Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Solving the Nonlinear Variable Order Fractional Differential Equations by Using Euler Wavelets

An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper.The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented.Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations.And the convergence of the Euler wavelets basis is given.The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

A SEMI-ANALYSIS METHOD OF DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS UNDER COMPLICATED BOUNDARY CONDITIONS

Based on a method of finite element model and combined with matrix theory, a method for solving differential equation with variable coefficients is proposed. With the method, it is easy to deal with the differential equations with variable coefficients. On most occasions and due to the nonuniformity nature, nonlinearity property can cause the equations of the kinds. Using the model, the satisfactory valuable results with only a few units can be obtained.

EXISTENCE OF MEROMORPHIC SOLUTIONS OF SOME HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS

This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.

Using reproducing kernel for solving a class of partial differential equation with variable-coefficients

How to solve the partial differential equation has been attached importance to by all kinds of fields. The exact solution to a class of partial differential equation with variable-coefficient is obtained in reproducing kernel space. For getting the approximate solution, give an iterative method, convergence of the iterative method is proved. The numerical example shows that our method is effective and good practicability.

ON THE DEPENDENT PROPERTY OF SOLUTIONS FOR HIGHER ORDER PERIODIC DIFFERENTIAL EQUATIONS

This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.

A numerical method based on boundary integral equations and radial basis functions for plane anisotropic thermoelastostatic equations with general variable coefficients

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.

INSTABILITY OF SOLUTION FOR THE FOURTH ORDER LINEAR DIFFERENTIAL EQUATION WITH VARIED COEFFICIENT

In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.

一类漂移系数分段连续的随机微分方程驯服Euler方法的L^(p)收敛率

本文研究一类漂移系数分段连续的标量随机微分方程的驯服Euler方法的L^(p)收敛率.更确切地说,本文在漂移系数是分段连续的并且呈多项式增长,扩散系数是Lipschitz连续的并且在漂移系数的间断点处不为0的假设下,证明方程具有唯一的强解,并且对于任意的p∈[1,∞),驯服Euler方法的L^(p)收敛阶都可以达到1/2.此外,本文还提供一个数值算例来验证理论结果.

Mean Square Numerical Methods for Initial Value Random Differential Equations

In this paper, the random Euler and random Runge-Kutta of the second order methods are used in solving random differential initial value problems of first order. The conditions of the mean square convergence of the numerical solutions are studied. The statistical properties of the numerical solutions are computed through numerical case studies.

Comparative Study of Radial Basis Functions for PDEs with Variable Coefficients

The radial basis functions(RBFs)play an important role in the numerical simulation processes of partial differential equations.Since the radial basis functions are meshless algorithms,its approximation is easy to implement and mathematically simple.In this paper,the commonly⁃used multiquadric RBF,conical RBF,and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients.Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems.It is shown that the conical RBF numerical results were more stable than the other two radial basis functions.From the comparison of three commonly⁃used RBFs,one may obtain the best numerical solutions for boundary value problems.

EXACT ANALYTIC METHOD FG(?) SOLVING VARIABLE COEFFICIENT DIFFERENTIAL EQUATION

Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation formal is obtained. Its convergence in proved. We can get analytic expressions which converge to exact solution and its higher order derivatives uniformy Four numerical examples are given, which indicate that satisfactory results can he obtanedby this method.

BOUND ON SOLUTIONS TO THE GENERAL SYSTEM OF VOLTERRA－TYPE LINEAR INTEGRAL INEQUALITIES IN SEVERAL VARIABLES AND ITS APPLICATIONS TO INTEGRO－PARTIAL DIFFERENTIAL EQUATIONS

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Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay

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.

The theoretical analysis of dynamic response on cantilever beam of variable stiffness

The paper presents the theoretical analysis of a variable stiffness beam. The bending stiffness EI varies continuously along the length of the beam. Dynamic equation yields differential equation with variable coefficients based on the model of the Euler-Bernoulli beam. Then differential equation with variable coefficients becomes that with constant coefficients by variable substitution. At last,the study obtains the solution of dynamic equation. The cantilever beam is an object for analysis. When the flexural rigidity at free end is a constant and that at clamped end is varied,the dynamic characteristics are analyzed under several cases. The results demonstrate that the natural angular frequency reduces as the flexural rigidity reduces. When the rigidity of clamped end is higher than that of free end,low-level mode contributes the larger displacement response to the total response. On the contrary,the contribution of low-level mode is lesser than that of high-level mode.

Contribution to the Analytical Equation Resolution Using Charts for Analysis and Design of Cylindrical and Conical Open Surge Tanks

In the event of an instantaneous valve closure, the pressure transmitted to a surge tank induces the mass fluctuations that can cause high amplitude of water-level fluctuation in the surge tank for a reasonable cross-sectional area. The height of the surge tank is then designed using this high water level mark generated by the completely closed penstock valve. Using a conical surge tank with a non-constant cross-sectional area can resolve the problems of space and height. When addressing issues in designing open surge tanks, key parameters are usually calculated by using complex equations, which may become cumbersome when multiple iterations are required. A more effective alternative in obtaining these values is the use of simple charts. Firstly, this paper presents and describes the equations used to design open conical surge tanks. Secondly, it introduces user-friendly charts that can be used in the design of cylindrical and conical open surge tanks. The contribution can be a benefit for practicing engineers in this field. A case study is also presented to illustrate the use of these design charts. The case study’s results show that key parameters obtained via successive approximation method required 26 iterations or complex calculations, whereas these values can be obtained by simple reading of the proposed chart. The use of charts to help surge tanks designing, in the case of preliminary designs, can save time and increase design efficiency, while reducing calculation errors.

OSCILLATIONS OF SOLUTIONS OF NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS AND DELAYS

Consider the neutral differential equations with variable coefficients and delays [x(t)-p(t)x(t-r(t))]'+ Qj(t)x(t-σj(t))=0. (1)We establish sufficient conditions for the oscillation of equation (1). Our condition is 'sharp' in the sense that when all the coefficients and delays of the equation are constants.Our conclusions improve and generalize some known results.

Fuzzy Modeling, Tracking Control and Synchronization of the Rossler's Chaotic System

In this paper, a novel method to model, track control and synchronize the Rossler’s chaotic system is proposed. The fuzzy logical system is used so that the fuzzy inference rule is transferred into a type of variable coefficient nonlinear ordinary differential equation. Consequently the model of the chaotic system is obtained. Then a fuzzy tracking control and a fuzzy synchronization for chaotic systems is proposed as well. First, a known tracking control for the Rossler’s system is used in this paper. We represent the Rossler’s chaotic and control systems into fuzzy inference rules. Then the variable coefficient nonlinear ordinary differential equation is also got. Simulation results show that such an approach is effective and has a high precision.

带Caputo导数的变分数阶随机微分方程的Euler-Maruyama方法

该文构造了Euler-Maruyama(EM)方法求解一类带Caputo导数的变分数阶随机微分方程.首先,证明了该方程的适定性;然后,详细推导出对应的EM方法,并对该方法进行了强收敛性的分析,通过使用EM方法的连续形式证明了其强收敛阶为β-0.5,其中β是Caputo导数的阶数,且满足0.5<β<1.最后,通过数值实验验证了理论分析结果的正确性.