Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some prop...Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some properties of meromorphic solutions, and we ob- tain some results, which are the improvements and extensions of some results in references. Examples show that our results are precise.展开更多
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
Dierential geometry play a fundamental role in discussing partial dierential equations（PDEs） in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the develo...Dierential geometry play a fundamental role in discussing partial dierential equations（PDEs） in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the development and application of discrete equivalents of the geometric notions and methods of dierential geometry. In this paper, a discrete theory of exterior dierential calculus and the analogue of the Poincar′e lemma for dierential-dierence complex are proposed. They provide an intrinsic idea for developing the theory to discuss the integrability of dierence equations.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
The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me...The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.展开更多
Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation wh...Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions(kink/antikink solitons, singular,periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform.展开更多
Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equat...Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.展开更多
基金supported by the National Natural Science Foundation of China(11171013)supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(16XNH117)
文摘Using Nevanlinna theory of the value distribution of meromorphic functions, we investigate the problem of the existence of meromorphic solutions of some types of complex differential-difference equations and some properties of meromorphic solutions, and we ob- tain some results, which are the improvements and extensions of some results in references. Examples show that our results are precise.
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)).
文摘Dierential geometry play a fundamental role in discussing partial dierential equations（PDEs） in mathematical physics. Recently discrete dierential geometry is an active mathematical terrain, which aims at the development and application of discrete equivalents of the geometric notions and methods of dierential geometry. In this paper, a discrete theory of exterior dierential calculus and the analogue of the Poincar′e lemma for dierential-dierence complex are proposed. They provide an intrinsic idea for developing the theory to discuss the integrability of dierence equations.
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.
文摘The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.
文摘Of recent increasing interest in the area of fractional calculus and nonlinear dynamics are fractional differential-difference equations. This study is devoted to a local fractional differential-difference equation which is related to a nonlinear electrical transmission line. Explicit traveling wave solutions(kink/antikink solitons, singular,periodic, rational) are obtained via the discrete tanh method coupled with the fractional complex transform.
基金the National Natural Science Foundation of China(No.10471065)the Natural Science Foundation of Guangdong Province(No.04010474)
文摘Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.
