In this work we re-examine a model of the nucleons that involve the weak interaction which was once considered by Heisenberg;that is a neutron may have the structure of a dwarf hydrogen-like atom. We formulate a quant...In this work we re-examine a model of the nucleons that involve the weak interaction which was once considered by Heisenberg;that is a neutron may have the structure of a dwarf hydrogen-like atom. We formulate a quantum dynamics for the Heisenberg model of the neutron associated with interaction that involves the beta decay in terms of a mixed Coulomb-Yukawa potential and the More General Exponential Screened Coulomb Potential (MGESCP), which has been studied and applied to various fields of physics. We show that all the components that form the MGESCP potential can be derived from a general system of linear first order partial differential equations similar to Dirac relativistic equation in quantum mechanics. There are many interesting features that emerge from the MGESCP potential, such as the MGESCP potential can be reduced to the potential that has been proposed to describe the interaction between the quarks for strong force in particle physics, and the energy spectrum of the bound states of the dwarf hydrogen-like atom is continuous with respect to distance. This result leads to an unexpected implication that a proton and an electron may also interact strongly at short distances. We also show that the Yukawa potential when restrained can generate and determine the mathematical structures of fundamental particles associated with the strong and weak fields.展开更多
In this paper, using Sadovskii’s fixed point theorem and properties of the measure of noncompactness, we obtain asymptotic stability results for solutions of nonlinear differential equation with variable delay. Resul...In this paper, using Sadovskii’s fixed point theorem and properties of the measure of noncompactness, we obtain asymptotic stability results for solutions of nonlinear differential equation with variable delay. Results presented in this paper extend some previous results due to Burton?[1], Becker and Burton?[2], Ardjouni and Djoudi [3], and Jin and Luo?[4].展开更多
Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained. Our results impr...Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained. Our results improve and complement some earlier results. Two examples are given to illustrate the importance of the topic and the main results obtained.展开更多
According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n−1.Solving the ordinary differential equat...According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n−1.Solving the ordinary differential equations numerically results in waves appearing in the figures.The ordinary differential equation is very simple;however,waves,including the regular amplitude and period,are drawn in the figure.In this study,the function for obtaining the wave is called the leaf function.Based on the leaf function,the exact solutions for the undamped and unforced Duffing equations are presented.In the ordinary differential equation,in the positive region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes negative.Therefore,in the case that the curves vary with the time under the conditionχ(t)>0,the gradient dχ(t)/d constantly decreases as time increases.That is,the tangential vector on the curve of the graph(with the abscissa and the ordinate χ(t)changes from the upper right direction to the lower right direction as time increases.On the other hand,in the negative region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes positive.The gradient d χ(t)/d constantly increases as time decreases.That is,the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases.Since the behavior occurring in the positive region of the variable χ(t)and the behavior occurring in the negative region of the variableχ(t)alternately occur in regular intervals,waves appear by these interactions.In this paper,I present seven types of damped and divergence exact solutions by combining trigonometric functions,hyperbolic functions,hyperbolic leaf functions,leaf functions,and exponential functions.In each type,I show the derivation method and numerical examples,as well as describe the features of the waveform.展开更多
Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical method...Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.展开更多
We consider the problem of viscosity solution of integro-partial differential equation( IPDE in short) with one obstacle via the solution of reflected backward stochastic dif ferential equations(RBSDE in short) with j...We consider the problem of viscosity solution of integro-partial differential equation( IPDE in short) with one obstacle via the solution of reflected backward stochastic dif ferential equations(RBSDE in short) with jumps. We show the existence and uniqueness of a continuous viscosity solution of equation with non local terms, if the generator is not monotonous and Levy's measure is infinite.展开更多
One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transfor...One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.展开更多
In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis fu...In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such,the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs,leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.展开更多
In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equ...In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.展开更多
This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps(NSFDEwPJs).The boundedness and almost sure exponential stability are also consi...This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps(NSFDEwPJs).The boundedness and almost sure exponential stability are also considered.In general,the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition.However,there are many equations that do not obey the linear growth condition.Therefore,our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role.Moreover,the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions.Finally,we present two examples to illustrate the effectiveness of our results.展开更多
In this article,we consider the Parabolic Anderson Model with constant initial condition,driven by a space-time homogeneous Gaussian noise,with general covariance function in time and spatial spectral measure satisfyi...In this article,we consider the Parabolic Anderson Model with constant initial condition,driven by a space-time homogeneous Gaussian noise,with general covariance function in time and spatial spectral measure satisfying Dalang's condition.First,we prove that the solution (in the Skorohod sense) exists and is continuous in Lp(Ω).Then,we show that the solution has a modification whose sample paths are H(o)lder continuous in space and time,under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time).This improves similar results which were obtained in [6,10] under more restrictive conditions,and with sub-optimal exponents for H(o)lder continuity.展开更多
This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multipl...This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multiple power series expansions and an iterative computation method to solve a kind of integro-differential equation. We introduce a symmetrized topology product area which is called a bicylinder. We expand functions and derivatives of them to power series. Moreover we obtain unknown functions by comparing coefficients of the series on both sides of equations. We express the approximation solutions by a regular product of matrixes.展开更多
In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove ...In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.展开更多
In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the loca...In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.展开更多
Preface This Special Issue in'Acta Mathematicae Applicatae Sinica(English Series)'gathers peer reviewed papers dedicated to Professor Philippe G.Ciarlet on the occasion of his eightieth birthday.Written by fri...Preface This Special Issue in'Acta Mathematicae Applicatae Sinica(English Series)'gathers peer reviewed papers dedicated to Professor Philippe G.Ciarlet on the occasion of his eightieth birthday.Written by friends and colleagues from France and Hong Kong/China,they stand as a testimony of their admiration for an exceptional human being and mathematician.展开更多
文摘In this work we re-examine a model of the nucleons that involve the weak interaction which was once considered by Heisenberg;that is a neutron may have the structure of a dwarf hydrogen-like atom. We formulate a quantum dynamics for the Heisenberg model of the neutron associated with interaction that involves the beta decay in terms of a mixed Coulomb-Yukawa potential and the More General Exponential Screened Coulomb Potential (MGESCP), which has been studied and applied to various fields of physics. We show that all the components that form the MGESCP potential can be derived from a general system of linear first order partial differential equations similar to Dirac relativistic equation in quantum mechanics. There are many interesting features that emerge from the MGESCP potential, such as the MGESCP potential can be reduced to the potential that has been proposed to describe the interaction between the quarks for strong force in particle physics, and the energy spectrum of the bound states of the dwarf hydrogen-like atom is continuous with respect to distance. This result leads to an unexpected implication that a proton and an electron may also interact strongly at short distances. We also show that the Yukawa potential when restrained can generate and determine the mathematical structures of fundamental particles associated with the strong and weak fields.
文摘In this paper, using Sadovskii’s fixed point theorem and properties of the measure of noncompactness, we obtain asymptotic stability results for solutions of nonlinear differential equation with variable delay. Results presented in this paper extend some previous results due to Burton?[1], Becker and Burton?[2], Ardjouni and Djoudi [3], and Jin and Luo?[4].
文摘Lyapunov direct method is employed to investigate the asymptotic behaviour and the boundedness of solutions to a certain third-order differential equation with delay and some new results are obtained. Our results improve and complement some earlier results. Two examples are given to illustrate the importance of the topic and the main results obtained.
文摘According to the wave power rule,the second derivative of a functionχ(t)with respect to the variable t is equal to negative n times the functionχ(t)raised to the power of 2n−1.Solving the ordinary differential equations numerically results in waves appearing in the figures.The ordinary differential equation is very simple;however,waves,including the regular amplitude and period,are drawn in the figure.In this study,the function for obtaining the wave is called the leaf function.Based on the leaf function,the exact solutions for the undamped and unforced Duffing equations are presented.In the ordinary differential equation,in the positive region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes negative.Therefore,in the case that the curves vary with the time under the conditionχ(t)>0,the gradient dχ(t)/d constantly decreases as time increases.That is,the tangential vector on the curve of the graph(with the abscissa and the ordinate χ(t)changes from the upper right direction to the lower right direction as time increases.On the other hand,in the negative region of the variableχ(t),the second derivative d^2χ(t)/dt^2 becomes positive.The gradient d χ(t)/d constantly increases as time decreases.That is,the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases.Since the behavior occurring in the positive region of the variable χ(t)and the behavior occurring in the negative region of the variableχ(t)alternately occur in regular intervals,waves appear by these interactions.In this paper,I present seven types of damped and divergence exact solutions by combining trigonometric functions,hyperbolic functions,hyperbolic leaf functions,leaf functions,and exponential functions.In each type,I show the derivation method and numerical examples,as well as describe the features of the waveform.
基金The research of L.Zhang was supported by the NNSFC(NOs.11601514,11771444,and 11801556)the research of C.Chen and J.Hong were supported by the NNSFC(NOs.91630312,11711530071,and 11871068)the research of L.Ji was supported by the NNSFC(NOs.11601032,and 11471310).
文摘Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.
文摘We consider the problem of viscosity solution of integro-partial differential equation( IPDE in short) with one obstacle via the solution of reflected backward stochastic dif ferential equations(RBSDE in short) with jumps. We show the existence and uniqueness of a continuous viscosity solution of equation with non local terms, if the generator is not monotonous and Levy's measure is infinite.
文摘One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.
基金the China Postdoctoral Science Foundation Funded Project (No.2017M620113)the National Natural Science Foundation of China (Nos.11801120,71773024 and 11771107)+4 种基金the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF.2019058)the Natural Science Foundation of Heilongjiang Province of China (No.G2018006)Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-014 and MOE2018-T2-1-059)National Science Foundation of China (No.11371376)the Innovation-Driven Project and Mathematics.
文摘In this paper,we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches,the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such,the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs,leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.
基金supported by National Natural Science Foundation of China(Grant No.11631004)Science and Technology Commission of Shanghai Municipality(Grant No.14XD1400400).
文摘In this paper, we give interior gradient and Hessian estimates for systems of semi-linear degenerate elliptic partial differential equations on bounded domains, using both tools of backward stochastic differential equations and quasi-derivatives.
基金National Natural Science Foundation of China(Grant Nos.61573156,61273126,61503142)the Ph.D.Start-up Fund of Natural Science Foundation of Guangdong Province(Grant No.2014A030310388)Fundamental Research Funds for the Central Universities(Grant No.x2zdD2153620).
文摘This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps(NSFDEwPJs).The boundedness and almost sure exponential stability are also considered.In general,the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition.However,there are many equations that do not obey the linear growth condition.Therefore,our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role.Moreover,the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions.Finally,we present two examples to illustrate the effectiveness of our results.
基金a grant from the Natural Sciences and Engineering Research Council of Canada, and the second author is supported by the grant MTM2015-67802P.
文摘In this article,we consider the Parabolic Anderson Model with constant initial condition,driven by a space-time homogeneous Gaussian noise,with general covariance function in time and spatial spectral measure satisfying Dalang's condition.First,we prove that the solution (in the Skorohod sense) exists and is continuous in Lp(Ω).Then,we show that the solution has a modification whose sample paths are H(o)lder continuous in space and time,under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time).This improves similar results which were obtained in [6,10] under more restrictive conditions,and with sub-optimal exponents for H(o)lder continuity.
基金This work was supported by the National Natural Science Foundation of China (11771357, 11171277)the Fundamental Research Funds for the Central Universities of Xiamen University (2010121002)the Science Foundation of Fujian province of China (S0850029, 2008J0206).
文摘This work deals with approximation solutions to a type of integro-differential equations in several complex variables. It concerns the Cauchy formula on higher dimensional domains. In our study, we make use of multiple power series expansions and an iterative computation method to solve a kind of integro-differential equation. We introduce a symmetrized topology product area which is called a bicylinder. We expand functions and derivatives of them to power series. Moreover we obtain unknown functions by comparing coefficients of the series on both sides of equations. We express the approximation solutions by a regular product of matrixes.
基金MATH-AmSud 18- MATH-07 SaSMoTiDep Project and HERMES project 41305. S. Torres is partially supported by the Project ECOS-CONICYT C15E05, REDES 15003& MATH-AmSud 18-MATH-07 SaSMoTiDep Project and Fondecyt (1171335)S. Tindel is supported by NSF (Grant DMS-1613163).
文摘In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H ∈(1/2,1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.
基金supported by National Natural Science Foundation of China (Grant No. 11571128).
文摘In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.
文摘Preface This Special Issue in'Acta Mathematicae Applicatae Sinica(English Series)'gathers peer reviewed papers dedicated to Professor Philippe G.Ciarlet on the occasion of his eightieth birthday.Written by friends and colleagues from France and Hong Kong/China,they stand as a testimony of their admiration for an exceptional human being and mathematician.