New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier–Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (\(1+1\))-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

     

A numerical scheme for solutions of the Chen system

In this paper, we will develop the Bessel collocation method to find approximate solutions of the Chen system, which is a three‐dimensional system of ODEs with quadratic nonlinearities. This scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. By help of the collocation points and the matrix operations of derivatives, the unknown coefficients of the Bessel polynomials are calculated. The accuracy and efficiency of the proposed approach are demonstrated by two numerical examples and performed with the aid of a computer code written in MAPLE. In addition, comparisons between our method and the homotopy perturbation method numerical solutions are made with the accuracy of solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Coexistence in a competitor–competitor–mutualist model

This paper is concerned with a system modeling a competitor–competitor–mutualist three-species Lotka–Volterra model. By Schauder fixed point theory, the existence of positive solutions to a strongly coupled elliptic system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed and a numerical simulation is also presented. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak.     

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho (A method to investigate bifurcation of periodic solution in retarded differential equations, J. Differ. Equ. Appl. 4 (1998), pp. 17–27) to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka–Valterra type; then a system of two equations in two unknowns that could model the interactions of two identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cooke and Ladeira (Applying Carvalho's method to find periodic solutions of difference equations, J. Differ. Equ. Appl. 2 (1996), pp. 105–115).

We first state Carvalho's result.     

Approximate solutions for time-fractional two-component evolutionary system of order $2$ using coupled fractional reduced differential transform method

In this paper, Coupled Fractional Reduced Differential Transform method is extended to apply to the generalized time-fractional two-component evolutionary system of order 2. By using this method, the solutions in the form of a generalized Taylor series are obtained. The graphics of numerical solutions together with the error analysis demonstrate that the present method is effective and accurate for obtaining approximate solutions of fractional coupled equations. Moreover, the results also indicate that the solutions obtained by residual power series method in previous literature (M. Alquran, Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method, J. Appl. Anal. Comput.,5(2015)(4), 589-599.) contain errors.     

Travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov () system

In this paper, travelling wave solutions for the nonlinear dispersion Drinfel’d–Sokolov system (called D(m,n) system) are studied by using the Weierstrass elliptic function method. As a result, more new exact travelling wave solutions to the D(m,n) system are obtained including not only all the known solutions found by Xie and Yan but also other more general solutions for different parameters m,n. Moreover, it is also shown that the D(m,1) system with linear dispersion possess compacton and solitary pattern solutions. Besides that, it should be pointed out that the approach is direct and easily carried out without the aid of mathematical software if compared with other traditional methods. We believe that the method can be widely applied to other similar types of nonlinear partial differential equations (PDEs) or systems in mathematical physics.     

Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions

A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n>1 dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.     

Construction and asymptotic stability of structurally stable internal layer solutions

We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined.

Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincaré mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues.

An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel's transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection.

We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.

     

Homoclinic Solutions for a Forced Liénard Type System

In this paper, we find a special class of homoclinic solutions which tend to 0 as t → ±∞, for a Liénard type system with a time-dependent force. Since it is not a small perturbation of a Hamiltonian system, we cannot employ the well-known Melnikov method to determine the existence of homoclinic solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. We prove that the sequence of those periodic solutions has an accumulation which gives an homoclinic solution of the forced Liénard type system.     

Chebyshev Wavelet collocation method for solving generalized Burgers–Huxley equation

In this paper, new and efficient numerical method, called as Chebyshev wavelet collocation method, is proposed for the solutions of generalized Burgers–Huxley equation. This method is based on the approximation by the truncated Chebyshev wavelet series. By using the Chebyshev collocation points, algebraic equation system has been obtained and solved. Approximate solutions of the generalized Burgers–Huxley equation are compared with exact solutions. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation solutions is quite high even in the case of a small number of grid points. Copyright © 2015 John Wiley & Sons, Ltd.     

The extended Jacobian elliptic function expansion method and its application in the generalized Hirota–Satsuma coupled KdV system

In this paper an extended Jacobian elliptic function expansion method, which is a direct and more powerful method, is used to construct more new exact doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system by using symbolic computation. As a result, sixteen families of new doubly periodic solutions are obtained which shows that the method is more powerful. When the modulus of the Jacobian elliptic functions m→1 or 0, the corresponding six solitary wave solutions and six trigonometric function (singly periodic) solutions are also found. The method is also applied to other higher-dimensional nonlinear evolution equations in mathematical physics.     

Generalized solutions of mixed problems for first-order partial functional differential equations

A theorem on the existence of solutions and their continuous dependence upon initial boundary conditions is proved. The method of bicharacteristics is used to transform the mixed problem into a system of integral functional equations of the Volterra type. The existence of solutions of this system is proved by the method of successive approximations using theorems on integral inequalities. Classical solutions of integral functional equations lead to generalized solutions of the original problem. Differential equations with deviated variables and differential integral problems can be obtained from the general model by specializing given operators. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 804–828, June, 2006.     

Dual solutions of boundary layer flow over an upstream moving plate

The homotopy analysis method is applied to study the boundary layer flow over a flat plate which has a constant velocity opposite in direction to that of the uniform mainstream. The dual solutions in series expressions are obtained with the proposed technique, which agree well with numerical results. Note that, by introducing a new auxiliary function β(z), the bifurcation of the solutions is obtained. This indicates that the homtopy analysis method is a open system, in the framework of this technique, we have great freedom to choose the auxiliary parameters or functions. As a result, complicated nonlinear problems may be resolved in a simple way. The present work shows that the homotopy analysis method is an effective tool for solving nonlinear problems with multiple solutions.     

New exact travelling wave solutions of two nonlinear physical models

In this paper, an improved tanh function method is used with a computerized symbolic computation for constructing new exact travelling wave solutions on two nonlinear physical models namely, the quantum Zakharov equations and the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions.The exact solutions are obtained which include new soliton-like solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and its applications are promising.     

Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G′/G)-expansion method

In this paper, we demonstrate the effectiveness of the so-called (G′/G)-expansion method by examining some nonlinear evolution equations with physical interest. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Explicit solutions of (2 + 1)-dimensional nonlinear KP-BBM equation by using Exp-function method

This paper is devoted to studying the (2 + 1)-dimensional KP-BBM wave equation. Exp-function method is used to conduct the analysis. The generalized solitary solutions, periodic solutions and other exact solutions for the (2 + 1)-dimensional KP-BBM wave equation are obtained via this method with the aid of symbolic computational system. It is also shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving other nonlinear evolution equations arising in mathematical physics.     

A Chebyshev spectral collocation method for solving Burgers’-type equations

In this paper, we elaborated a spectral collocation method based on differentiated Chebyshev polynomials to obtain numerical solutions for some different kinds of nonlinear partial differential equations. The problem is reduced to a system of ordinary differential equations that are solved by Runge–Kutta method of order four. Numerical results for the nonlinear evolution equations such as 1D Burgers’, KdV–Burgers’, coupled Burgers’, 2D Burgers’ and system of 2D Burgers’ equations are obtained. The numerical results are found to be in good agreement with the exact solutions. Numerical computations for a wide range of values of Reynolds’ number, show that the present method offers better accuracy in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.