In this paper we discuss the continuous piecewise polynomial spline collocation method for a kind of integral operator equations, which include smooth kernel Fredholm equations and Volterra equations as well as Green...In this paper we discuss the continuous piecewise polynomial spline collocation method for a kind of integral operator equations, which include smooth kernel Fredholm equations and Volterra equations as well as Green’s kernel integral equations. It will be shown that the collocation solution itself may admit an ideal error expansion at the knots. Based on this expanison, the multilevel corrected global estimates can be obtained by using the "higher order interpolation" technique.展开更多
Using the technique of integration within an ordered product (IWOP) of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation tha...Using the technique of integration within an ordered product (IWOP) of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation that is an operator generalization of the solution of thermo conduction equation. Then we seach for the solution of operator Fredholm integration equations, which provides us with a new approach for deriving some operator identities.展开更多
Based on the bipartite entangled state representation and using the technique of integration within an ordered product (IWOP) of operators we construct the corresponding operator Fredholm equations and then derive t...Based on the bipartite entangled state representation and using the technique of integration within an ordered product (IWOP) of operators we construct the corresponding operator Fredholm equations and then derive their solutions. As its application we deduce some new bosonic operator identities and new relations about the two-variable Hermite polynomials.展开更多
To describe the physical reality, there are two ways of constructing the dynamical equation of field, differential formalism and integral formalism. The importance of this fact is firstly emphasized by Yang in case of...To describe the physical reality, there are two ways of constructing the dynamical equation of field, differential formalism and integral formalism. The importance of this fact is firstly emphasized by Yang in case of gauge field [Phys. Rev. Lett. 33 (1974) 44fi], where the fact has given rise to a deeper understanding for Aharonov-Bohm phase and magnetic monopole [Phys. Rev. D 12 (1975) 3846]. In this paper we shall point out that such a fact also holds in general wave function of matter, it may give rise to a deeper understanding for Berry phase. Most importantly, we shall prove a point that, for general wave function of matter, in the adiabatic limit, there is an intrinsic difference between its integral formalism and differential formalism. It is neglect of this difference that leads to an inconsistency of quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93 (2004) 160408]. It has been widely accepted that there is no physical difference of using differential operator or integral operator to construct the dynamical equation of field. Nevertheless, our study shows that the Schroedinger differential equation (i.e., differential formalism for wave function) shall lead to vanishing Berry phase and that the Schroedinger integral equation (i.e., integral formalism for wave function), in the adiabatic limit, can satisfactorily give the Berry phase. Therefore, we reach a conclusion: There are two ways of describing physical reality, differential formalism and integral formalism; but the integral formalism is a unique way of complete description.展开更多
Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1...Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.展开更多
In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using...In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using natural boundary reduction.The governing equation is first discretized in time, leading to a time-stepping scheme,where an exterior elliptic problem has to be solved in each time step. By Fourier ex-pansion, we derive a natural integral equation of the elliptic problem related to timestep and Poisson integral integral formula over exterior circular domain. Finite elementdiscretization of the natural integral equation is employed to solve this problem. Thecomputational aspects of this method are discussed. Numerical results are presented toillustrate feasibility and efficiency of our method.展开更多
By using the strong continuous semigroup theory of linear operators we prove the existence of a unique positive time-dependent solution of the model describing a re-pairable, standby, human & machine system.
In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, ...In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, the existence of at least six distinct fixed points of nonlinear operators is proved. The theoretical results are then applied to nonlinear system of Hammerstein integral equations.展开更多
The Hirota-Satsuma coupled KdV equations associated 2 x 2 matrix spectral problem is discussed by the dressing method, which is based on the factorization of integral operator on a line into a product of two Volterra ...The Hirota-Satsuma coupled KdV equations associated 2 x 2 matrix spectral problem is discussed by the dressing method, which is based on the factorization of integral operator on a line into a product of two Volterra integral operators. A new solution is obtained by choosing special kernel of integral operator.展开更多
文摘In this paper we discuss the continuous piecewise polynomial spline collocation method for a kind of integral operator equations, which include smooth kernel Fredholm equations and Volterra equations as well as Green’s kernel integral equations. It will be shown that the collocation solution itself may admit an ideal error expansion at the knots. Based on this expanison, the multilevel corrected global estimates can be obtained by using the "higher order interpolation" technique.
基金The project supported by the President Foundation of the Chinese Academy of Sciences
文摘Using the technique of integration within an ordered product (IWOP) of operators we construct intermediate coordinate-momentum representation, with which we build a type of operator Fredholm integration equation that is an operator generalization of the solution of thermo conduction equation. Then we seach for the solution of operator Fredholm integration equations, which provides us with a new approach for deriving some operator identities.
基金The project supported by the Specialized Research Fund for the Doctorial Progress of Higher Education of China under Grant No. 20040358019 and National Natural Science Foundation of China under Grant No. 10475056
文摘Based on the bipartite entangled state representation and using the technique of integration within an ordered product (IWOP) of operators we construct the corresponding operator Fredholm equations and then derive their solutions. As its application we deduce some new bosonic operator identities and new relations about the two-variable Hermite polynomials.
文摘To describe the physical reality, there are two ways of constructing the dynamical equation of field, differential formalism and integral formalism. The importance of this fact is firstly emphasized by Yang in case of gauge field [Phys. Rev. Lett. 33 (1974) 44fi], where the fact has given rise to a deeper understanding for Aharonov-Bohm phase and magnetic monopole [Phys. Rev. D 12 (1975) 3846]. In this paper we shall point out that such a fact also holds in general wave function of matter, it may give rise to a deeper understanding for Berry phase. Most importantly, we shall prove a point that, for general wave function of matter, in the adiabatic limit, there is an intrinsic difference between its integral formalism and differential formalism. It is neglect of this difference that leads to an inconsistency of quantum adiabatic theorem pointed out by Marzlin and Sanders [Phys. Rev. Lett. 93 (2004) 160408]. It has been widely accepted that there is no physical difference of using differential operator or integral operator to construct the dynamical equation of field. Nevertheless, our study shows that the Schroedinger differential equation (i.e., differential formalism for wave function) shall lead to vanishing Berry phase and that the Schroedinger integral equation (i.e., integral formalism for wave function), in the adiabatic limit, can satisfactorily give the Berry phase. Therefore, we reach a conclusion: There are two ways of describing physical reality, differential formalism and integral formalism; but the integral formalism is a unique way of complete description.
文摘Making use of upper and lower solutions and analytical method, the author studies theexistence of positive solution for the singular equation x + f(t, z) = 0 satisfying nonlinear boundary conditions: x (0) = 0, h(x (1), x’ (1)) = 0, g (z (0), x’(0)) = 0, and x (1) = 0,which extends the result of J. V. Baxley.
基金National Natural Science Foundation of China(19701001)
文摘In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using natural boundary reduction.The governing equation is first discretized in time, leading to a time-stepping scheme,where an exterior elliptic problem has to be solved in each time step. By Fourier ex-pansion, we derive a natural integral equation of the elliptic problem related to timestep and Poisson integral integral formula over exterior circular domain. Finite elementdiscretization of the natural integral equation is employed to solve this problem. Thecomputational aspects of this method are discussed. Numerical results are presented toillustrate feasibility and efficiency of our method.
基金This research is supported by the Tianyuan Mathematics Foundation (No. 10226007) and the Science Foundation of Xinjiang University
文摘By using the strong continuous semigroup theory of linear operators we prove the existence of a unique positive time-dependent solution of the model describing a re-pairable, standby, human & machine system.
基金This research is supported by NSFC (10071042)NSFSP (Z2000A02).
文摘In this paper, the famous Amann three-solution theorem is generalized. Multiplicity question of fixed points for nonlinear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, the existence of at least six distinct fixed points of nonlinear operators is proved. The theoretical results are then applied to nonlinear system of Hammerstein integral equations.
基金Supported by the National Natural Science Foundation of China under Grant No.11001250
文摘The Hirota-Satsuma coupled KdV equations associated 2 x 2 matrix spectral problem is discussed by the dressing method, which is based on the factorization of integral operator on a line into a product of two Volterra integral operators. A new solution is obtained by choosing special kernel of integral operator.