Trigonometric Function Solutions Research Articles

Analytical solutions of the (2+1)-dimensional space–time fractional Bogoyavlenskii’s breaking soliton equation

In this paper, the improved fractional sub-equation method is applied to obtain analytical solutions of the (2+1)-dimensional space–time fractional Bogoyavlenskii’s breaking soliton equation involving Jumarie’s modified Riemann–Liouville derivative in mathematical physics. As a result, explicit analytical solutions are successfully obtained, which contain more generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions. The graphical representations have been used for demonstrating the nature of the obtained results.

The $$\phi ^{6}$$ ϕ 6 -model expansion method for solving the nonlinear conformable time-fractional Schrödinger equation with fourth-order dispersion and parabolic law nonlinearity

The $$\phi ^{6}$$ -model expansion method combined with the conformable time-fractional derivative is applied in this paper for finding many new exact solutions including Jacobi elliptic function solutions, solitary wave solutions, trigonometric function solutions and other solutions to the nonlinear conformable time-fractional Schrodinger equation with fourth-order dispersion and parabolic law nonlinearity. This method presents a wider applicability for handling the nonlinear partial differential equations. Comparing our results with the well-known results are given.

Improved tan $$\left( {\frac{{\phi \left(\varvec{\xi}\right)}}{2}} \right)$$ ϕ ξ 2 -expansion method for (2 + 1)-dimensional KP–BBM wave equation

An improved tan $$\left( {\frac{\phi \left( \xi \right)}{2}} \right)$$ -expansion method has been implemented to solve (2 + 1)-dimensional KP–BBM wave equation. Several types of exact particular solutions with parameters are obtained. Those includes hyperbolic function solutions, trigonometric function solutions, and rational solutions. From our work it is evident that the improved technique provide more verities of solutions and that too with less computational work.

Abundant closed form solutions of the conformable time fractional Sawada-Kotera-Ito equation using (G′/G)-expansion method

In current study, we use the (G′/G)-expansion method to construct the closed form solutions of the seventh order time fractional Sawada-Kotera-Ito (TFSKI) equation based on conformable fractional derivative. As a result, trigonometric, hyperbolic and rational functions solutions with arbitrary constants are obtained. When the arbitrary constants are taken some special values, the periodic and soliton solutions are obtained from the travelling wave solutions. The obtained solutions are new and not found elsewhere. The effect of the fractional order on some of these solutions are represented graphically to illustrate the behavior of the exact solutions when the parameter take some special choose.

New closed form soliton and other solutions of the Kundu–Eckhaus equation via the extended sinh-Gordon equation expansion method

In this paper, the extended sinh-Gordon equation expansion method (ShGEEM) is applied to look for new complex hyperbolic and complex trigonometric function solutions, especially dark, bright, combined dark–bright, singular, combined singular soliton and other soliton solutions are obtained of the Kundu–Eckhaus (KE) equation which demonstrates the physical explanation in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. By choosing randomly the special values of free parameters, the solitary wave solutions are shown different patterns. The reported results might play an important role in the quantum field theory and nonlinear optics for explaining the physical meaning of the studied model. The obtained results also demonstrate that the extended ShGEEM is more competent than the other applied method applied by authors in the literature. Therefore, it is shown that the applied method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations arises in mathematical physics and nonlinear optics.

Soliton solutions of fractional complex Ginzburg–Landau equation with Kerr law and non-Kerr law media

In this study, new soliton solutions of the fractional complex Ginzburg–Landau equation, that models soliton propagation in the presence of detuning factor, have been constructed. The Kerr law, power law, dual-power law and log law nonlinearity have been considered. The exp(−ϕ(ξ))-expansion method has been utilized for finding new exact solutions of fractional complex Ginzburg–Landau equation. Different forms of solutions, including the hyperbolic, trigonometric and rational function solutions are formally extracted. The method suggests a useful and efficient technique to look for the exact solutions of a wide range of nonlinear fractional partial differential equations.

Some new analytical solutions of the variant Boussinesq equations

In this manuscript, the exact analytical solutions for the variant Boussinesq equations are constructed by utilizing the improved $$\tanh (\phi (\xi )/2)$$ -expansion method (ITEM) and the improved $$(G'{/}G)$$ -expansion method (IGEM). Using wave transformation, we converted the problem under consideration to an ordinary differential equation and then employed the ITEM and IGEM for hyperbolic function, trigonometric function and rational function solutions. The Mathematica software is used throughout for the solution of the system of algebraic equations obtained along the way and also for the graphical illustrations, respectively.

Complexiton and solitary wave solutions of the coupled nonlinear Maccari’s system using two integration schemes

In this paper, we consider a coupled nonlinear Maccari’s system (CNMS) which describes the motion of isolated waves localized in a small part of space. There are some integration tools that are adopted to retrieve the solitary wave solutions. They are the modified F-Expansion and the generalized projective Riccati equation methods. Topological, non-topological, complexiton, singular and trigonometric function solutions are derived. A comparison between the results in this paper and the well-known results in the literature is also given. The derived structures of the obtained solutions offer a rich platform to study the nonlinear CNMS. Numerical simulation of the obtained solutions are presented with interesting figures showing the physical meaning of the solutions.

Traveling Wave Solutions of Two Nonlinear Wave Equations by (G′/G)-Expansion Method

We employ the (G′/G)-expansion method to seek exact traveling wave solutions of two nonlinear wave equations—Padé-II equation and Drinfel’d-Sokolov-Wilson (DSW) equation. As a result, hyperbolic function solution, trigonometric function solution, and rational solution with general parameters are obtained. The interesting thing is that the exact solitary wave solutions and new exact traveling wave solutions can be obtained when the special values of the parameters are taken. Comparing with other methods, the method used in this paper is very direct. The (G′/G)-expansion method presents wide applicability for handling nonlinear wave equations.

Exact solutions of perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity by improved $${\textbf{tan}} \left( {\frac{{\boldsymbol{\phi}} \left( {\boldsymbol{\xi}} \right)}{{\textbf{2}}}} \right)$$tanϕξ2-expansion method

This paper carries out exact solutions of the perturbed nonlinear Schodinger’s equation withy Kerr law nonlinearity by using the improved tan $$\left( {\frac{\phi \left( \xi \right)}{2}} \right)$$ -expansion method. The exact solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. The method appears to be easier and faster by means of symbolic computational system and can be applied to the other nonlinear evolution equations in mathematical physics.

Exact Solutions for nonlinear partial differential equation by modified F-expansion method

In this Paper, we use modified F-expansion method to construct new exact traveling wave solutions of nonlinear partial differential equation, The (2+1) – dimensional zoomeron equation, The obtained solutions include soliton – like solutions, trigonometric function solutions, rational solutions and exponential solutions.

Stability Analysis, Numerical and Exact Solutions of the (1+1)-Dimensional NDMBBM Equation

A newly propose mathematical approach is presented in this study. We utilize the new approach in investigating the solutions of the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation. The new analytical technique is based on the popularly known sinh-Gordon equation and a wave transformation. In developing this new technique at each every steps involving integration, the integration constants are considered to not be zero which gives rise to new form of travelling wave solutions. The (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony is used in modelling an approximation for surface long waves in nonlinear dispersive media. We construct some new trigonometric function solution to this equation. Moreover, the finite forward difference method is utilized in investigating the numerical behavior of this equation by taking one of the obtained analytical solutions into consideration. We finally, give a comprehensive conclusions.

New travelling wave solutions for fractional regularized long-wave equation and fractional coupled Nizhnik-Novikov-Veselov equation

In this paper, solitary-wave ansatz and the (G?/G)-expansion methods have been used to obtain exact solutions of the fractional regularized long-wave (RLW) and coupled Nizhnik-Novikov-Veselov (NNV) equation. As a result, three types of exact analytical solutions such as rational function solutions, trigonometric function solutions, hyperbolic function solutions are formally derived these equations. Proposed methods are more powerful and can be applied to other fractional differential equations arising in mathematical physics.

Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations

In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the exp(-phi (z))-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.

Implementation of three reliable methods for finding the exact solutions of (2 + 1) dimensional generalized fractional evolution equations

In this study, we have implemented the three methods namely extended \((G^{\prime}/G)\)-expansion, extended \((1/G^{\prime})\)-expansion and \((G^{\prime}/G,\,\,1/G)\)-expansion methods to determine exact solutions for the (2 + 1) dimensional generalized time–space fractional differential equations. We use Conformable fractional derivative and its properties in this research to convert fractional differential equations to ordinary differential equations with integer order. By using above mentioned methods, three types of traveling wave solutions are successfully obtained which have been expressed by the hyperbolic, trigonometric, and rational function solutions. The considered methods and transformation techniques are efficient and consistent for solving nonlinear time and space-fractional differential equations.

Variable separation method for a nonlinear time fractional partial differential equation with forcing term

In this paper, a variable-coefficient nonlinear time fractional partial differential equation (PDE) with initial and boundary conditions is solved by using the variable separation method. As a result, some new explicit and exact solutions of the time fractional PDE are obtained including Airy function solution, hyperbolic function solution, trigonometric function solution and rational solution. It is shown that the variable separation method can provide a useful mathematical tool for solving some other nonlinear time fractional PDEs in science and engineering.

Analytical solutions of Phi-four equation

This study bases attention on new analytical solutions of Phi-four equation. The modified exp -expansion function method (MEFM) has been used to obtain analytical solutions of the Phi-four equation. By using this method, dark soliton solutions and trigonometric function solution of the Phi-four equation have been found.

The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative

In this paper, we construct exact solutions for the space–time nonlinear conformable fractional Bogoyavlenskii equations by using the first integral method, and with the help of Maple. As a result, generalized hyperbolic function solutions, generalized trigonometric function solutions and rational function solutions with free and deformation parameters are obtained. The method is very suitable, easy and effective handling of the solution process of nonlinear conformable fractional equations.

Jacobi elliptic solutions, solitons and other solutions for the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity

Based on several integration tools, namely the Riccati equation method, the Bernoulli equation method, the extended auxiliary equation method, the new mapping method and the $\phi^{6}$ -model expansion method, we obtain many exact solutions including the optical bright-dark-singular soliton solutions, Jacobi elliptic solutions and trigonometric function solutions of the nonlinear Schrodinger equation (NLSE) with fourth-order dispersion and cubic-quintic nonlinearity, self-steeping and self-frequency shift effects which describes the propagation of an optical pulse in optical fibers. We compare the results yielding from these integration tools together with each others. Also, a comparison between our results in this paper and the well-known results are given.

On the exact solutions of the fractional (2+1)- dimensional Davey - Stewartson equation system

In this work, we construct the exact traveling wave solutions of the fractional (2+1)-dimensional Davey-Stewartson equation system (D-S) that is complex equation system using the Modified Trial Equation Method (MTEM). We obtained trigonometric function solutions by this method that are new in literature.