In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equat...In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equations with variable coefficients. The properties of the Legendre polynomials are used to reduce the proposed problems to the solution of non-linear system of algebraic equations using Newton iteration method. We give numerical results to satisfy the accuracy and the applicability of the proposed schemes.展开更多
Infinite dimensional Riccati integral equations with all coefficients indefinite are studied in this paper. Relationship between this sort of Riccati integral equations and Fredholm integral equations is established.
An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise in...An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise integration method （PIM） for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix. The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE. Based ,on the error analysis, the criterion for choosing two parameters of the PIM is given. Three kinds of IPIMs for solving the DRE are proposed. The numerical examples machine accuracy solutions. show that the IPIM is stable and gives the展开更多
Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.
In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation （ODE） method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin-Bo...In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation （ODE） method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin-Bona-Mahoney equation. As a result, some traveling wave solutions for the two nonlinear equations are established successfully. Also we make a comparison between the two methods. It turns out that the Riccati sub-ODE method is more effective than the first integral method in handling the proposed problems, and more general solutions are constructed by the Riccati sub-ODE method.展开更多
By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed....By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed.Some sufficient oscillation criteria for previous equations were built up.Some oscillation criteria have been expanded and strengthened in some other known results.展开更多
文摘In this paper, we propose and analyze some schemes of the integral collocation formulation based on Legendre polynomials. We implement these formulae to solve numerically Riccati, Logistic and delay differential equations with variable coefficients. The properties of the Legendre polynomials are used to reduce the proposed problems to the solution of non-linear system of algebraic equations using Newton iteration method. We give numerical results to satisfy the accuracy and the applicability of the proposed schemes.
文摘Infinite dimensional Riccati integral equations with all coefficients indefinite are studied in this paper. Relationship between this sort of Riccati integral equations and Fredholm integral equations is established.
基金Project supported by the National Natural Science Foundation of China(Nos.10902020 and 10721062)
文摘An improved precise integration method （IPIM） for solving the differential Riccati equation （DRE） is presented. The solution to the DRE is connected with the exponential of a Hamiltonian matrix, and the precise integration method （PIM） for solving the DRE is connected with the scaling and squaring method for computing the exponential of a matrix. The error analysis of the scaling and squaring method for the exponential of a matrix is applied to the PIM of the DRE. Based ,on the error analysis, the criterion for choosing two parameters of the PIM is given. Three kinds of IPIMs for solving the DRE are proposed. The numerical examples machine accuracy solutions. show that the IPIM is stable and gives the
文摘Oscillation criteria for semilinear elliptic differential equations are obtained. The results are extensions of integral averaging technique of Kamenev. General means are employed to establish our results.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10571110)the Natural Science Foundation of Shandong Province of China (Grant Nos. ZR2009AM011 and ZR2010AZ003)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20103705110003)
文摘In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation （ODE） method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin-Bona-Mahoney equation. As a result, some traveling wave solutions for the two nonlinear equations are established successfully. Also we make a comparison between the two methods. It turns out that the Riccati sub-ODE method is more effective than the first integral method in handling the proposed problems, and more general solutions are constructed by the Riccati sub-ODE method.
文摘By the generalized Riccati transformation and the integral averaging technique, some sufficient conditions of oscillation of the solutions for second order nonlinear differential equations with damping were discussed.Some sufficient oscillation criteria for previous equations were built up.Some oscillation criteria have been expanded and strengthened in some other known results.
