In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled b...In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.展开更多
The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and ...The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.展开更多
We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.
In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is de?ned and its basic properties are proved. The...In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is de?ned and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1,the authors ?rst give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.展开更多
Based on a new linear, continuous and bounded operator (PGOPO), a more effective approach and optimal control algorithm than by the block-pulse functions and Walsh functions to design the linear servomechanism of time...Based on a new linear, continuous and bounded operator (PGOPO), a more effective approach and optimal control algorithm than by the block-pulse functions and Walsh functions to design the linear servomechanism of time-varying systems with time-delay is proposed in the paper. By means of the operator, the differential equation is transferred to a more explicit algebraic form which is much easier than the numerical integration of nonlinear TPBVP derived from Pantryagin's maximum principle method. Furthermore, the method is established strictly based on the theory of convergence in the mean square and it is convenient and simple in computation. So the method can be applied to industry control and aeronautics and astronautics field which is frequently mixed with time varying and time delay. Some illustrative numerical examples are interpreted to support the technique.展开更多
Most studies of the time-reversibility are limited to a linear or an affine involution.In this paper,the authors consider the case of a quadratic involution.For a polynomial differential system with a linear part in t...Most studies of the time-reversibility are limited to a linear or an affine involution.In this paper,the authors consider the case of a quadratic involution.For a polynomial differential system with a linear part in the standard form(-y,x)in R~2,by using the method of regular chains in a computer algebraic system,the computational procedure for finding the necessary and sufficient conditions of the system to be time-reversible with respect to a quadratic involution is given.When the system is quadratic,the necessary and sufficient conditions can be completely obtained by this procedure.For some cubic systems,the necessary and sufficient conditions for these systems to be time-reversible with respect to a quadratic involution are also obtained.These conditions can guarantee the corresponding systems to have a center.Meanwhile,a property of a center-focus system is discovered that if the system is time-reversible with respect to a quadratic involution,then its phase diagram is symmetric about a parabola.展开更多
We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.
In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the...In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the quadratic systems with degenerate infinity, and obtain some similar properties to 2-dimensional quadratic systems.展开更多
This paper studies triangular differential systems arising from various decompositions of partial differential polynomial systems. In theoretical aspects, we emphasizeon translating differential problems into purely a...This paper studies triangular differential systems arising from various decompositions of partial differential polynomial systems. In theoretical aspects, we emphasizeon translating differential problems into purely algebraic ones. Rosenfeld’s lemma is extended to a more general setting; relations between passivity and coherence are clarified;regular systems and simple systems are generalized and proposed, respectively. In algorithmic aspects, we review the Ritt-Wu and Seidenberg algorithms, and outline a methodfor decomposing a differential polynomial system into simple ones. Some applications arealso discussed.展开更多
文摘In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.
文摘The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.
文摘We present an introduction to the Darboux integrability theory of planar complex and real polynomial differential systems containing some improvements to the classical theory.
基金supported by NKRDPC under Grant No.2018YFA0306702the National Natural Science Foundation of China under Grant No.11688101
文摘In this paper, a Ritt-Wu characteristic set method for Laurent partial differential polynomial systems is presented. The concept of Laurent regular differential chain is de?ned and its basic properties are proved. The authors give a partial method to decide whether a Laurent differential chain A is Laurent regular. The decision for whether A is Laurent regular is reduced to the decision of whether a univariate differential chain A1 is Laurent regular. For a univariate differential chain A1,the authors ?rst give a criterion for whether A1 is Laurent regular in terms of its generic zeros and then give partial results on deciding whether A1 is Laurent regular.
基金National Natural Science Foundation of China(69934010)
文摘Based on a new linear, continuous and bounded operator (PGOPO), a more effective approach and optimal control algorithm than by the block-pulse functions and Walsh functions to design the linear servomechanism of time-varying systems with time-delay is proposed in the paper. By means of the operator, the differential equation is transferred to a more explicit algebraic form which is much easier than the numerical integration of nonlinear TPBVP derived from Pantryagin's maximum principle method. Furthermore, the method is established strictly based on the theory of convergence in the mean square and it is convenient and simple in computation. So the method can be applied to industry control and aeronautics and astronautics field which is frequently mixed with time varying and time delay. Some illustrative numerical examples are interpreted to support the technique.
基金partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education(SRFDP,China)under Grant No.20115134110001。
文摘Most studies of the time-reversibility are limited to a linear or an affine involution.In this paper,the authors consider the case of a quadratic involution.For a polynomial differential system with a linear part in the standard form(-y,x)in R~2,by using the method of regular chains in a computer algebraic system,the computational procedure for finding the necessary and sufficient conditions of the system to be time-reversible with respect to a quadratic involution is given.When the system is quadratic,the necessary and sufficient conditions can be completely obtained by this procedure.For some cubic systems,the necessary and sufficient conditions for these systems to be time-reversible with respect to a quadratic involution are also obtained.These conditions can guarantee the corresponding systems to have a center.Meanwhile,a property of a center-focus system is discovered that if the system is time-reversible with respect to a quadratic involution,then its phase diagram is symmetric about a parabola.
文摘We show how to use the Lucas polynomials of the second kind in the solution of a homogeneous linear differential system with constant coefficients, avoiding the Jordan canonical form for the relevant matrix.
文摘In this paper we show the distribution of critical points at infinity of n- dimensional polynomial differential systems, and give the conditions, under which the system is degenerate at infinity. Also, we discuss the quadratic systems with degenerate infinity, and obtain some similar properties to 2-dimensional quadratic systems.
文摘This paper studies triangular differential systems arising from various decompositions of partial differential polynomial systems. In theoretical aspects, we emphasizeon translating differential problems into purely algebraic ones. Rosenfeld’s lemma is extended to a more general setting; relations between passivity and coherence are clarified;regular systems and simple systems are generalized and proposed, respectively. In algorithmic aspects, we review the Ritt-Wu and Seidenberg algorithms, and outline a methodfor decomposing a differential polynomial system into simple ones. Some applications arealso discussed.
基金Project supported by the Natural Science Foundation of China(No.60572093)Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20050004016)NSFC-KOSEF Joint Research Project and IITA Professorship Program of Gwangju Instiute of Science and Technology
