Exact Solutions Of Nonlinear Equations Research Articles

Resonant excitation of the bushes of nonlinear vibrational modes in monoatomic chains

We present methods for excitation of bushes of nonlinear normal modes in the dynamical system, whose symmetry is described by any point or space group, with the aid of acting on the individual atoms periodic external forces with the frequency of some normal mode of the considered system. It is shown that these methods allow one to excite all bushes of nonlinear modes possible in this system. The application of these methods is illustrated by the excitation of bushes in the chains with the Lennard-Jones interatomic interactions. Each bush corresponds to some exact solution of non-linear equations of motion, which lies on a certain invariant manifold distinguished by symmetry of the considered dynamical system. The value of the concept of bushes of nonlinear modes is determined by the fact that they do not depend on the interatomic interactions, as well as on the degree of nonlinearity. Questions often arising in studying the bush theory, in particular, the problem of their stability, are discussed.

Cauchy invariants and exact solutions of nonlinear equations of hydrodynamics

Cauchy invariants and exact solutions of nonlinear equations of hydrodynamics

Some new exact solutions of (4+1)-dimensional Davey–Stewartson-Kadomtsev–Petviashvili equation

Exact solutions of nonlinear equations have got formidable attraction of researchers because these solutions demonstrate the physical behaviour of a model. In this paper, we focus on extracting some new exact solutions of a (4+1)-dimensional Davey–Stewartson-Kadomtsev–Petviashvili (DSKP) equation. To find new travelling wave solutions of the DSKP equation, we use (G′G′+G+A)-expansion technique. The obtained solutions are in the form of the exponential and trigonometric functions. We obtain different kinds of waves solutions for specific values of parameters. We simulate the achieved solutions in 3D and 2D plots.

Retraction Note to: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations

Retraction Note to: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations

Exact solutions of nonlinear delay reaction–diffusion equations with variable coefficients

A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized form of nonlinear equations of reaction-diffusion type with delay and which are nonlinear and associated with variable coefficients. A novel technique is used in this study to obtain the exact solutions which are new and are of the form of traveling-wave solutions. Arbitrary functions are present in the solutions and they also contain free parameters, which make them suitable for usage in solving certain modeling problems, testing numerical and approximate analytical methods. The results of this study also find applications in obtaining the exact solutions of other forms of partial differential equations which are more complex. Specific examples of nonlinear equations of reaction-diffusion type with delay are given and their exact solutions are presented. Solutions of certain reaction-diffusion equations are also displayed graphically.

Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

Exact Solutions of Nonlinear Equations in Mathematical Physics via Negative Power Expansion Method

Some Methods about Finding the Exact Solutions of Nonlinear Modified BBM Equation

Finding exact solutions of nonlinear equations plays an important role in nonlinear science, especially in engineering and mathematical physics. In this paper, we employed the complex method to get eight exact solutions of the modified BBM equation for the first time, including two elliptic function solutions, two simply periodic solutions, and four rational function solutions. We used the exp − ϕ z -expansion methods to get fourteen forms of solutions of the modified BBM equation. We also used the sine-cosine method to obtain eight styles’ exact solutions of the modified BBM equation. Only the complex method can obtain elliptic function solutions. We believe that the complex method presented in this paper can be more effectively applied to seek solutions of other nonlinear evolution equations.

Analytical Methods for Nonlinear Evolution Equations in Mathematical Physics

In this article, we will apply some of the algebraic methods to find great moving solutions to some nonlinear physical and engineering questions, such as a nonlinear (1 + 1) Ito integral differential equation and (1 + 1) nonlinear Schrödinger equation. To analyze practical solutions to these problems, we essentially use the generalized expansion approach. After various W and G options, we get several clear means of estimating the plentiful nonlinear physics solutions. We present a process like-direct expansion process-method of expansion. In the particular case of W′=λG, G′=μW in which λ and μ are arbitrary constants, we use the expansion process to build some new exact solutions for nonlinear equations of growth if it fulfills the decoupled differential equations.

RETRACTED ARTICLE: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations

The main objective of this paper is to solve non-linear equations for engineering and science applications. Various emerging engineering and science applications are using non-linear equation to represent the entire problem. Several earlier research works have been focused on solving linear and non-linear equation using different methods but they are application based. In recent days applications are described and represented mathematically in accordance to the non-linear, discreate and non-discrete data. Few research works have used homogeneous balance method for solving multi-dimension equations. Some of the earlier approaches have used homogeneous balance method for investigating WBKL, Kaup–Kupershmidt, and CSNLPD equations, whereas all the methods are not complexity reduced. In order to provide a complexity reduced, fast and highly suitable method for solving non-linear equations representing recent engineering applications. Also, it is well known that the homogeneous balance method is a powerful technique to symbolically compute traveling wave solutions of some nonlinear wave and evolution equations. Hence, in this paper the exact solutions of Gardner equation and Burgers equation are obtained by using homogeneous balance method and verified using MATHEMATICA.

On one method for constructing exact solutions of nonlinear equations of mathematical physics

A new method for constructing exact solutions of nonlinear equations of mathematical physics, which is based on nonlinear integral type transformations in combination with the splitting principle, is proposed. The effectiveness of the method is illustrated on nonlinear equations of the reaction-diffusion type, which depend on two or three arbitrary functions. New exact functional separable solutions and generalized traveling wave solutions are described.

On One Method for Constructing Exact Solutions of Nonlinear Equations of Mathematical Physics

On One Method for Constructing Exact Solutions of Nonlinear Equations of Mathematical Physics

Admissible equivalence transformations for linearization of nonlinear wave type equations

In the present paper, we consider a general family of two‐dimensional wave equations, which represents a great variety of linear and nonlinear equations within the framework of the transformations of equivalence groups. We have investigated the existence problem of point transformations that lead mappings between linear and nonlinear members of particular families and determined the structure of the nonlinear terms of linearizable equations. We have also given examples about some equivalence transformations between linear and nonlinear equations and obtained exact solutions of nonlinear equations via the linear ones.

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp (−φ(η))-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (−φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.

A generalized Kudryashov method to some nonlinear evolution equations in mathematical physics

Nonlinear evolution equations form the most fundamental theme in mathematical physics. The search for exact solutions of nonlinear equations has been of interest in recent years. In this paper, we obtain exact solutions of the nonlinear Jaulent–Miodek hierarchy and (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff equation by using the generalized Kudryashov method. All calculations in this study have been made and checked back with the aid of the Maple packet program.

A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

Traveling Wave Solution of (3+1) Dimensional Potential-YTSF Equation by Bernoulli Sub-ODE Method

In this paper, we derive exact traveling wave soluti-ons of (3+1) dimensional potential-YTSF equation by a proposed Bernoulli sub-ODE method. The method appears to be efficient in seeking exact solutions of nonlinear equations. We also make a comparison between the present method and the known (G’/G) expansion method.

Traveling Wave Solution of (2+1) Dimensional Breaking Soliton Equation by Bernoulli Sub-ODE Method

In this paper, we derive exact traveling wave soluti-ons of (2+1) dimensional breaking soliton equation by a proposed Bernoulli sub-ODE method. The method appears to be efficient in seeking exact solutions of nonlinear equations. We also make a comparison between the present method and the known (G’/G) expansion method.

A class of exact solutions of second grade fluid

This paper describes a class of exact solutions of the equations of motion for an unsteady, incompressible non-Newtonian second grade fluid in the Cartesian co-ordinates. The exact solutions of non-linear equations governing the flow of second grade fluids are obtained through generalized separation variables method by assuming certain form of stream functions. The expressions for velocity profile and vorticity stream function are constructed. Key words: Exact solutions, vorticity functions, second grade fluids, separation of variables.

Generalized separation of variables and exact solutions of nonlinear equations

We consider a generalized procedure of separation of variables in nonlinear hyperbolic-type equations and Korteweg–de-Vries-type equations. We construct a broad class of exact solutions of these equations that cannot be obtained by the Lie method and method of conditional symmetries.

Exact Solutions of Nonlinear Equation of Rod Deflections Involving the Lauricella Hypergeometric Functions

The stress induced in a loaded beam will not exceed some threshold, but also its maximum deflection, as for all the elastic systems, will be controlled. Nevertheless, the linear beam theory fails to describe the large deflections; highly flexible linear elements, namely, rods, typically found in aerospace or oil applications, may experience large displacements—but small strains, for not leaving the field of linear elasticity—so that geometric nonlinearities become significant. In this article, we provide analytical solutions to large deflections problem of a straight, cantilevered rod under different coplanar loadings. Our researches are led by means of the elliptic integrals, but the main achievement concerns the Lauricella hypergeometric functions use for solving elasticity problems. Each of our analytic solutions has been individually validated by comparison with other tools, so that it can be used in turn as a benchmark, that is, for testing other methods based on the finite elements approximation.