Analytical and Approximate Solutions for Complex Nonlinear Schr?dinger Equation via Generalized Auxiliary Equation and Numerical Schemes

This article studies the performance of analytical, semi-analytical and numerical scheme on the complex nonlinear Schr¨odinger(NLS) equation. The generalized auxiliary equation method is surveyed to get the explicit wave solutions that are used to examine the semi-analytical and numerical solutions that are obtained by the Adomian decomposition method, and B-spline schemes(cubic, quantic, and septic). The complex NLS equation relates to many physical phenomena in different branches of science like a quantum state, fiber optics, and water waves. It describes the evolution of slowly varying packets of quasi-monochromatic waves, wave propagation, and the envelope of modulated wave groups, respectively. Moreover, it relates to Bose-Einstein condensates which is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. Some of the obtained solutions are studied under specific conditions on the parameters to constitute and study the dynamical behavior of this model in two and three-dimensional.

New exact solutions for the time fractional coupled Boussinesq-Burger equation and approximate long water wave equation in shallow water

The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq-Burger and approximate long water wave equations by using the generalized Kudryashov method.The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann-Liouville derivative sense.Applying the generalized Kudryashov method through with symbolic computer maple package,numerous new exact solutions are successfully obtained.All calculations in this study have been established and verified back with the aid of the Maple package program.The executed method is powerful,effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.

Semi-analytical and numerical simulations of the modified Benjamin-Bona-Mahony model

In this article,semi-analytical and numerical simulations of the well-known modified Benjamin-Bona-Mahony(mBBM)equation are processed.This study targets to check the accuracy of the obtained analytical solutions of the mBBM model that have been obtained in[1]through three recent analytical schemes(extended simplest equation(ESE)method,modified kudryashov(mKud)method,and SechTanh(ST)expansion method).The considered model describes the propagation of long waves in the nonlinear dispersive media in a visual illusion.The Homotopy iteration(HI)method,exponential cubic-B-spline(ECBS)method,and trigonometric-quantic-B-spline(TQBS)method are employed to construct novel semi-analytical and accurate numerical solutions.The obtained solutions’accuracy has been checked through some different types of two-dimensional graphs.

Solitary wave solutions for the generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation

In this paper,we utilize the exp(−ϕ(ξ))-expansion method to find exact and solitary wave solutions of the generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation.The generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony nonlinear evolution equation describes the model for the propagation of long waves that mingle with nonlinear and dissipative impact.This model is used in the analysis of the surface waves of long wavelength in hydro magnetic waves in cold plasma,liquids,acoustic waves in harmonic crystals and acoustic-gravity waves in compressible fluids.By using this method,seven different kinds of traveling wave solutions are successfully obtained for this model.The considered method and transformation techniques are efficient and consistent for solving nonlinear evolution equations and obtain exact solutions that are applied to the science and engineering fields.

Numerical solutions of nonlinear fractional Wu-Zhang system for water surface versus three approximate schemes

This paper examines the effects of three distinct numerical schemes(Adomian Decomposition,quintic&septic Spline methods)to investigate semi-analytical and approximate solutions on Wu-Zhang(ZW)system.It describes the(1+1)-dimensional dispersive long wave in two horizontal directions on shallow waters.The ZW model is one of the fractional nonlinear partial differential equations.Conformable derivatives properties are employed to convert the nonlinear fractional partial differential equation into an ordinary differential equation with integer order so as to obtain the approximate solutions for this model.The solutions obtained for each technique were compared to reveal their relationship to their characteristics illustrated under the suitable choice of the parameters values.The obtained solutions showed the power,easiness,and effectiveness of these methods on nonlinear partial differential equations.

Plenty of analytical and semi-analytical wave solutions of shallow water beneath gravity

This article studies novel soliton wave solutions of the nonlinear fractional ill-posed Boussinesq(NLFIPB)dynamic wave equation by applying the extended Riccati-expansion(ERE)method.Jacques Hadamard has formulated the investigated model to figure out the dynamic characterizations of waves in shallow water under gravity.The obtained solutions are explained through some sketches in 2D and 3D and contour plots.At the same time,the results’accuracy is checked by comparing the obtained solutions with semianalytical solutions through the well-known Adomian decomposition(AD)method.The superiority of the ERE method over the original method is explained.All constructed solutions are checked by submitting them back into the original model through Mathematica 12 software.