Modified Simple Equation Method Research Articles

Assorted Spatial Optical Dynamics of a Generalized Fractional Quadruple Nematic Liquid Crystal System in Non-Local Media

Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model.

Lump, interaction of lump and kink and solitonic solution of nonlinear evolution equation which describe incompressible viscoelastic Kelvin–Voigt fluid

In this script, we consider the modified Oskolkov equation in incompressible viscoelastic Kelvin–Voigt fluid and fluid dynamics. A dominant direct algebraic method namely modified simple equation method (MSE) uses to retrieve various dynamical structural solutions of the nonlinear models. As a results, we can derive a disguise version of analytic structural solutions in exponential, hyperbolic, trigonometric and Jacobi elliptic functions with some appreciate parameters. In this work, we depict the physical explanation of cross periodic lump wave, interaction of lump and kink, lump and anti-kink shape solution, double periodic wave, kink, anti-kink shape solution, disguise version of soliton solutions with 3D contour plot and 2D plot.

Impressive and accurate solutions to the generalized Fokas-Lenells model

In this article, we utilize the generalized full nonlinearity perturbed complex Fokas-Lenells model (GFLM) which is a general dynamics representation of modern electronic communications "Internet blogs, Facebook communication and Twitter comments". The modified simple equation method (MSEM) has been applied effectively to generate closed form solution. On the other hand, the Riccati-Bernoulli Sub-ODE method (RPSOM) which reduces the steps of calculation has been applied perfectly to achieve accurate solution to this equation. We established the solutions achieved by these distinct manners in same vein and parallel.

New perception of the exact solutions of the 3D-fractional Wazwaz-Benjamin-Bona-Mahony (3D-FWBBM) equation

In this article we will utilize new perception of the Benjamin-Bona-Mahony equation that represents developed stretch for the Korteweg-de Varies equation which denotes to the unidirectional propagation of small amplitude long waves on the surface of hydro magnetic and acoustic waves in channel significantly for shallow water. The main idea of this research realizing new exact and hence solitary solutions of this new perception with its spatial and temporal variables and fractional order whose called 3D-fractional Wazwaz-Benjamin-Bona-Mahony model (3D -FWBBM). Two important different techniques are invited for the first time to proposed this new perception of the exact solutions, the first one is the modified simple equation method (MSEM) and the second is the Riccati-Bernolli Sub-ODE method (RBSOM) whose are used effectively to obtain accurate description of the wave solutions to this equation.

Soliton solutions to voltage analysis in nonlinear electrical transmission lines and electric signals in telegraph lines

Soliton solutions of couple of nonlinear evolution equations (NLEEs) ascending for voltage analysis in nonlinear electric transmission lines and electric signals in telegraph lines have been investigated through the modified simple equation (MSE) method. We establish general closed form soliton solutions associated with transmission line parameters and integral constants, for definite values of which some of the existing solutions in the literature are rediscovered and some new solutions are originated. The 3D and contour plots of the obtained solutions are depicted to illustrate the voltage characteristics in the transmission line and in the optical fibers. The obtained solutions might play important role in science and engineering fields. It is noteworthy to observe that the MSE method is easy, compatible and powerful mathematical tool to extract soliton solutions.

Chiral Solitons with Bohm Potential by Modified Simple Equation Method and Trial Equation Scheme

Chiral Solitons with Bohm Potential by Modified Simple Equation Method and Trial Equation Scheme

Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications

AbstractThe aim of this article is to construct some new traveling wave solutions and investigate localized structures for fourth-order nonlinear Ablowitz-Kaup-Newell-Segur (AKNS) water wave dynamical equation. The simple equation method (SEM) and the modified simple equation method (MSEM) are applied in this paper to construct the analytical traveling wave solutions of AKNS equation. The different waves solutions are derived by assigning special values to the parameters. The obtained results have their importance in the field of physics and other areas of applied sciences. All the solutions are also graphically represented. The constructed results are often helpful for studying several new localized structures and the waves interaction in the high-dimensional models.

Closed form traveling wave solutions of non-linear fractional evolution equations through the modified simple equation method

In this article, the modified simple equation (MSE) method is introduced to examine the closed form wave solutions of the fractional non-linear Cahn-Allen equation and of the fractional generalized reaction Duffing equation. The fractional derivatives are delineated in the sense of Jumarie?s modified Riemann-Liouville derivative. A fractional complex transformation is used to transform the fractional-order PDE into integer order ODE. The reduced equations are then examined by using the MSE method and some new and further general solutions of these equations are successfully established. The approach of this method is simple, standard and the obtained solutions are highly encouraging. It is also powerful, reliable and effective.

Closed Form Exact Solutions to the Higher Dimensional Fractional Schrodinger Equation via the Modified Simple Equation Method

In this article, we investigate some exact wave solutions to the higher dimensional time-fractional Schrodinger equation, an important equation in quantum mechanics. The fractional Schrodinger equation further precisely describes the quantum state of a physical system changes in time. In order to determine the solutions a suitable transformation is considered to transmute the equations into a simpler ordinary differential equation (ODE) namely fractional complex transformation. We then use the modified simple equation (MSE) method to obtain new and further general exact wave solutions. The MSE method is more powerful and can be used in other works to establish completely new solutions for other kind of nonlinear fractional differential equations arising in mathematical physics. The affect of obtaining parameters for its definite values which are examined from the solutions of two dimensional and three dimensional time-fractional Schrodinger equations are discussed and therefore might be useful in different physical applications where the equations arise in this article.

New exact traveling wave solutions of the (4+1)-dimensional Fokas equation

In this paper, two distinct methods are applied to look for exact traveling wave solutions of the (4+1)-dimensional nonlinear Fokas equation, namely the modified simple equation method (MSEM) and the extended simplest equation method (ESEM). Some new exact traveling wave solutions involving some parameters are obtained. The solitary wave solutions can be extracted by assigning special values of these parameters. The obtained solutions show the simplicity and efficiency of the used approaches that can be applied for nonlinear equations as well as linear ones.

New solitary wave in shallow water, plasma and ion acoustic plasma via the GZK-BBM equation and the RLW equation

This work obtains the disguise version of exact solitary wave solutions of the generalized (2+1)-dimensional Zakharov-Kuznetsov–Benjamin-Bona-Mahony and the regularized long wave equation with some free parameters via modified simple equation method (MSE). Usually the method does not give any solution if the balance number is more than one, but we apply MSE method successfully in different way to carry out the solutions of nonlinear evolution equation with balance number two. Finally some graphical results of the velocity profiles are presented for different values of the material constants. It is shown that this method, without help of any symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear evolution equation.

Closed form solutions of complex wave equations via the modified simple equation method

AbstractThe Kundu–Eckhaus equation and the derivative nonlinear Schrodinger equation describe various physical processes in nonlinear optics, plasma physics, fluid mechanics, magneto-hydrodynamic equation in the presence of the Hall Effect. Thus, closed form solutions of these equations are very important to realize the obscurity of the phenomena. The modified simple equation (MSE) method is highly effective and competent mathematical tool to examine closed form wave solutions of nonlinear evolution equations (NLEEs) arising in mathematical physics, applied mathematics and engineering. In this article, the MSE method is suggested and executed to construct closed form wave solutions of the above-mentioned equations involving parameters. When the parameters receive special values, impressive solitary wave solutions are derived from the exact solutions.

A New Modification in Simple Equation Method and its applications on nonlinear equations of physical nature

Abstract The modification of the expansion method, namely, the New Modified Simple Equation Method is presented and subsequently implemented on nonlinear partial differential equations. The implementation of the new modification is demonstrated by solving Landau-Ginburg-Higgs equation and Cahn-Allen equation, as a result many new and more general multiple wave solution consisting of Hyperbolic function solutions, trigonometric function solutions and rational function solutions are obtained. Also double solitary wave solutions are found by substituting the different values for some parameters. The method appears to be straightforward and powerful by means of symbolic computational system.

Traveling Wave Solutions of Nonlinear Evolution Equations Via the Modified Simple Equation Method

Exact solutions of nonlinear evolution equations (NLEEs) are very crucial to realize the obscurity of many physical phenomena in mathematical science. The modified simple equation (MSE) method is especially effective and highly proficient mathematical instrument to obtaining exact traveling wave solutions to NLEEs arising in science, engineering and mathematical physics. Though the modified simple equation method effectively provides exact traveling wave solutions to nonlinear evolution equations for balance number less or equal 2 but the method did not applied to solve if the balance number is greater than 2. In this article, the MSE method is executed to construct exact traveling wave solutions of nonlinear evolution equations namely Kuramoto-Sivashinsky equation with balance number equal to 3. The obtained solutions are expressed in terms of exponential and trigonometric functions including solitary and periodic solutions. Moreover the procedure of this method reduces huge volume of calculations.

Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation Via the Modified Simple Equation Method

Solitary Wave Solutions to the ZKBBM Equation and the KPBBM Equation Via the Modified Simple Equation Method

Solitary Wave Solutions to the Sharma-Tasso-Olver Equation and the Similar Hirota-Satsuma KdV System through the Modified Simple Equation Method

Solitary Wave Solutions to the Sharma-Tasso-Olver Equation and the Similar Hirota-Satsuma KdV System through the Modified Simple Equation Method

The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations

Abstract The modified simple equation method, the exp-function method, and the method of soliton ansatz for solving nonlinear partial differential equations are presented. Based on these three different methods, we obtain the exact solutions and the bright–dark soliton solutions with parameters of the long-short wave resonance equations which describe the resonance interaction between the long wave and the short wave. When these parameters take special values, the solitary wave solutions are derived from the exact solutions. We compare the results obtained using the three methods. Also, a comparison between our results and the well-known results is given.

Exact solutions of the generalized ([formula omitted])-dimensional nonlinear evolution equations via the modified simple equation method

In this paper, the modified simple equation method (MSEM) is applied to construct exact solutions of the generalized (2+1)-dimensional nonlinear evolution equations (NLEEs) involving parameters via the (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation, the (2+1)-dimensional breaking soliton equation and the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. The solitary wave solutions are derived from the exact solutions by assigning special values of the parameters.

The Modified Simple Equation Method and Its Application to Solve NLEEs Associated with Engineering Problem

The Modified Simple Equation Method and Its Application to Solve NLEEs Associated with Engineering Problem

Exact and Solitary Wave Solutions to the Generalized Fifth-order KdV Equation by Using the Modified Simple Equation Method

Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solve NLEEs for balance number two. In order to verify the process, the generalized fifth-order KdV equation has been solved. By means of this scheme, we found some fresh traveling wave solutions to the above mentioned equation. When the parameters receive special values, solitary wave solutions are derived from the exact solutions. We analyze the solitary wave properties by the graphs of the solutions. This shows the validity, usefulness, and necessity of the process.