In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions ...In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.展开更多
Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relati...Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relation between difference equations with continuous arguments and discrete ones. However, the oscillation results in ref. [1] need the hypothesis liminf p_i(t)】0, which is an essential condition.t→∞ In this note, we compare eq. (1) with certain delay differential equation, and thereby establish some new sufficient conditions for the oscillation of eqs. (1) and (2). These conditions are integral conditions which do not need the hypothesis liminf p_i(t)】0.展开更多
This paper discusses all cases of second order linear singular defferential difference equations with delay and different coefficients, and prensents the conditionsfor existence and uniqueness of solutions nearly in a...This paper discusses all cases of second order linear singular defferential difference equations with delay and different coefficients, and prensents the conditionsfor existence and uniqueness of solutions nearly in all cases.展开更多
In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions ar...In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.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.
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
In this paper, the numerical solution of the boundary value problem that is two-order fuzzy linear differential equations is discussed. Based on the generalized Hukuhara difference, the fuzzy differential equation is ...In this paper, the numerical solution of the boundary value problem that is two-order fuzzy linear differential equations is discussed. Based on the generalized Hukuhara difference, the fuzzy differential equation is converted into a fuzzy difference equation by means of decentralization. The numerical solution of the boundary value problem is obtained by calculating the fuzzy differential equation. Finally, an example is given to verify the effectiveness of the proposed method.展开更多
The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+ b(t) x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer...The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+ b(t) x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer and [·] denotes the greatest integer function.Some new oscillation and nonoscillation conditions are obtained.展开更多
A differential evolution based methodology is introduced for the solution of elliptic partial differential equations (PDEs) with Dirichlet and/or Neumann boundary conditions. The solutions evolve over bounded domains ...A differential evolution based methodology is introduced for the solution of elliptic partial differential equations (PDEs) with Dirichlet and/or Neumann boundary conditions. The solutions evolve over bounded domains throughout the interior nodes by minimization of nodal deviations among the population. The elliptic PDEs are replaced by the corresponding system of finite difference approximation, yielding an expression for nodal residues. The global residue is declared as the root-mean-square value of the nodal residues and taken as the cost function. The standard differential evolution is then used for the solution of elliptic PDEs by conversion to a minimization problem of the global residue. A set of benchmark problems consisting of both linear and nonlinear elliptic PDEs has been considered for validation, proving the effectiveness of the proposed algorithm. To demonstrate its robustness, sensitivity analysis has been carried out for various differential evolution operators and parameters. Comparison of the differential evolution based computed nodal values with the corresponding data obtained using the exact analytical expressions shows the accuracy and convergence of the proposed methodology.展开更多
文摘In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.
基金Project supported by the National Natural Science Foundation of China.
文摘Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relation between difference equations with continuous arguments and discrete ones. However, the oscillation results in ref. [1] need the hypothesis liminf p_i(t)】0, which is an essential condition.t→∞ In this note, we compare eq. (1) with certain delay differential equation, and thereby establish some new sufficient conditions for the oscillation of eqs. (1) and (2). These conditions are integral conditions which do not need the hypothesis liminf p_i(t)】0.
文摘This paper discusses all cases of second order linear singular defferential difference equations with delay and different coefficients, and prensents the conditionsfor existence and uniqueness of solutions nearly in all cases.
基金The project supported by the State Key Basic Research Program of China under Grant No 2004CB318000
文摘In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.
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 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In this paper, the numerical solution of the boundary value problem that is two-order fuzzy linear differential equations is discussed. Based on the generalized Hukuhara difference, the fuzzy differential equation is converted into a fuzzy difference equation by means of decentralization. The numerical solution of the boundary value problem is obtained by calculating the fuzzy differential equation. Finally, an example is given to verify the effectiveness of the proposed method.
文摘The delay differential equation with piecewise constant argument x′(t)+a(t)x(t)+ b(t) x([t-k])=0 is considered,where a(t) and b(t) are continuous functions on [-k,∞),b(t)≥0,k is a positive integer and [·] denotes the greatest integer function.Some new oscillation and nonoscillation conditions are obtained.
文摘A differential evolution based methodology is introduced for the solution of elliptic partial differential equations (PDEs) with Dirichlet and/or Neumann boundary conditions. The solutions evolve over bounded domains throughout the interior nodes by minimization of nodal deviations among the population. The elliptic PDEs are replaced by the corresponding system of finite difference approximation, yielding an expression for nodal residues. The global residue is declared as the root-mean-square value of the nodal residues and taken as the cost function. The standard differential evolution is then used for the solution of elliptic PDEs by conversion to a minimization problem of the global residue. A set of benchmark problems consisting of both linear and nonlinear elliptic PDEs has been considered for validation, proving the effectiveness of the proposed algorithm. To demonstrate its robustness, sensitivity analysis has been carried out for various differential evolution operators and parameters. Comparison of the differential evolution based computed nodal values with the corresponding data obtained using the exact analytical expressions shows the accuracy and convergence of the proposed methodology.
