Tanh Coth Method Research Articles

Comparison between the generalized tanh–coth and the (G′/G)-expansion methods for solving NPDEs and NODEs

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.

A new analytical approach to solve some of the fractional-order partial differential equations

The aim of the present paper is to present an analytical method for the time fractional biological population model, time fractional Burgers, time fractional Cahn–Hilliard, space–time fractional Whitham–Broer–Kaup, space–time fractional Fokas equations by using the generalized tanh–coth method. The fractional derivative is described in the sense of the modified Riemann–Liouville derivatives. The method gives an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. We have obtained the exact solutions for the aforementioned nonlinear fractional equations. A generalized fractional complex transform is appropriately used to convert these fractional equations to ordinary differential equations which subsequently resulted into number of exact solutions.

New exact solutions for the Khokhlov-Zabolotskaya-Kuznetsov, the Newell-Whitehead-Segel and the Rabinovich wave equations by using a new modification of the tanh-coth method

The family of the tangent hyperbolic function methods is one of the most powerful method to find the solutions of the nonlinear partial differential equations. In the mathematical literature, there are a great deal of tanh-methods completing each other. In this article, the unified tanh-function method as a unification of the family of tangent hyperbolic function methods is introduced and implemented to find traveling wave solutions for three important physical models, namely the Khoklov–Zabolotskaya–Kuznetsov (KZK) equation, the Newell–Whitehead–Segel (NWS) equation, and the Rabinovich wave equation with nonlinear damping. Various exact traveling wave solutions of these physical structures are formally derived.

Exact Solutions of Gardner Equations through tanh-coth Method

In this paper, we apply the tanh-coth method and traveling wave transformation method for solving Gardner equations, including (1 + 1)-Gardner and (2 + 1)- Gardner equations. The tanh-coth method proved to be reliable and effective in handling a large number of nonlinear dispersive and disperse equations. Through tanh-coth method, we get analytical expressions of soliton solutions of Gardner equations. The one-soliton solution is characterized by an infinite wing or infinite tail.

The solution of the sinh-Gordon wave equation using lattice-Boltzmann and the tanh-coth method

The solution of the sinh-Gordon wave equation using lattice-Boltzmann and the tanh-coth method

Construction of exact solutions to the modified forms of DP and CH equations by analytical methods

In this work, we establish the exact solutions to the modified forms of Degasperis–Procesi (DP) and Camassa–Holm (CH) equations. The generalized (G’/G)-expansion and generalized tanh-coth methods were used to construct solitary wave solutions of nonlinear evolution equations. The generalized (G’/G)-expansion method presents a wider applicability for handling nonlinear wave equations. It is shown that the (G’/G)-expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

Construction of exact solutions to the modified forms of DP and CH equations by analytical methods

In this work, we establish the exact solutions to the modified forms of Degasperis–Procesi (DP) and Camassa–Holm (CH) equations. The generalized (G’/G)-expansion and generalized tanh-coth methods were used to construct solitary wave solutions of nonlinear evolution equations. The generalized (G’/G)-expansion method presents a wider applicability for handling nonlinear wave equations. It is shown that the (G’/G)-expansion method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

NEW TRAVELLING WAVE SOLUTIONS FOR STOCHASTIC CUBIC NON LINEAR KLEIN-GORDON EQUATION

This paper is devoted to give white noise functional solutions for stochastic cubic nonlinear Klein-Gordon equation. Abundant exact travelling wave solutions for Wick-type generalized stochastic Klein-Gordon equation are obtained. By using white noise analysis, Hermite transform, modified Riccati equation and modified tanh-coth method many exact travelling wave solutions are given. Detailed computations and implemented examples for the investi- gated model are explicitly provided.

A nonlinear fractional Sharma–Tasso–Olver equation: New exact solutions

Due to its importance in scientific applications the fractional nonlinear Sharma–Tasso–Olver equation has been studied by several authors from point of view of exact solutions. In this paper, new explicit and exact solutions are obtained by using the fractional complex transform and the improved extended tanh-coth method. The results can be compared with other methods used to solve the same model.

Exact travelling wave solutions of a variety of Boussinesq-like equations

In this paper, we obtain exact traveling wave solutions of a variety of Boussinesq-like equations by using two distinct methods with symbolic computation. The Boussinesq equations play an important role in physical applications, such as in nonlinear lattice waves, acoustic waves, iron sound waves in a plasma, and vibrations in a nonlinear string. More precisely, the modified tanh-coth method is employed to obtain single soliton solutions, and the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions. Further, it is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions. The employed approaches are quite efficient for the determination of the solutions, and are practically well suited for solving nonlinear evolution equations arising in physics.

New Travelling Wave Solutions of Two Nonlinear Physical Models by Using a Modified Tanh-Coth Method

In this work, a modified tanh – coth method is used to derive travelling wave solutions for (2 + 1)-dimensional Zakharov-Kuznetsov (ZK) equation and (3 + 1)-dimensional Burgers equation. A new variable is used to solve these equations and established new travelling wave solutions.

The generalized tanh-coth method applied to biological model referent to nano-solitons of ionic wave

Based on two previous reports related with soliton-solutions for two equations: the rst a biological model and the other a physic model, we consider the generalized tanh-coth method for obtain exact solutions for the rst in a more general form that those obtained in previous reports. Additionally, we solve a second nonlinear equation whose relevance consists on the fact that solutions of it can be used for obtain solutions of the rst and to other equation related with nano-ionic currents associated with transmission lines. Finally, some conclusion are given.

Exact solution of the Bogoyavlenskii equation using the improved tanh-coth method

An improved tanh-coth method is used to obtain exact solutions of the Bogoyavlenskii equation. The obtained solutions are given in a more general form than those obtained in previous literature. One of the most important facts in the paper is that we consider the arbitrary constant derived from a rst integration when the equation is ordinary which is a frequent error in many works which use similar computational methods. Mathematics Subject Classication: 35C05

Exact Solutions for the Integrable Sixth-Order Drinfeld-Sokolov-Satsuma-Hirota System by the Analytical Methods.

We establish exact solutions including periodic wave and solitary wave solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system. We employ this system by using a generalized (G′/G)-expansion and the generalized tanh-coth methods. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

Exact Solutions for the KdV Equation with Forcing Term by the Generalized tanh-coth Method and the (G′/G)-Expansion Method

An application of the generalized tanh-coth method and the (G′/G)-expansion method to search for exact solutions of nonlinear partial differential equations is analyzed. These methods are used for the KdV equation with forcing term. The generalized tanh-coth method and the (G′/G)-expansion method were used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. It is shown that the generalized tanh-coth method and the (G′/G)-expansion method, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear problems.

Closed form solutions for a generalized Benjamin–Bona–Mahony–Burgers equation with higher-order nonlinearity

We obtain closed form solutions for a generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation with higher-order nonlinearity which belong to a class of so called nonlinear pseudo-parabolic or type Sobolev equations. The generalized tanh–coth method is used here; with the aid of Mathematica, periodic and soliton solutions are formally derived. Finally some conclusions are given.

Exact travelling wave solutions of the symmetric regularized long wave (SRLW) using analytical methods

In this article, we establish exact travelling wave solutions of the symmetric regularized long wave (SRLW) by using analytical methods. The analytical methods are: the tanh-coth method and the sech 2 method which used to construct solitary wave solutions of nonlinear evolution equations. With the help of symbolic computation, we show that aforementioned methods provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

Travelling Wave Solutions for the Coupled IBq Equations by Using the tanh-coth Method

Based on the availability of symbolic computation, the tanh-coth method is used to obtain a number of travelling wave solutions for several coupled improved Boussinesq equations. The abundant new solutions can be seen as improvement of the previously known data. The obtained results in this work also demonstrate the efficiency of the method.

On a KdV6 with forcing term: exact solutions

The homogeneous sixth-order KdV equation (KdV6) has been studied by several authors from the point of view of its integrability and exact solutions. However, in the case that a forcing term F(t) is considered, the results presented here are new. In this work, the improved generalized tanh-coth method is used to obtain exact solutions to the sixth-order bidirectional Kaup–Kupershmidt equation with forcing term (bKKf).

Exact Solutions for Stochastic Fractional Zakharov-Kuznetsov Equations

The main aim of this paper is to give exact solutions for the space-time-fractional Zakharov-Kuznetsov equation modified with variable coefficients. Our proposed technique is based on employing white noise analysis, Hermite transform, the homogeneous balance principle, and a modified tanh-coth method. Some white noise functional solutions for Wick-type stochastic space-time-fractional Zakharov-Kuznetsov equations with variable coefficients are established. Detailed computations and an implemented example are explicitly provided.