In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff E...In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff Equations, and two examples are presented to illustrate the application of the results.展开更多
This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a n...This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.展开更多
This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T (where, b ∈AT, and T is an arbitrary but fixed subspace of C<sup>m</sup>) and th...This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T (where, b ∈AT, and T is an arbitrary but fixed subspace of C<sup>m</sup>) and the generalized in A<sub>T,S</sub> For the special case when b ∈AT and dim(T)=dim(AT), this splitting iterative methverse A<sub>T,S</sub> hod converges to A<sub>T,S</sub>b (the unique solution of the general restricted system Ax=bx ∈T).展开更多
In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schr&#168;odinger equat...In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schr&#168;odinger equation to illustrate the validity and ad-vantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.展开更多
In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noet...In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noether'sconserved quantity,the new form conserved quantity,and the Hojman's conserved quantity of system are derived fromthem.Finally,an example is given to illustrate the application of the result.展开更多
The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf(xkv(y)k^-1)dwK(k)= Φ g(x)h(y), x, y ∈ G,where G is any topological group...The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf(xkv(y)k^-1)dwK(k)= Φ g(x)h(y), x, y ∈ G,where G is any topological group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of G and f, g, h are continuous complex-valued functions.Consequently, we have generalized the results of stability for d'Alembert's and Wilson's equations by R. Badora, J. Baker, B. Bouikhalene, P. Gavruta, S. Kabbaj, Pl. Kannappan, G. H.Kim, J.M. Rassias, A. Roukbi, L. Sz′ekelyhidi, D. Zeglami, etc.展开更多
Presented here are the Generalized BCS Equations incorporating Fermi Energy for the study of the {Δ, Tc, jc(T)} values of both elemental and composite superconductors (SCs) for all T ≤ Tc, where Δ, Tc and jc(T) den...Presented here are the Generalized BCS Equations incorporating Fermi Energy for the study of the {Δ, Tc, jc(T)} values of both elemental and composite superconductors (SCs) for all T ≤ Tc, where Δ, Tc and jc(T) denote, respectively, one of the gap values, the critical temperature and the T-dependent critical current density. This framework, which extends our earlier study that dealt with the {Δ0, Tc, jc(0)} values of an SC, is also shown to lead to T-dependent values of several other related parameters such as the effective mass of electrons, their number density, critical velocity, Fermi velocity (VF), coherence length and the London penetration depth. The extended framework is applied to the jc(T) data reported by Romijn et al. for superconducting Aluminium strips and is shown not only to provide an alternative to the explanation given by them, but also to some novel features such as the role of the Sommerfeld coefficient γ(T) in the context of jc(T) and the role of VF(T) in the context of a recent finding by Plumb et al. about the superconductivity of Bi-2212.展开更多
The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is nece...The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.展开更多
To study Lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms, the determining equations of the Lie symmetry of the system is derived. A con- served quantity of Hojman' s typ...To study Lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms, the determining equations of the Lie symmetry of the system is derived. A con- served quantity of Hojman' s type and a Noether' s conserved quantity are deduced by the Lie symme- try under some conditions. One example is given to illustrate the application of the result.展开更多
In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressi...In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.展开更多
This paper focuses on studying the generalized geometry theory of constrained rotating relativistic Birkhoffian systems. Based on the fact that relativistic rotating inertia is embedded in the Birkhoffian systems, the...This paper focuses on studying the generalized geometry theory of constrained rotating relativistic Birkhoffian systems. Based on the fact that relativistic rotating inertia is embedded in the Birkhoffian systems, the Pfaff action of rotating relativistic Birkhoffian systems was defined. The Pfaff-Birkhoff principles and Birkhoff's equations of the constrained rotating relativistic systems were obtained. The geometrical description, the exact properties and their forms on R T^*M for the constrained rotating relativistic Birkhoffian systems are given. The global analyses of the autonomous, semi-autonomous and non-autonomous constrained relativistic Birkhoff's equations as well as the geometrical properties of energy change for the constrained rotating relativistic Birkhoffian systems were also conducted.展开更多
In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is ...In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is showed under some local integrability condition on the diffusion and drift coefficients. The strong comparison theorem for solutions is also established.展开更多
This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers(BBM-Burgers) equation to the corresponding degenerate boundary layer solutions in the half-s...This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers(BBM-Burgers) equation to the corresponding degenerate boundary layer solutions in the half-space.It is shown that the convergence rate is t-(α/4) as t →∞ provided that the initial perturbation lies in H α 1 for α 〈 α(q):= 3 +(2/q),where q is the degeneracy exponent of the flux function.Our analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant introduced in [1]展开更多
In this paper, an improved algorithm for the solution of Generalized Burger-Fisher’s Equation is presented. A Maple code is generated for the algorithm and simulated. It was observed that the algorithm gives the solu...In this paper, an improved algorithm for the solution of Generalized Burger-Fisher’s Equation is presented. A Maple code is generated for the algorithm and simulated. It was observed that the algorithm gives the solution with less computation. The solution gives a better result when compared with the numerical solutions in the existing literature.展开更多
In this paper, equilibrium stability of generalized Birkhoff's autonomous system is discussed. First, equilibrium equations of generalized Birkhoff's autonomous system are set up, and the n the linear approx...In this paper, equilibrium stability of generalized Birkhoff's autonomous system is discussed. First, equilibrium equations of generalized Birkhoff's autonomous system are set up, and the n the linear approximate method and direct method of stability in equilibrium st ate are studied. Some results on equilibrium of generalized Birkhoff's autonomou s system are obtained on the basis of Lyapunov's thorem. Last, the applica tion of the results is illustrated with an example.展开更多
In the present paper, three kinds of forms for Noether’s conservation laws of hol-onomic nonconservative dynamical systems in generalized mechanics are given.
In this paper, we investigate the existence, uniqueness and the asymptotic equiv- alence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of...In this paper, we investigate the existence, uniqueness and the asymptotic equiv- alence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type (DEPCAG). This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.展开更多
In this paper,the authors study the monotoneity and convexity of certain combinations and composites defined in terms of the generalized Grotzsch ring function μa (r), which appears in Ramanujan' s generalized mo...In this paper,the authors study the monotoneity and convexity of certain combinations and composites defined in terms of the generalized Grotzsch ring function μa (r), which appears in Ramanujan' s generalized modular equations,and obtain some inequalities for this function.展开更多
For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-al...For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.展开更多
基金the National Natural Science Foundation of China(10572021 and 10772025)the Doctoral Program Foundation of Institution of Higher Education,China(20040007022)
文摘In this article, generalized Birkhoff equations are put forward by adding supplementary terms to the Birkhoff equations. A conformal invariance of the Birkhoff equations can be used to study the generalized Birkhoff Equations, and two examples are presented to illustrate the application of the results.
基金supported by the National Natural Science Foundation of China (Grant Nos 10572021 and 10772025)
文摘This paper presents a Poisson theory of the generalized Birkhoff equations, including the algebraic structure of the equations, the sufficient and necessary condition on the integral and the conditions under which a new integral can be deduced by a known integral as well as the form of the new integral.
基金This project is supported by Science and Technology Foundation of Shanghai Higher Eduction,Doctoral Program Foundation of Higher Education in China.National Nature Science Foundation of China and Youth Science Foundation of Universities in Shanghai.
文摘This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T (where, b ∈AT, and T is an arbitrary but fixed subspace of C<sup>m</sup>) and the generalized in A<sub>T,S</sub> For the special case when b ∈AT and dim(T)=dim(AT), this splitting iterative methverse A<sub>T,S</sub> hod converges to A<sub>T,S</sub>b (the unique solution of the general restricted system Ax=bx ∈T).
基金The NSF(11001042) of ChinaSRFDP(20100043120001)FRFCU(09QNJJ002)
文摘In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher's equation, the nonlinear schr&#168;odinger equation to illustrate the validity and ad-vantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.
基金National Natural Science Foundation of China under Grant No.10272034the Doctoral Program Foundation of China
文摘In this paper,the form invariance and the Lie symmetry of Lagrange's equations for nonconservativesystem in generalized classical mechanics under the infinitesimal transformations of group are studied,and the Noether'sconserved quantity,the new form conserved quantity,and the Hojman's conserved quantity of system are derived fromthem.Finally,an example is given to illustrate the application of the result.
文摘The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf(xkv(y)k^-1)dwK(k)= Φ g(x)h(y), x, y ∈ G,where G is any topological group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of G and f, g, h are continuous complex-valued functions.Consequently, we have generalized the results of stability for d'Alembert's and Wilson's equations by R. Badora, J. Baker, B. Bouikhalene, P. Gavruta, S. Kabbaj, Pl. Kannappan, G. H.Kim, J.M. Rassias, A. Roukbi, L. Sz′ekelyhidi, D. Zeglami, etc.
文摘Presented here are the Generalized BCS Equations incorporating Fermi Energy for the study of the {Δ, Tc, jc(T)} values of both elemental and composite superconductors (SCs) for all T ≤ Tc, where Δ, Tc and jc(T) denote, respectively, one of the gap values, the critical temperature and the T-dependent critical current density. This framework, which extends our earlier study that dealt with the {Δ0, Tc, jc(0)} values of an SC, is also shown to lead to T-dependent values of several other related parameters such as the effective mass of electrons, their number density, critical velocity, Fermi velocity (VF), coherence length and the London penetration depth. The extended framework is applied to the jc(T) data reported by Romijn et al. for superconducting Aluminium strips and is shown not only to provide an alternative to the explanation given by them, but also to some novel features such as the role of the Sommerfeld coefficient γ(T) in the context of jc(T) and the role of VF(T) in the context of a recent finding by Plumb et al. about the superconductivity of Bi-2212.
基金Supported by the National Natural Science Foundation of China(10972151,11272227)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province(CXZZ11_0949)the Innovation Program for Postgraduate of Suzhou University of Science and Technology(SKCX11S_050)
文摘The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model.With the successful use of fractional calculus in many areas of science and engineering,it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s equations.The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics.Here,the fractional Pfaff-Birkhoff variational problem is presented and studied.The definitions of fractional derivatives,the formulae for integration by parts and some other preliminaries are firstly given.Secondly,the fractional Pfaff-Birkhoff principle and the fractional Birkhoff′s equations in terms of RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.
基金Supported by the National Natural Science Foundation of China(10772025)the Key Program of the National Natural Science Foundation of China(10932002)
文摘To study Lie symmetry and the conserved quantity of a generalized Birkhoff system with additional terms, the determining equations of the Lie symmetry of the system is derived. A con- served quantity of Hojman' s type and a Noether' s conserved quantity are deduced by the Lie symme- try under some conditions. One example is given to illustrate the application of the result.
基金This work is supported by the National Natural Science Foundation of China(Nos.11672069,11702059,11232003,11672062)The Ph.D.Programs Foundation of Ministry of Education of China(No.20130041110050)+3 种基金the Research Startup Project Plan for Liaoning Doctors(No.20141119)the Fundamental Research Funds for the Central Universities(20000101)the Natural Science Foundation of Liaoning Province(No.20170540199)111 Project(B08014).
文摘In this paper,a nonlinear wave equation with variable coefficients is studied,interestingly,this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities.With the aid of Lou’s direct method1,the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained.The corresponding numerical examples of exact solutions are presented by using different coefficients.Particularly,while the variable coefficients are taken as some special constants,the nonlinear wave equation with variable coefficients reduces to the one with constant coefficients,which can be used to describe the propagation of the travelling waves in general cylindrical rods composed of generally hyperelastic materials.Using the same method to solve the nonlinear wave equation,the validity and rationality of this method are verified.
基金Project supported by the National Natural Science Foundation of China (Grant Nos.10672143, 10372053), and the Natural Science Foundation of Henan Province (Grant Nos.03011011400, 05011022200)
文摘This paper focuses on studying the generalized geometry theory of constrained rotating relativistic Birkhoffian systems. Based on the fact that relativistic rotating inertia is embedded in the Birkhoffian systems, the Pfaff action of rotating relativistic Birkhoffian systems was defined. The Pfaff-Birkhoff principles and Birkhoff's equations of the constrained rotating relativistic systems were obtained. The geometrical description, the exact properties and their forms on R T^*M for the constrained rotating relativistic Birkhoffian systems are given. The global analyses of the autonomous, semi-autonomous and non-autonomous constrained relativistic Birkhoff's equations as well as the geometrical properties of energy change for the constrained rotating relativistic Birkhoffian systems were also conducted.
文摘In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is showed under some local integrability condition on the diffusion and drift coefficients. The strong comparison theorem for solutions is also established.
基金supported by the "Fundamental Research Funds for the Central Universities"the National Natural Science Foundation of China (10871151)
文摘This paper is concerned with the convergence rates of the global solutions of the generalized Benjamin-Bona-Mahony-Burgers(BBM-Burgers) equation to the corresponding degenerate boundary layer solutions in the half-space.It is shown that the convergence rate is t-(α/4) as t →∞ provided that the initial perturbation lies in H α 1 for α 〈 α(q):= 3 +(2/q),where q is the degeneracy exponent of the flux function.Our analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant introduced in [1]
文摘In this paper, an improved algorithm for the solution of Generalized Burger-Fisher’s Equation is presented. A Maple code is generated for the algorithm and simulated. It was observed that the algorithm gives the solution with less computation. The solution gives a better result when compared with the numerical solutions in the existing literature.
文摘In this paper, equilibrium stability of generalized Birkhoff's autonomous system is discussed. First, equilibrium equations of generalized Birkhoff's autonomous system are set up, and the n the linear approximate method and direct method of stability in equilibrium st ate are studied. Some results on equilibrium of generalized Birkhoff's autonomou s system are obtained on the basis of Lyapunov's thorem. Last, the applica tion of the results is illustrated with an example.
文摘In the present paper, three kinds of forms for Noether’s conservation laws of hol-onomic nonconservative dynamical systems in generalized mechanics are given.
文摘In this paper, we investigate the existence, uniqueness and the asymptotic equiv- alence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type (DEPCAG). This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.
文摘In this paper,the authors study the monotoneity and convexity of certain combinations and composites defined in terms of the generalized Grotzsch ring function μa (r), which appears in Ramanujan' s generalized modular equations,and obtain some inequalities for this function.
文摘For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.