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Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method
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作者 Ayad R. Khudair S. A. M. Haddad Sanaa L. Khalaf 《Open Journal of Applied Sciences》 2016年第4期287-297,共11页
The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results sh... The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results show that DTM is an efficient and accurate technique for finding exact and approximate solutions. 展开更多
关键词 random differential equations Stochastic differential equation differential Transformation Method
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Finite Difference Scheme for Solving Parabolic Partial Differential Equations with Random Variable Input under Mean Square Sense
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作者 M. A. Sohaly W. W. Mohammed 《Journal of Mathematics and System Science》 2016年第7期263-275,共13页
This study deal with seven points finite difference method to find the approximation solutions in the area of mean square calculus solutions for linear random parabolic partial differential equations. Several numerica... This study deal with seven points finite difference method to find the approximation solutions in the area of mean square calculus solutions for linear random parabolic partial differential equations. Several numerical examples are presented to show the ability and efficiency of this method. 展开更多
关键词 Mean Square Convergence random Partial differential equations Finite Difference Technique.
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Random Crank-Nicolson Scheme for Random Heat Equation in Mean Square Sense 被引量:1
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作者 M. T. Yassen M. A. Sohaly Islam Elbaz 《American Journal of Computational Mathematics》 2016年第2期66-73,共8页
The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for s... The goal of computational science is to develop models that predict phenomena observed in nature. However, these models are often based on parameters that are uncertain. In recent decades, main numerical methods for solving SPDEs have been used such as, finite difference and finite element schemes [1]-[5]. Also, some practical techniques like the method of lines for boundary value problems have been applied to the linear stochastic partial differential equations, and the outcomes of these approaches have been experimented numerically [7]. In [8]-[10], the author discussed mean square convergent finite difference method for solving some random partial differential equations. Random numerical techniques for both ordinary and partial random differential equations are treated in [4] [10]. As regards applications using explicit analytic solutions or numerical methods, a few results may be found in [5] [6] [11]. This article focuses on solving random heat equation by using Crank-Nicol- son technique under mean square sense and it is organized as follows. In Section 2, the mean square calculus preliminaries that will be required throughout the paper are presented. In Section 3, the Crank-Nicolson scheme for solving the random heat equation is presented. In Section 4, some case studies are showed. Short conclusions are cleared in the end section. 展开更多
关键词 random Partial differential equations (RPDEs) Mean Square Sense (m.s) Second Order random Variable (2r.v.'s) random Crank-Nicolson Scheme CONVERGENCE CONSISTENCY Stability
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Some Notes of p-Moment Boundedness of Nonlinear Differential Equation with Pandom Impulses
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作者 赵佃立 《Journal of Shanghai Jiaotong university(Science)》 EI 2006年第3期384-388,共5页
A model of nonlinear differential systems with impulsive effect on random moments is considered. The extensions of qualitative analysis of the model is mainly focused on and three modified sufficient conditions are pr... A model of nonlinear differential systems with impulsive effect on random moments is considered. The extensions of qualitative analysis of the model is mainly focused on and three modified sufficient conditions are presented about p-moment boundedness in the process by Liapunov method with nonlinear item dependent on the impulsive effects, which may gain wider use in industrial engineering, physics, etc. At last, an example is given to show an theoretical application of the obtained results. 展开更多
关键词 p-moment boundedness nonlinear differential equation with random impulses Liapunov function
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A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients
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作者 Junlong Lyu Zhiwen Zhang 《Communications in Mathematical Research》 CSCD 2020年第2期154-192,共39页
We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,... We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation. 展开更多
关键词 random partial differential equations(RPDEs) uncertainty quantification(UQ) multiscale finite element method(MsFEM) multilevel Monte Carlo(MLMC) reduced basis convergence analysis
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On Existence and Uniqueness of Random Impulsive Differential Equations 被引量:2
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作者 ZHANG Shuorui SUN Jitao 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2016年第2期300-314,共15页
This paper studies existence and uniqueness for random impulsive differential equations.The authors first generalize a random fixed point theorem of Schaefer's type.Then the authors shall rely on the generalized S... This paper studies existence and uniqueness for random impulsive differential equations.The authors first generalize a random fixed point theorem of Schaefer's type.Then the authors shall rely on the generalized Schaefer's type random fixed point theorem to discuss the existence of the system.In addition,the authors study the existence and uniqueness of random impulsive differential equations by applying random Banach fixed point theorem and obtain some less conservative results.Finally,an example is given to illustrate the effectiveness of the results. 展开更多
关键词 EXISTENCE IMPULSES random differential equations random fixed point uniqueness.
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Quasimonotone random and stochastic functional differential equations with applications
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作者 Xiaoming Bai Jifa Jiang Tianyuan Xu 《Science China Mathematics》 SCIE CSCD 2023年第9期2021-2056,共36页
In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random ... In this paper, we study monotone properties of random and stochastic functional differential equations and their global dynamics. First, we show that random functional differential equations(RFDEs)generate the random dynamical system(RDS) if and only if all the solutions are globally defined, and establish the comparison theorem for RFDEs and the random Riesz representation theorem. These three results lead to the Borel measurability of coefficient functions in the Riesz representation of variational equations for quasimonotone RFDEs, which paves the way following the Smith line to establish eventual strong monotonicity for the RDS under cooperative and irreducible conditions. Then strong comparison principles, strong sublinearity theorems and the existence of random attractors for RFDEs are proved. Finally, criteria are presented for the existence of a unique random equilibrium and its global stability in the universe of all the tempered random closed sets of the positive cone. Applications to typical random or stochastic delay models in monotone dynamical systems,such as biochemical control circuits, cyclic gene models and Hopfield-type neural networks, are given. 展开更多
关键词 quasimonotone random and stochastic functional differential equations eventual strong mono-tonicity strong comparison theorems strong sublinearity random attractor global convergence to random equilibrium
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GAUSS WHITE NOISE PERTURBATIONS OF NONHOLONOMIC MECHANICAL SYSTEMS
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作者 申泽淳 刘风丽 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1993年第10期945-952,共8页
The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper. It is proved that the differential equations of the first-order moments of the solution process coincide wit... The perturbations of nonholonomic mechanical systems under the Gauss white noises are studied in this paper. It is proved that the differential equations of the first-order moments of the solution process coincide with the corresponding equations in the non-perturbational case, and that there are e2 -terms but no e-terms in the differential equations of the second-order moments. Two propositions are obtained. Finally, an example is given to illustrate the application of the results. 展开更多
关键词 nonholonomic systems white noises random differential equations
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Efficient Collocational Approach for Parametric Uncertainty Analysis 被引量:10
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作者 Dongbin Xiu 《Communications in Computational Physics》 SCIE 2007年第2期293-309,共17页
A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented.The uncertain parameters are modeled as random variables,and the governing equations are treated as sto... A numerical algorithm for effective incorporation of parametric uncertainty into mathematical models is presented.The uncertain parameters are modeled as random variables,and the governing equations are treated as stochastic.The solutions,or quantities of interests,are expressed as convergent series of orthogonal polynomial expansions in terms of the input random parameters.A high-order stochastic collocation method is employed to solve the solution statistics,and more importantly,to reconstruct the polynomial expansion.While retaining the high accuracy by polynomial expansion,the resulting“pseudo-spectral”type algorithm is straightforward to implement as it requires only repetitive deterministic simulations.An estimate on error bounded is presented,along with numerical examples for problems with relatively complicated forms of governing equations. 展开更多
关键词 Collocation methods pseudo-spectral methods stochastic inputs random differential equations uncertainty quantification.
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