Solutions Of Nonlinear Evolution Equations Research Articles

New applications of the two variable (G′/G, 1/G)-expansion method for closed form traveling wave solutions of integro-differential equations

Most fundamental themes in mathematical physics and modern engineering are investigated by the closed form traveling wave solutions of nonlinear evolution equations. In our research, we ascertain abundant new closed form traveling wave solution of the nonlinear integro-differential equations via Ito equation, integro-differential Sawada–Kotera equation, first integro-differential KP hierarchy equation and second integro-differential KP hierarchy equation by two variable (G′/G, 1/G)-expansion method with the help of computer package like Mathematica. Some shape of solutions like, bell profile solution, anti-king profile solution, soliton profile solution, periodic profile solution etc. are obtain in this investigation. Trigonometric function solution, hyperbolic function solution and rational function solution are established by using our eminent method and comparing with our results to all of the well-known results which are given in the literature. By means of free parameters, plentiful solitary solutions are derived from the exact traveling wave solutions. The method can be easier and more applicable to investigate such type of nonlinear evolution models.

Exact Traveling Wave Solutions of Nonlinear Evolution Equations: Indeterminant Homogeneous Balance and Linearizability

Exact traveling (solitary) wave solutions of nonlinear partial differential equations (NLPDEs) are analyzed for third-order nonlinear evolution equations. These equations have indeterminant homogenous balance and therefore cannot be solved by the Power Index Method (PIM). Some evolution equations are linearizable where solutions are transferred from those of a linear PDE. For other evolution equations transforming to a NLPDE which has a homogenous balance gives rise to possible solutions by the PIM. The solutions for evolution equations that are not linearizable are developed here.

The closed form solutions of simplified MCH equation and third extended fifth order nonlinear equation

The investigation of closed form solutions for nonlinear evolution equations (NLEEs) is being an attractive subject in the different branches of mathematical and physical sciences. In this article, the enhanced (G′/G)-expansion method has been applied to find the closed form solutions for NLEEs, such as the simplified MCH equation and third extended fifth order nonlinear equations which are very important in mathematical physics. Plentiful closed form solutions with arbitrary parameters are successfully obtained by this method which are expressed in terms of hyperbolic and trigonometric functions. It is shown that the obtained solutions are more general and fresh and can be helpful to analyze the NLEES in mathematical physics and engineering problems.

Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications

In this paper, we introduce a generalized viscosity explicit method (GVEM) for nonexpansive mappings in the setting of Banach spaces and, under some new techniques and mild assumptions on the control conditions, prove some strong convergence theorems for the proposed method, which converge to a fixed point of the given mapping and a solution of the variational inequality. As applications, we apply our main results to show the existence of fixed points of strict pseudo-contractions and periodic solutions of nonlinear evolution equations and Fredholm integral equations. Finally, we give some numerical examples to illustrate the efficiency and implementation of our method.

Testing efficiency of the generalised $$\left( {{{G}'}/G} \right) $$ G ′ / G -expansion method for solving nonlinear evolution equations

In this investigation, we employ the generalised $$(G'{/}G)$$ -expansion method to test its efficiency in extracting travelling wave solutions of nonlinear evolution equations (NLEEs). As test cases, the modified Kuramoto–Sivashinsky (mK-S) and the modified Burgers–Korteweg–de Vries (mB-KdV) equations are considered because of their importance in soliton theory. The general solutions are obtained in hyperbolic, trigonometric and rational function forms for both the equations. Taking specific parametric values in the corresponding general solutions, some new exact travelling waves in trigonometric and hyperbolic forms and only in hyperbolic form are obtained for the mK-S and mB-KdV equations, respectively. The obtained results are checked to see whether the criticism made by Parkes (Comput. Fluids 42, 108 (2011)), that the so-called ‘new’ solutions derived by the $$(G'{/}G)$$ -expansion method are often erroneous and are merely disguised versions of previously known solutions, is justified also for the generalised $$(G'{/}G)$$ -expansion method. The solutions were checked with Maple by putting them back into their corresponding equations. With specific values of parameters, some of our obtained solutions satisfied directly and some solutions never satisfied the considered NLEEs. Among the satisfactory solutions, some are found to be in disguised versions of some results obtained in this study.

Simplest exp-function method for exact solutions of Mikhauilov-Novikov-Wang equations

In this paper, the simplest exp-function method which combines the exp-function method with a direct algorithm is used to exactly solve the Mikhauilov-Novikov-Wang equations. As a result, two explicit and exact solutions are obtained. It is shown that the simplest exp-function method provides a simpler but more effective mathematical tool for constructing exact solutions of non-linear evolution equations in fluids.

Applications of modified mathematical method on some nonlinear water wave dynamical models

The extended simple equation method is applied to construct solitary wave solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM), Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM), Breaking soliton (BS) and (2 + 1) Maccari system waves system of equations. These models have prevalent usage in modern science. This technique can also be functional to solve different kinds of nonlinear evolution problems in contemporary areas of research. It is an effective and powerful mathematical tool in finding solitary wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics.

Exact Solutions of Non-Linear Evolution Models in Physics and Biosciences Using the Hyperbolic Tangent Method

There has been considerable interest in seeking exact solutions of non-linear evolution equations that describe important physical and biological processes. Nonetheless, it is a difficult undertaking to determine closed form solutions of mathematical models that describe natural phenomena. This is because of their high non-linearity and the huge number of parameters of which they consist. In this article we determine, using the hyperbolic tangent (tanh) method, travelling wave solutions to non-linear evolution models of interest in biology and physics. These solutions have recognizable properties expected of other solutions and thus can be used to deduce properties of the general solutions.

Searching for traveling wave solutions of nonlinear evolution equations in mathematical physics

This paper deals with the analytical solutions for two models of special interest in mathematical physics, namely the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation and the (3+1)-dimensional generalized Boiti-Leon-Manna-Pempinelli equation. Using a modified version of the Fan sub-equation method, more new exact traveling wave solutions including triangular solutions, hyperbolic function solutions, Jacobi and Weierstrass elliptic function solutions have been obtained by taking full advantage of the extended solutions of the general elliptic equation, showing that the modified Fan sub-equation method is an effective and useful tool to search for analytical solutions of high-dimensional nonlinear partial differential equations.

Solitary and periodic wave solutions of nonlinear wave equations via the functional variable method

In this letter, the functional variable method has been discussed for finding the solitary and periodic wave solutions of nonlinear evolution equations (NLEEs). We use this method to construct aforementioned solutions to the coupled nonlinear Klein-Gordon (CNKG) equations and the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona Mahony (DMBBM) equation. Each of the obtained solutions contain an explicit function of the variables in the considered equations. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering fields without the help of computer algebra system.

Noncoercive mixed equilibrium problems under pseudomonotone perturbations and applications to nonlinear evolution equations with lack of coercivity

ABSTRACTIn this paper, we study the existence of solutions for noncoercive mixed equilibrium problems which are described by the sum of a maximal monotone bifunction and a pseudomonotone (or quasimonotone) bifunction in the sense of Brézis. Our approach is based on recession analysis and on recent results established by the authors for the existence of solutions of mixed equilibrium problems under pseudomonotone perturbations. As an application, we study the existence of solutions for nonlinear evolution equations associated with a noncoercive time-dependent pseudomonotone (or quasimonotone) operator.

Closed form Solutions of New Fifth Order Nonlinear Equation and New Generalized Fifth Order Nonlinear Equation via the Enhanced (G'/G)-expansion Method

Closed form solutions of nonlinear evolution equations (NLEEs) are very imperative in order to better understand the inner mechanism and complexity of complex physical phenomena. The enhanced (/)GGexpansion′− method is a effectual and proficient mathematical tool which can be used to discover the closed form solutions of NLEEs arising in mathematical physics, applied mathematics and engineering. In this article, the enhanced (/)GGexpansion′− method is recommended and carry out to investigate the closed form solutions of the new fifth order non-linear equation and the new generalized fifth order non-linear equation. The performance of this method is reliable, proficient and possible to obtain a lot of new exact solutions than the existing other methods.

Exact periodic and explicit solutions of higher dimensional equations with fractional temporal evolution

The term soliton has been used for a pulse like nonlinear wave (solitary wave) which leaves an interaction with unaltered shape and speed. To date, no less than seven particular wave frameworks or systems have been found to show such solutions. This speaks to an extensive variety of utilizations in applied science. The Exp(−ϕ(ξ))-expansion technique is utilized to find generalized solitary solutions and intermittent or periodic solutions for nonlinear evolution equations emerging in mathematical physics with the use of the enhanced time conformable equation. The technique is direct and succinct, and its applications are promising for other nonlinear mathematical physics.

Computing exact solutions for conformable time fractional generalized seventh-order KdV equation by using $${\left( {{\varvec{G}}}^{\prime }/{{\varvec{G}}}\right) }$$ G ′ / G -expansion method

In this paper, the authors have established the \(\left( G^{\prime }/G\right)\)-expansion method to find exact solutions for conformable time fractional generalized seventh-order KdV equation (FGKdV7). This method is an effective method in finding exact traveling wave solutions of nonlinear evolution equations in mathematical physics. The effectiveness of this manageable method has been shown by applying it to several particular cases of the FGKdV7. The present approach has the potential to be applied to other nonlinear fractional differential equations. All of the numerical calculations in the present study have been performed on a PC applying some programs written in Mathematica.

Iterative Method for a New Class of Evolution Equations with Non-instantaneous Impulses

In this paper, we are concerned with the existence of mild solutions for the initial value problem to a new class of abstract evolution equations with non-instantaneous impulses on ordered Banach spaces. The existence and uniqueness theorem of mild solution for the associated linear evolution equation with non-instantaneous impulses is established. With the aid of this theorem, the existence of mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained by using perturbation technique and iterative method under the situation that the corresponding solution semigroup $T(\cdot)$ and non-instantaneous impulsive function $g_k$ are compact, $T(\cdot)$ is not compact and $g_k$ is compact, $T(\cdot)$ and $g_k$ are not compact, respectively. The results obtained in this paper essentially improve and extend some related conclusions on this topic. Two concrete examples to parabolic partial differential equations with non-instantaneous impulses are given to illustrate that our results are valuable.

Application of the enhanced modified simple equation method for Burger-Fisher and modified Volterra equations

The enhanced modified simple equation method plays a vital role in finding an exact traveling wave solution of nonlinear evolution equations (NLEEs) in engineering and mathematical physics. In this article, we use the enhanced modified simple equation method to find the exact solutions of NLEEs via the Burger-Fisher equation and the modified Volterra equations and achieve exact solutions involving parameters. When the parameters receive special values, the solitary wave solutions are derived from the exact solutions. It is established that the enhanced modified simple equation method offers a further influential mathematical tool for constructing exact solutions of NLEEs in mathematical physics.

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the (2+1)-dimensional Sawada–Kotera (SK) equation and (3+1)-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota’s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

Exact solutions for a coupled discrete nonlinear Schrödinger system with a saturation nonlinearity

Deriving exact solutions of nonlinear evolution equations has recently received tremendous attention in mathematics and physics. In this paper, using the ( G ′ ∕ G ) -expansion method, we obtain some exact solutions for a coupled discrete nonlinear Schrödinger system with a saturation nonlinearity. These exact solutions include the hyperbolic function type, trigonometric function type and rational function type, of which the hyperbolic function type solutions may generate kink and anti-kink solitons.

Application of the Exp-Function Method for Solving a Partial Differential Equation Arising in Problems of Hydrodynamics

The aim of present paper is to obtain the analytical solution of Harry Dym equation by using Exp-function method (EFM). He’s Exp-function method is used to construct solitary and soliton solutions of nonlinear evolution equation. The method is straightforward and concise, and its applications are promising. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations. Also, it is shown that the EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations.

Unified Riccati equation expansion method and its application to two new classes of Benjamin–Bona–Mahony equations

The Backlund transformations and the superposition formulas of the Riccati equation with constant coefficients are constructed. Two fractional type solutions of the Riccati equation are obtained from its Backlund transformations. The equivalence relations between fractional solutions and previous known solutions are proved. A so-called unified Riccati equation expansion method for generating infinite number of exact traveling wave solutions for nonlinear evolution equations is then developed on the basis of the fractional solutions. With the method, infinitely many exact traveling wave solutions of two new classes of Benjamin–Bona–Mahony equations are presented.