In this paper,we mainly investigate the forms of entire solutions for certain Fermattype partial differential-difference equations in C^(2)by using Nevanlinna’s theory of several complex variables.
The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the applicat...The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.展开更多
In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex d...In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations...A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.展开更多
In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional co...In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.展开更多
In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formu...In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and point out that the boundary layer at t = 0 has a great influence upon the interior layer at t = a. At the same time, on the basis of differential inequality techniques, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of our result. The result of this article is new and it complements the previously known ones.展开更多
Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried ...Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.展开更多
A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability ...A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectie map and a completely integrable tinite-dimensionai Hamiltonian system.展开更多
The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381(2011) 963] quite recently, related to the discrete...The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381(2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic,trigonometric and rational, which have not been reported before.展开更多
In this paper,we mainly investigate entire solutions of the following two non-linear differential-difference equations f^(n)(z)+ωf^(n-1)(z)f′(z)+f^((k))(z+c)=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥5 and f^(n)(z)+ωf^(n-1)...In this paper,we mainly investigate entire solutions of the following two non-linear differential-difference equations f^(n)(z)+ωf^(n-1)(z)f′(z)+f^((k))(z+c)=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥5 and f^(n)(z)+ωf^(n-1)(z)f′(z)+q(z)f^((k))(z+c)e^(Q(z))=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥4,where k≥0 is an integer,c,ω,p_(1),p_(2),α_(1),α_(2)are non-zero constants,q(z)is a non-vanishing polynomial and Q(z)is a non-constant polynomial.Under some additional hypotheses,we analyze the existence and expressions of transcendental entire solutions of the above equations.展开更多
A new expanded approach is presented to find exact solutions of nonlinear differential-difference equations. As its application, the soliton solutions and periodic solutions of a lattice equation are obtained.
By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-differen...By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-difference equations, two interesting results are obtained. And it extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L（z, f） =n∑i=0m∑j=0Aij（z）f^（j）（z ＋ ci） = 0 or F（z）with entire or meromorp...The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L（z, f） =n∑i=0m∑j=0Aij（z）f^（j）（z ＋ ci） = 0 or F（z）with entire or meromorphic coefficients, and ci, i = 0,..., n being distinct complex numbers,where there is only one dominant coefficient.展开更多
On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear ...On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].展开更多
In this paper,based on the symbolic computing system Maple,the direct method for Lie symmetry groupspresented by Sen-Yue Lou [J.Phys.A:Math.Gen.38 (2005) L129] is extended from the continuous differential equationsto ...In this paper,based on the symbolic computing system Maple,the direct method for Lie symmetry groupspresented by Sen-Yue Lou [J.Phys.A:Math.Gen.38 (2005) L129] is extended from the continuous differential equationsto the differential-difference equations.With the extended method,we study the well-known differential-difference KPequation,KZ equation and (2+1)-dimensional ANNV system,and both the Lie point symmetry groups and the non-Liesymmetry groups are obtained.展开更多
In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for three nonlinear differential-difference equations are obtained.
In this paper,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f(z)is a finite order transcendental meromorp...In this paper,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f(z)is a finite order transcendental meromorphic solution of complex linear differential-difference equation:W_(1)(z)f'(z+1)+W_(2)(z)f(z)=W_(3)(z),where W_(1)(z),W_(2)(z),W_(3)(z) are nonzero meromorphic functions,with their orders of growth being less than one,such that W_(1)(z)+W_(2)(z)■0.If f(z) and a meromorphic function g(z) share 0,1,∞ CM,then either f(z)≡g(z) or f(z)+g(z)≡f(z)g(z) or f^(2)(z)(g(z)-1)^(2)+g^(2)(z)(f(z)-1)^(2)≡f(z)g(z)(f(z)g(z)-1) or there exists a polynomial φ(z)=az+b_(0) such that ■ where a(≠0),a_(0),b_(0) are constants with e^(a_(0))≠e^(b_(0)).展开更多
In this paper,the Bernstein polynomials method is proposed for the numerical solution of Fredholm integro-differential-difference equation with variable coefficients and mixed conditions.This method is using a simple ...In this paper,the Bernstein polynomials method is proposed for the numerical solution of Fredholm integro-differential-difference equation with variable coefficients and mixed conditions.This method is using a simple computational manner to obtain a quite acceptable approximate solution.The main characteristic behind this method lies in the fact that,on the one hand,the problem will be reduced to a system of algebraic equations.On the other hand,the efficiency and accuracy of the Bernstein polynomials method for solving these equations are high.The existence and uniqueness of the solution have been proved.Moreover,an estimation of the error bound for this method will be shown by preparing some theorems.Finally,some numerical experiments are presented to show the excellent behavior and high accuracy of this algorithm in comparison with some other well-known methods.展开更多
In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.
Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference...Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11971344).
文摘In this paper,we mainly investigate the forms of entire solutions for certain Fermattype partial differential-difference equations in C^(2)by using Nevanlinna’s theory of several complex variables.
基金the State Key Programme of Basic Research of China under，高等学校博士学科点专项科研项目
文摘The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.
文摘In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金supported by National Natural Science Foundation of China and the Natural Science Foundation of Shandong Province
文摘A new approach is presented by means of a new general ansitz and some relations among Jacobian elliptic functions, which enables one to construct more new exact solutions of nonlinear differential-difference equations. As an example, we apply this new method to Hybrid lattice, diseretized mKdV lattice, and modified Volterra lattice. As a result, many exact solutions expressible in rational formal hyperbolic and elliptic functions are conveniently obtained with the help of Maple.
文摘In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.
基金Supported by the National Natural Science Funds (11071075)the Natural Science Foundation of Shanghai(10ZR1409200)+1 种基金the National Laboratory of Biomacromolecules,Institute of Biophysics,Chinese Academy of Sciencesthe E-Institutes of Shanghai Municipal Education Commissions(E03004)
文摘In this article, the interior layer for a second order nonlinear singularly perturbed differential-difference equation is considered. Using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and point out that the boundary layer at t = 0 has a great influence upon the interior layer at t = a. At the same time, on the basis of differential inequality techniques, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved. Finally, an example is given to demonstrate the effectiveness of our result. The result of this article is new and it complements the previously known ones.
文摘Differential-difference equations are considered to be hybrid systems because the spatial variable n is discrete while the time t is usually kept continuous.Although a considerable amount of research has been carried out in the field of nonlinear differential-difference equations,the majority of the results deal with polynomial types.Limited research has been reported regarding such equations of rational type.In this paper we present an adaptation of the(G /G)-expansion method to solve nonlinear rational differential-difference equations.The procedure is demonstrated using two distinct equations.Our approach allows one to construct three types of exact traveling wave solutions(hyperbolic,trigonometric,and rational) by means of the simplified form of the auxiliary equation method with reduced parameters.Our analysis leads to analytic solutions in terms of topological solitons and singular periodic functions as well.
基金Supported by the Science and Technology Plan Projects of the Educational Department of Shandong Province of China under GrantNo. J08LI08
文摘A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectie map and a completely integrable tinite-dimensionai Hamiltonian system.
文摘The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381(2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic,trigonometric and rational, which have not been reported before.
基金the National Natural Science Foundation of China(11971344)。
文摘In this paper,we mainly investigate entire solutions of the following two non-linear differential-difference equations f^(n)(z)+ωf^(n-1)(z)f′(z)+f^((k))(z+c)=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥5 and f^(n)(z)+ωf^(n-1)(z)f′(z)+q(z)f^((k))(z+c)e^(Q(z))=p_(1)e^(α1 z)+p_(2)e^(α2 z),n≥4,where k≥0 is an integer,c,ω,p_(1),p_(2),α_(1),α_(2)are non-zero constants,q(z)is a non-vanishing polynomial and Q(z)is a non-constant polynomial.Under some additional hypotheses,we analyze the existence and expressions of transcendental entire solutions of the above equations.
基金the National Natural Science Foundation of China (No. 60773119)
文摘A new expanded approach is presented to find exact solutions of nonlinear differential-difference equations. As its application, the soliton solutions and periodic solutions of a lattice equation are obtained.
文摘By using the Nevanlinna value distribution theory, we will mainly investigate the form of entire solutions with finite order on a type of system of differential-difference equations and a type of differential-difference equations, two interesting results are obtained. And it extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1130123311171119)+1 种基金the Natural Science Foundation of Jiangxi Province(Grant No.20132BAB211002)the Youth Science Foundation of Education Bureau of Jiangxi Province(Grant No.GJJ14271)
文摘The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L（z, f） =n∑i=0m∑j=0Aij（z）f^（j）（z ＋ ci） = 0 or F（z）with entire or meromorphic coefficients, and ci, i = 0,..., n being distinct complex numbers,where there is only one dominant coefficient.
基金Supported by the National Natural Science Foundation of China(12261023,11861023)the Foundation of Science and Technology project of Guizhou Province of China([2018]5769-05)。
文摘On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].
文摘In this paper,based on the symbolic computing system Maple,the direct method for Lie symmetry groupspresented by Sen-Yue Lou [J.Phys.A:Math.Gen.38 (2005) L129] is extended from the continuous differential equationsto the differential-difference equations.With the extended method,we study the well-known differential-difference KPequation,KZ equation and (2+1)-dimensional ANNV system,and both the Lie point symmetry groups and the non-Liesymmetry groups are obtained.
基金The project supported by National Natural Science Foundation of China under Grant Nos.90511009 and 40305006
文摘In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for three nonlinear differential-difference equations are obtained.
基金Supported by the National Natural Science Foundation of China (Grant No. 12001211)the Natural Science Foundation of Fujian Province,China (Grant No. 2021J01651)。
文摘In this paper,we mainly study the uniqueness of transcendental meromorphic solutions for a class of complex linear differential-difference equations.Specially,suppose that f(z)is a finite order transcendental meromorphic solution of complex linear differential-difference equation:W_(1)(z)f'(z+1)+W_(2)(z)f(z)=W_(3)(z),where W_(1)(z),W_(2)(z),W_(3)(z) are nonzero meromorphic functions,with their orders of growth being less than one,such that W_(1)(z)+W_(2)(z)■0.If f(z) and a meromorphic function g(z) share 0,1,∞ CM,then either f(z)≡g(z) or f(z)+g(z)≡f(z)g(z) or f^(2)(z)(g(z)-1)^(2)+g^(2)(z)(f(z)-1)^(2)≡f(z)g(z)(f(z)g(z)-1) or there exists a polynomial φ(z)=az+b_(0) such that ■ where a(≠0),a_(0),b_(0) are constants with e^(a_(0))≠e^(b_(0)).
基金Supported by the Shiraz University of Technology,Shiraz,Iran.
文摘In this paper,the Bernstein polynomials method is proposed for the numerical solution of Fredholm integro-differential-difference equation with variable coefficients and mixed conditions.This method is using a simple computational manner to obtain a quite acceptable approximate solution.The main characteristic behind this method lies in the fact that,on the one hand,the problem will be reduced to a system of algebraic equations.On the other hand,the efficiency and accuracy of the Bernstein polynomials method for solving these equations are high.The existence and uniqueness of the solution have been proved.Moreover,an estimation of the error bound for this method will be shown by preparing some theorems.Finally,some numerical experiments are presented to show the excellent behavior and high accuracy of this algorithm in comparison with some other well-known methods.
文摘In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.
基金supported by the National Natural Science Foundation of China(10471067)NSF of Guangdong Province(04010474)
文摘Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj（z）f1（λj1）（z＋cj）=R2（z,f2（z））,j=1∑nβj（z）f2（λj2）（z＋cj）=R1（z,f1（z））. where λij （j = 1, 2,…, n; i = 1, 2） are finite non-negative integers, and cj （j = 1, 2,… , n） are distinct, nonzero complex numbers, αj（z）, βj（z） （j = 1,2,… ,n） are small functions relative to fi（z） （i =1, 2） respectively, Ri（z, f（z）） （i = 1, 2） are rational in fi（z） （i =1, 2） with coefficients which are small functions of fi（z） （i = 1, 2） respectively.
