Numerical solution of systems of random differential equations with Gaussian statistics

A direct approach is used to compute a numerical solution for a system of (nonlinear) ordinary differential equations with Gaussian statistics. It is shown that for a large class of problems the accuracy of the solution can be made as great as desired by taking the step size h sufficiently small.     

Numerical solution of systems of random differential equations with Gaussian statistics

A direct approach is used to compute a numerical solution for a system of (nonlinear) ordinary differential equations with Gaussian statistics. It is shown that for a large class of problems the accuracy of the solution can be made as great as desired by taking the step size h sufficiently small.     

Numerical solution of stochastic differential equations with random structure on supercomputers

In this paper, the accuracy of estimating the expectation of solutions to stochastic differential equations with random structure is investigated. This accuracy is shown as a function of the integration step in the generalized Euler method and the number of simulated trajectories. A drastic decrease in the accuracy at deterministic or random times of changing the SDE structure is shown by a simple equation as an example. This could be improved by the use of supercomputers for statistical modeling. The results of some numerical experiments carried out at the Siberian Supercomputer Center are presented.     

Numerical solutions of random mean square Fisher-KPP models with advection

This paper deals with the construction of numerical stable solutions of random mean square Fisher-Kolmogorov-Petrosky-Piskunov (Fisher-KPP) models with advection. The construction of the numerical scheme is performed in two stages. Firstly, a semidiscretization technique transforms the original continuous problem into a nonlinear inhomogeneous system of random differential equations. Then, by extending to the random framework, the ideas of the exponential time differencing method, a full vector discretization of the problem addresses to a random vector difference scheme. A sample approach of the random vector difference scheme, the use of properties of Metzler matrices and the logarithmic norm allow the proof of stability of the numerical solutions in the mean square sense. In spite of the computational complexity, the results are illustrated by comparing the results with a test problem where the exact solution is known.     

Numerical solution of stochastic differential equations

Numerical solution of stochastic differential equations, P. E. Kioeden and E. Platen, Springer-Verlag (1992). 632 pp. $59.00 ISBN 0-386-54062-8     

Solving the random diffusion model in an infinite medium: A mean square approach

This paper deals with the construction of an analytic-numerical mean square solution of the random diffusion model in an infinite medium. The well-known Fourier transform method, which is used to solve this problem in the deterministic case, is extended to the random framework. Mean square operational rules to the Fourier transform of a stochastic process are developed and stated. The main statistical moments of the stochastic process solution are also computed. Finally, some illustrative numerical examples are included.     

Explicit criteria for mean square exponential stability of stochastic differential equations

Using a novel approach, we present new explicit criteria for the mean square exponential stability of general non-linear stochastic differential equations. An application to stochastic neural networks is given.     

Numerical diagnostics of solution blowup in differential equations

New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.     

Numerical solution of stochastic differential equations on Supercomputers

This paper deals with some problems of accuracy of algorithms for the numerical solutions of stochastic differential equations (SDEs) versus the size of the ensemble of trajectories simulated. The problems of accuracy arise in estimating functionals of SDE solutions with increasing variance, highly asymmetric distributions, and an indefinite time of arrival of trajectories of solutions at the boundaries of given domains. Some ways of parallelization of statistical algorithms on a multiprocessor cluster are described. Results of numerical experiments performed on a supercomputer available at the Siberian Supercomputer Center are presented.     

Exponential mean square stability of numerical methods for systems of stochastic differential equations

This paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split-step theta method for stochastic systems. First, we consider linear autonomous systems. Under a sufficient and necessary condition for exponential mean square stability of the exact solution, it is proved that the two classes of theta methods with θ≥0.5$\theta \ge 0.5$ are exponentially mean square stable for all positive step sizes and the methods with θ<0.5$\theta <0.5$ are stable for some small step sizes. Then, we study the stability of the methods for nonlinear non-autonomous systems. Under a coupled condition on the drift and diffusion coefficients, it is proved that the split-step theta method with θ>0.5$\theta >0.5$ still unconditionally preserves the exponential mean square stability of the underlying systems, but the stochastic theta method does not have this property. Finally, we consider stochastic differential equations with jumps. Some similar results are derived.     

Numerical solution of random differential initial value problems: Multistep methods

This paper deals with the construction of numerical methods of random initial value problems. Random linear multistep methods are presented and sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical approach to differential equations of fractional order

In this paper, the variational iteration method and the Adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. This paper presents a numerical comparison between the two methods for solving systems of fractional differential equations. Numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order.     

Numerical solution of fuzzy differential equations under generalized differentiability

In this paper, we interpret a fuzzy differential equation by using the strongly generalized differentiability concept. Utilizing the Generalized Characterization Theorem, we investigate the problem of finding a numerical approximation of solutions. Then we show that any suitable numerical method for ODEs can be applied to solve numerically fuzzy differential equations under generalized differentiability. The generalized Euler approximation method is implemented and its error analysis, which guarantees pointwise convergence, is given. The method’s applicability is illustrated by solving a linear first-order fuzzy differential equation.     

Numerical solution of fractional order differential equations by extrapolation

We present an extrapolation type algorithm for the numerical solution of fractional order differential equations. It is based on the new result that the sequence of approximate solutions of these equations, computed by means of a recently published algorithm by Diethelm [6], possesses an asymptotic expansion with respect to the stepsize. From this we conclude that the application of extrapolation is justified, and we obtain a very efficient differential equation solver with practically no additional numerical costs. This is also illustrated by a number of numerical examples. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mean square convergent three points finite difference scheme for random partial differential equations

In this paper, the random finite difference method with three points is used in solving random partial differential equations problems mainly: random parabolic, elliptic and hyperbolic partial differential equations. The conditions of the mean square convergence of the numerical solutions are studied. The numerical solutions are computed through some numerical case studies.     

A new approach to monotone iterative technique for solution of differential equations

We discuss a monotone interative technique for initial value problems. The novelty of the approach is the use of g-monotonicity, i.e. of monotonicity of the ratio of two functions. This is in contrast to the more usual notion of monotonicity