Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space,disturbed by Gaussian processes. Part I

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.     

Orthogonal expansions of the nonlinear functionals of the Gaussian process and their application to statistical analysis

Orthogonal expansion of the space of quadratically integrable nonlinear functionals of the Gaussian random process was considered. The problem of explicit construction of the expansion terms was restated as generation of the differential equations for its coordinate functions. The method of canonical expansions from the linear statistical analysis was generalized to the nonlinear case on the basis of the results obtained.     

Electromagnetic imaging of buried perfectly conducting cylinder targets using the dynamic differential evolution

Dynamic differential evolution (DDE) for shape reconstruction of perfect conducting cylinder buried in a half‐space is presented. Assume that a conducting cylinder of unknown shape is buried in one half‐space and scatters the field incident from another half‐space where the scattered filed is measured. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated into an optimization problem. The inverse problem is resolved by an optimization approach, and the global searching scheme DDE is then used to search the parameter space. Numerical results demonstrate that even when the initial guess is far away from the exact one, good reconstruction can be obtained by using DDE both with and without the additive Gaussian noise. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2012.     

Probabilistic solutions of some multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise

In this paper, the state-space-split method is extended for the dimension reduction of some high-dimensional Fokker–Planck–Kolmogorov equations or the nonlinear stochastic dynamical systems in high dimensions subject to external excitation which is the filtered Gaussian white noise governed by the second order stochastic differential equation. The selection of sub state variables and then the dimension-reduction procedure for a class of nonlinear stochastic dynamical systems is given when the external excitation is the filtered Gaussian white noise. The stretched Euler–Bernoulli beam with hinge support at two ends, point-spring supports, and excited by uniformly distributed load being filtered Gaussian white noise governed by the second-order stochastic differential equation is analyzed and numerical results are presented. The results obtained with the presented procedure are compared with those obtained with the Monte Carlo simulation and equivalent linearization method to show the effectiveness and advantage of the state-space-split method and exponential polynomial closure method in analyzing the stationary probabilistic solutions of the multi-degree-of-freedom nonlinear stochastic dynamical systems excited by filtered Gaussian white noise.     

Simulation of constraints by compliant joints

Application of models of compliant (flexible) joints instead of constraint equations for systems with closed-loop kinematic chains is considered; this makes it possible to replace differential-algebraic motion equations by stiff ordinary differential equations. For substantiation of this replacement, methods of analysis of singularly perturbed equations are used. Examples of mathematical models of compliant joints and expressions for approximate Jacobian matrices corresponding to these force interactions are given. Numerical solutions of direct and inverse dynamics equations for a number of controlled mechanisms are used for testing the method and its program realization.     

多模态多目标差分进化算法求解非线性方程组

针对当前算法在求解非线性方程组时面临解的个数不完整、精确度不高、收敛速度慢等问题进行了研究,提出一种多模态多目标差分进化算法。首先将非线性方程组转换为多模态多目标优化问题,初始化一个随机种群并对种群中全部个体进行评价;然后通过非支配解排序和决策空间拥挤距离选择机制,挑选种群中的一半优质个体进行变异;接着在变异过程中采用一种新的变异策略和边界处理方法以增加解的多样性;最后通过交叉和选择机制使优质个体进行进化,直到搜索到全部最优解。在所选测试函数集和工程实例上的实验结果表明,该算法能有效地搜索到非线性方程组的解,并通过与当前四个算法进行比较,该算法在解的数量和成功率上具有优越性。     

Perturbation theorems for systems with impulse effect

In this paper two systems of ordinary differential equations with an impulse effect and their respective perturbed systems are considered. Sufficient conditions are found for which some solutions of the perturbed systems tend to zero as t→∞. Assuming that these systems have zero solutions, their stability is proved.     

Direct solutions of nth-order IVPs by homotopy-perturbation method

In this paper, a class of linear and nonlinear nth-order initial value problems (IVPs) is considered. The solutions of these IVPs are obtained by the homotopy-perturbation method (HPM). The HPM can be considered as one of the new methods belonging to the general classification of perturbation methods. Generally, the HPM deals with exact solvers for linear differential equations and approximative solvers for nonlinear equations. Several test cases are chosen to demonstrate the efficiency of HPM.     

On the mathematical modelling of physical systems by ordinary differential stochastic equations

This paper deals with the problem of defining and analysing a general mathematical model for the analysis of physical systems described by ordinary stochastic differential equations with random coefficients and initial conditions. The existence, continuity and stability of the evolution process defined by the considered class of evolution equations is here considered. Some actual models of physical systems are also considered and related to the general mathematical model which is proposed and mathematically analyzed in this work.     

The Exp-function Method for Some Time-fractional Differential Equations

In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie's. This approach can also be applied to other nonlinear time and space fractional differential equations.      

Formal solutions for a class of stochastic pursuit-evasion games

A class of differential pursuit-evasion games is examined in which the dynamics are linear and perturbed by additive white Gaussian noise, the performance index is quadratic, and both players receive measurements perturbed independently by additive white Gaussian noise. Linear minimax solutions are characterized in terms of a set of implicit integro-differential equations. A game of this type also possesses a "certainty-coincidence" property, meaning that its minimax behavior coincides with that of the corresponding deterministic game in the event that all noise values are zero. This property is used to decompose the minimax strategies into sums of a certainty-equivalent term and error terms.     

Alternative Method of Solution of the Regulator Equation: L2‐Space Approach

An alternative method for the proof of solvability of the differential equation that is a part of the regulator equation which arises from the solution of the output regulation problem. The proof uses the L²‐space based theory of solutions of partial differential equations for the case of the linear output regulation problem. In the nonlinear case, a sequence of linear equations is defined so that their solutions converge to the solution of the nonlinear problem. This is proved using the Banach Contraction Theorem. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Evolution of Probability Distribution in Time for Solutions of Hyperbolic Equations

We investigate the evolution of the probability distribution function in time for some wave and Maxwell equations in random media for which the parameters, e.g. permeability, permittivity, fluctuate randomly in space; more precisely, two different media interface randomly in space. We numerically compute the probability distribution and density for output solutions. The underlying numerical and statistical techniques are the so-called polynomial chaos Galerkin projection, which has been extensively used for simulating partial differential equations with uncertainties, and the Monte Carlo simulations.     

测度性奇异脉冲微分系统的某些基本理论的研究

具有脉冲解的奇异测度微分系统的基本理论至今尚未建立，本文对这一问题进行了研究，首先建立了奇异测度微分系统与相应的奇异测度积分系统的等性，然后借助地Ｄｒａｚｉｎ逆，在初始条件相容的情况下给出了线性奇异测度微分系统解的表达式，最后得到了一类非性奇异测度微分系统的常数变易公式。     

Stochastic differential equations with variable structure driven by multiplicative Gaussian noise and sliding mode dynamic

This work is concerned with existence of weak solutions to discontinuous stochastic differential equations driven by multiplicative Gaussian noise and sliding mode control dynamics generated by stochastic differential equations with variable structure, that is with jump nonlinearity. The treatment covers the finite dimensional stochastic systems and the stochastic diffusion equation with multiplicative noise.     

New complex solutions for some special nonlinear partial differential systems

In this present work, we explore new applications of direct algebraic method for some special nonlinear partial differential systems and equations. Then new types of complex travelling wave solutions are obtained to the Davey-Stewartson system, the coupled Higgs system and the perturbed nonlinear Schrodinger’s equation; the balance number of it is not a positive integer.     

Asymptotic equilibrium and stability of fuzzy differential equations

The local existence and uniqueness theorems and the global existence of solutions were investigated in [1–3], respectively, for the Cauchy problem of fuzzy-valued functions of a real variable whose values are in the fuzzy number space (Eⁿ, D). In this paper, we first study the asymptotic equilibrium for fuzzy evolution equations. Then, the stability properties of the trivial fuzzy solution of the perturbed semilinear fuzzy evolution equations are investigated by extending the Lyapunov's direct method.     

Solving random diffusion models with nonlinear perturbations by the Wiener–Hermite expansion method

This paper deals with the construction of approximate series solutions of random nonlinear diffusion equations where nonlinearity is considered by means of a frank small parameter and uncertainty is introduced through white noise in the forcing term. For the simpler but important case in which the diffusion coefficient is time independent, we provide a Gaussian approximation of the solution stochastic process by taking advantage of the Wiener–Hermite expansion together with the perturbation method. In addition, approximations of the main statistical functions associated with a solution, such as the mean and variance, are computed. Numerical values of these functions are compared with respect to those obtained by applying the Runge–Kutta second-order stochastic scheme as an illustrative example.     

Computational stochastic dynamic programming on a vectormultiprocessor

Numerical methods which have been developed to solve optimal feedback control problems for nonlinear, continuous-time dynamical systems, perturbed by Poisson as well as by Gaussian random white noise, are discussed. Predictor-corrector methods are modified for the nonstandard functional partial differential equation of stochastic dynamic programming to treat nonlinearities attributable to quadratic costs and nonsmoothness attributable to control switching. This numerical formulation is highly suitable for vectorization and parallelization techniques. Advanced computing techniques and hardware are used to help alleviate Bellman's curse of dimensionality in dynamic programming computations     

Solutions generating technique for Abel-type nonlinear ordinary differential equations

A second-order nonlinear differential equation describing the dynamics of a bulk viscous fluid filled general relativistic Friedmann-Robertson-Walker Universe is considered. By means of appropriate transformations, the evolution equation of the space-time is transformed into a third-order polynomial Abel differential equation. With the use of some particular solutions of this equation, repeated classes of exact solutions are generated.