To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’...To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.展开更多
This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to st...This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to stochastic delay differential equations. Based on the Caratheodory approximate procedure, it was proved that stochastic delay differential equations have unique solution and established that the Caratheodory approximate solution converges to the unique solution of stochastic delay differential equations under the Cauchy sequence and initial condition. This Caratheodory approximate procedure and Euler method both converge at the same rate. This is achieved by replacing the present state with past state. The existence and uniqueness of an approximate solution of the stochastic delay differential equation were shown and the approximate solution to the unique solution was also shown. .展开更多
The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of soluti...The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.展开更多
A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpos...A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.展开更多
This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment ...This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.展开更多
A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.
In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in ste...In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.展开更多
In this paper, a class of non-autonomous functional integro-differential stochastic equations in a real separable Hilbert space is studied. When the operators A（t） satisfy Acquistapace-Terreni conditions, and with s...In this paper, a class of non-autonomous functional integro-differential stochastic equations in a real separable Hilbert space is studied. When the operators A（t） satisfy Acquistapace-Terreni conditions, and with some suitable assumptions, the existence and uniqueness of a square-mean almost periodic mild solution to the equations are obtained.展开更多
This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solutio...This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.展开更多
An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the ex...An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.展开更多
In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the conv...In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.展开更多
This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the stand...This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.展开更多
This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters.First,we prove the existence and uniqueness of these equations under non-Lipschitz conditions.Second...This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters.First,we prove the existence and uniqueness of these equations under non-Lipschitz conditions.Second,we construct maximum likelihood estimators of these parameters and then discuss their strong consistency.Third,a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered.Finally,we estimate the errors between solutions of these equations and that of their numerical equations.展开更多
A variation of the direct Taylor expansion algorithm is suggested and applied to several linear and nonlinear differential equations of interest in physics and engineering, and the results are compared with those obta...A variation of the direct Taylor expansion algorithm is suggested and applied to several linear and nonlinear differential equations of interest in physics and engineering, and the results are compared with those obtained from other algorithms. It is shown that the suggested algorithm competes strongly with other existing algorithms, both in accuracy and ease of application, while demanding a shorter computation time.展开更多
In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarant...In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarantee of computations with a given precision. The equations of programmed constraints and those of constraint perturbations are defined. The stability of the programmed manifold for numerical solutions of the kinematical and dynamical equations is obtained by corresponding construction of the constraint perturbation equations. The dynamical equations of system with programmed constraints are set up in the form of Lagrange’s equations in generalized coordinates. Certain inverse problems of rigid body dynamics are examined.展开更多
The numerical analysis of heat transfer of laminar nanofluid flow over a fiat stretching sheet is presented. Two sets of boundary conditions （BCs） axe analyzed, i.e., a constant （Case 1） and a linear streamwise va...The numerical analysis of heat transfer of laminar nanofluid flow over a fiat stretching sheet is presented. Two sets of boundary conditions （BCs） axe analyzed, i.e., a constant （Case 1） and a linear streamwise variation of nanopaxticle volume fraction and wall temperature （Case 2）. The governing equations and BCs axe reduced to a set of nonlinear ordinary differential equations （ODEs） and the corresponding BCs, respectively. The dependencies of solutions on Prandtl number Pr, Lewis number Le, Brownian motion number Nb, and thermophoresis number Nt are studied in detail. The results show that the reduced Nusselt number and the reduced Sherwood number increase for the BCs of Case 2 compared with Case 1. The increases of Nb, Nt, and Le numbers cause a decrease of the reduced Nusselt number, while the reduced Sherwood number increases with the increase of Nb and Le numbers. For low Prandtl numbers, an increase of Nt number can cause to decrease in the reduced Sherwood number, while it increases for high Prandtl numbers.展开更多
Nonlinear stochastic modeling plays a significant role in disciplines such as psychology,finance,physical sciences,engineering,econometrics,and biological sciences.Dynamical consistency,positivity,and boundedness are ...Nonlinear stochastic modeling plays a significant role in disciplines such as psychology,finance,physical sciences,engineering,econometrics,and biological sciences.Dynamical consistency,positivity,and boundedness are fundamental properties of stochastic modeling.A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation.Well-known explicit methods such as Euler Maruyama,stochastic Euler,and stochastic Runge–Kutta are investigated for the stochastic model.Regrettably,the above essential properties are not restored by existing methods.Hence,there is a need to construct essential properties preserving the computational method.The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model.The comparison of the results of deterministic and stochastic models is also presented.Our proposed efficient computational method well preserves the essential properties of the model.Comparison and convergence analyses of the method are presented.展开更多
With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the stead...With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steadystate diffusion equation.In this paper,time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process.Based on the original steady-state diffusion equation,we first propose a stochastic partial differential equationmodel.The numerical solution to the stochastic partial differential model is carried out by using the finite element method.When the time resolution is high enough,the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation,which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality.In addition,the process of generating Cerenkov and penetrating in vitro imaging of 18 F radionuclide inmuscle tissue are also first proposed by GEANT4Monte Carlomethod.The result of the GEANT4 simulation is compared with the numerical solution of the corresponding stochastic partial differential equations,which shows that the stochastic partial differential equation can simulate the corresponding process.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
In this paper, a new one-step explicit method of fourth order is derived. The new method is proved to be A-stable and L-stable, and it gives exact results when applied to the test equation y’=λy with Re(λ)【0, Also...In this paper, a new one-step explicit method of fourth order is derived. The new method is proved to be A-stable and L-stable, and it gives exact results when applied to the test equation y’=λy with Re(λ)【0, Also several numerical examples are included.展开更多
文摘To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.
文摘This study is focused on the approximate solution for the class of stochastic delay differential equations. The techniques applied involve the use of Caratheodory and Euler Maruyama procedures which approximated to stochastic delay differential equations. Based on the Caratheodory approximate procedure, it was proved that stochastic delay differential equations have unique solution and established that the Caratheodory approximate solution converges to the unique solution of stochastic delay differential equations under the Cauchy sequence and initial condition. This Caratheodory approximate procedure and Euler method both converge at the same rate. This is achieved by replacing the present state with past state. The existence and uniqueness of an approximate solution of the stochastic delay differential equation were shown and the approximate solution to the unique solution was also shown. .
文摘The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.
基金National Natural Science Foundations of China(Nos.11401261,11471071)Qing Lan Project of Jiangsu Province,China(No.2012)+2 种基金Natural Science Foundation of Higher Education Institutions of Jiangsu Province(No.13KJB110005)the Grant of Jiangsu Second Normal University(No.JSNU-ZY-02)the Jiangsu Government Overseas Study Scholarship,China
文摘A class of stochastic differential equations with random jump magnitudes( SDEwRJMs) is investigated. Under nonLipschitz conditions,the convergence of semi-implicit Euler method for SDEwRJMs is studied. The main purpose is to prove that the semi-implicit Euler solutions converge to the true solutions in the mean-square sense. An example is given for illustration.
文摘This paper proves that, under the local Lipschitz condition, the stochastic functional differential equations with infinite delay have global solutions without the linear growth condition. Furthermore, the pth moment exponential stability conditions are given. Finally, one example is presented to illustrate our theory.
基金National Natural Science Foundation of China(No.71171003)Natural Science Foundation of Anhui Province of China(No.090416225)Natural Science Foundation of Universities of Anhui Province of China(No.KJ2010A037)
文摘A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.
基金Foundation item: Hubei University Youngth Foundations (099206).
文摘In this paper we prove the pathwise uniqueness of a kind of two-parameter Volterra type stochastic differential equations under the coefficients satisfy the non-Lipschitz conditions. We use a martingale formula in stead of Ito formula, which leads to simplicity the process of proof and extends the result to unbounded coefficients case.
基金Acknowledgement This article is funded by the National Natural Science Foundation of China （11161052）, Guangxi Natural Science Foundation of China （201 ljjA10044） and Guangxi Education Hall Project （201012MS183）
文摘In this paper, a class of non-autonomous functional integro-differential stochastic equations in a real separable Hilbert space is studied. When the operators A（t） satisfy Acquistapace-Terreni conditions, and with some suitable assumptions, the existence and uniqueness of a square-mean almost periodic mild solution to the equations are obtained.
文摘This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤0.where aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.
基金National Natural Science Foundation of China(No.11371087)
文摘An initial value problem was considered for a coupled differential system with multi-term Caputo type fractional derivatives. By means of nonlinear alternative of Leray-Schauder and Banach contraction principle,the existence and uniqueness of solutions for the system were derived. Using a fractional predictorcorrector method, a numerical method was presented for the specified system. An example was given to illustrate the obtained results.
文摘In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.
基金partially supported by the National Science and Engineering Research Council of Canada(NSERC)the start-up funds from the University of Calgary
文摘This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.
基金supported by NSF of China(11001051,11371352,12071071)China Scholarship Council(201906095034).
文摘This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters.First,we prove the existence and uniqueness of these equations under non-Lipschitz conditions.Second,we construct maximum likelihood estimators of these parameters and then discuss their strong consistency.Third,a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered.Finally,we estimate the errors between solutions of these equations and that of their numerical equations.
文摘A variation of the direct Taylor expansion algorithm is suggested and applied to several linear and nonlinear differential equations of interest in physics and engineering, and the results are compared with those obtained from other algorithms. It is shown that the suggested algorithm competes strongly with other existing algorithms, both in accuracy and ease of application, while demanding a shorter computation time.
基金Supported by Russian Fund of Fund amental Investigations(Pr.990101064)and Russian Minister of Educatin
文摘In this paper the method of design of kinematical and dynamical equations of mechanical systems, applied to numerical ealization, is proposed. The corresponding difference equations, which are obtained, give a guarantee of computations with a given precision. The equations of programmed constraints and those of constraint perturbations are defined. The stability of the programmed manifold for numerical solutions of the kinematical and dynamical equations is obtained by corresponding construction of the constraint perturbation equations. The dynamical equations of system with programmed constraints are set up in the form of Lagrange’s equations in generalized coordinates. Certain inverse problems of rigid body dynamics are examined.
文摘The numerical analysis of heat transfer of laminar nanofluid flow over a fiat stretching sheet is presented. Two sets of boundary conditions （BCs） axe analyzed, i.e., a constant （Case 1） and a linear streamwise variation of nanopaxticle volume fraction and wall temperature （Case 2）. The governing equations and BCs axe reduced to a set of nonlinear ordinary differential equations （ODEs） and the corresponding BCs, respectively. The dependencies of solutions on Prandtl number Pr, Lewis number Le, Brownian motion number Nb, and thermophoresis number Nt are studied in detail. The results show that the reduced Nusselt number and the reduced Sherwood number increase for the BCs of Case 2 compared with Case 1. The increases of Nb, Nt, and Le numbers cause a decrease of the reduced Nusselt number, while the reduced Sherwood number increases with the increase of Nb and Le numbers. For low Prandtl numbers, an increase of Nt number can cause to decrease in the reduced Sherwood number, while it increases for high Prandtl numbers.
基金the Research and initiative center COVID-19-DES-2020-65,Prince Sultan University.
文摘Nonlinear stochastic modeling plays a significant role in disciplines such as psychology,finance,physical sciences,engineering,econometrics,and biological sciences.Dynamical consistency,positivity,and boundedness are fundamental properties of stochastic modeling.A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation.Well-known explicit methods such as Euler Maruyama,stochastic Euler,and stochastic Runge–Kutta are investigated for the stochastic model.Regrettably,the above essential properties are not restored by existing methods.Hence,there is a need to construct essential properties preserving the computational method.The non-standard approach of finite difference is examined to maintain the above basic features of the stochastic model.The comparison of the results of deterministic and stochastic models is also presented.Our proposed efficient computational method well preserves the essential properties of the model.Comparison and convergence analyses of the method are presented.
基金National Science Foundation of China(NSFC)(61671009,12171178).
文摘With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steadystate diffusion equation.In this paper,time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process.Based on the original steady-state diffusion equation,we first propose a stochastic partial differential equationmodel.The numerical solution to the stochastic partial differential model is carried out by using the finite element method.When the time resolution is high enough,the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation,which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality.In addition,the process of generating Cerenkov and penetrating in vitro imaging of 18 F radionuclide inmuscle tissue are also first proposed by GEANT4Monte Carlomethod.The result of the GEANT4 simulation is compared with the numerical solution of the corresponding stochastic partial differential equations,which shows that the stochastic partial differential equation can simulate the corresponding process.
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In this paper, a new one-step explicit method of fourth order is derived. The new method is proved to be A-stable and L-stable, and it gives exact results when applied to the test equation y’=λy with Re(λ)【0, Also several numerical examples are included.
