE) with Kudryashov's sextic power-law the nonlinear refractive index (SPLawNRI). Then, we were able to produce a system of linear equations in polynomial form by using the approach. Different solution sets including the values of the parameters of the studied equation and the suggested approach were produced by solving the linear system of equations. Findings: We acquired the optical soliton solutions of the main equation after inserting the sets and wave transformation into the solution functions suggested by the approach. The constraint conditions for the related solutions were suggested. We proved that the gained solutions satisfied the NLSE with Kudryashov's SPLawNRI under the suggested constraint conditions. Originality: We present contour, 3D and 2D depictions in various simulations in figures to comment the obtained solution functions. Besides, we investigate the effects of the power-law nonlinearity parameter that expresses SPM in the main equation and the parameter that are the group velocity dispersion (GVD) or chromatic dispersion (CD) on soliton behavior. The results suggest that the utilized approach is efficient, reliable, and powerful to be readily applied to various NLSEs with higher-order or higher pLawNLs that characterize real-life problems.

Purpose: When it comes to third and higher-order dispersion, the Schrödinger–Hirota equation is one of the main models developed outside the classical NLSE management models for optical soliton transmission. The cubic–quartic Fokas–Lenells equation is also one of the recently developed equations, which has importance in the field of telecommunications regarding optical soliton transmission in the absence of chromatic dispersion. In this study, in order to examine the optical solitons, the Schrödinger–Hirota equation in the presence of the chromatic dispersion and the cubic–quartic Fokas–Lenells equation discarding the chromatic dispersion were investigated. For this intent, by obtaining certain soliton types using the unified Riccati equation expansion method (UREEM), optical soliton solutions were obtained for both models and graphical representations and comments were made. Methodology: By developing appropriate computer algorithms and applying UREEM in the following ways, symbolic calculation software was made and analytical optical soliton solutions were obtained. Findings: Through computer algebra software, we plotted the obtained results via 3D, 2D views and we also illustrated the investigation of wave behavior caused by parameter change on 2D graphics. Originality: Different soliton behavior under the parameters effect of the Schrödinger–Hirota equation having chromatic dispersion and the cubic–quartic Fokas-Lenells equation is investigated and the obtained results are reported.

In this study, the Jacobi wavelet collocation method is studied to derive a solution of the time-fractional Fisher?s equation in Caputo sense. Jacobi wavelets can be considered as a generalization of the wavelets since the Gegenbauer, and thus also Chebyshev and Legendre polynomials are a special type of the Jacobi polynomials. So, more accurate and fast convergence solutions can be possible for some kind of problems thanks to Jacobi wavelets. After applying the proposed method to the considered equation and discretizing the equation at the collocation points, an algebraic equation system is derived and solving the equation system is quite sim?ple rather than solving a non-linear PDE. The obtained values of our method are checked against the other numerical and analytic solution of considered equation in the literature and the results are visualized by using graphics and tables so as to reveal whether the method is effectiveness or not. The obtained results evince that the wavelet method is quite proper because of its simple algorithm, high accuracy and less CPU time for solving the considered equation.