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
We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are a...We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given bydv/dy= f(y)v3-n+ k(y)v3-m+ g(y)v2+ h(y)v3, with v = 1/y′.As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation,corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.展开更多
In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results a...In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.展开更多
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
The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreove...The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreover, we examine the properties of the kernel function of this integral representation and obtain the partial differential equation provided by this kernel function.展开更多
In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space whic...In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.展开更多
In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate pa...In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.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.
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We present some exact integrability cases of the extended Liénard equation y′′+ f(y)(y′)n +k(y)(y′)m + g(y)y′+ h(y) = 0, with n > 0 and m > 0 arbitrary constants, while f(y), k(y), g(y), and h(y) are arbitrary functions. The solutions are obtained by transforming the equation Liénard equation to an equivalent first kind first order Abel type equation given bydv/dy= f(y)v3-n+ k(y)v3-m+ g(y)v2+ h(y)v3, with v = 1/y′.As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Liénard equation,corresponding to n = 2 and m = 3, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Liénard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Liénard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with g(y) ≡ 0 and h(y) ≡ 0, and arbitrary n and m, thus allowing to obtain the general solution of the corresponding Liénard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.
文摘In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.
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.
基金supported by the Scientific and Technological Research Council of Turkey(TüBìTAK)
文摘The aim of this paper is to construct the integral representation of the solution of Sturm-Liouville equation with eigenparameter-dependent discontinuity conditions at an interior point of the finite interval. Moreover, we examine the properties of the kernel function of this integral representation and obtain the partial differential equation provided by this kernel function.
文摘In this paper the concepts of the boundary value problem of abstract kinetic equation with the first kind of critical parameter γ 0 and generalized periodic boundary conditions are introduced in a Lebesgue space which consists of functions with vector valued in a general Banach space, and then describe the solution of these abstract boundary value problem by the abstract linear integral operator of Volterra type. We call this process the integral operator solving process.
文摘In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.
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.
