Stochastic Ordinary Differential Equations Research Articles

Numerical solution of stochastic ordinary differential equations using HAAR wavelet collocation method

In the present investigation, we developed an efficient method for numerical solution of stochastic ordinary differential equations (SODE) using Haar wavelets. First, we are analyzing the properties of SODE and Haar wavelets. Next, in order to find the numerical solution of SODE Haar wavelets coefficient and operational matrix of integration is generated and employed. Convergence and error analysis of the proposed method is presented. To demonstrate the strength and effectiveness of the proposed scheme, we give some examples. The numerical computational is conducted to ensure that the method is accurate. The numerical and graphical results obtained are reported, the scheme proposed for studying and finding the solution of SODE is very realistic and accurate computationally.

Predicting the effectiveness of chemotherapy using stochastic ODE models of tumor growth

Ordinary differential equation (ODE) models of cancer growth are often used to predict tumor growth and form the basis for more complex models used in personalized medicine. Unfortunately, ODE models provide predictions of the average behavior of the cell population neglecting the fact that cells are discrete objects subject to discrete events. This kind of stochasticity can dramatically change the time course of tumor growth, particularly when the cell population is small. Here, we investigate stochastic versions of seven common ODE models of cancer growth to determine the role of stochasticity in eradicating tumors via chemotherapy. We find that stochasticity leads to differences in predictions among the different models of both the level of chemotherapy needed to cure a tumor and the time it takes to achieve a cure. Our results highlight the need for more investigation of which model provides the best description of cancer growth.

Lie Point Symmetries of Autonomous Scalar First-Order Itô Stochastic Delay Ordinary Differential Equations

Lie Point Symmetries of Autonomous Scalar First-Order Itô Stochastic Delay Ordinary Differential Equations

Numerical Investigation of Thermal Stability of Catalyst Granules with Internal Heat Generation in a Random Temperature Field

Method for numerical simulation of temperature of granules with internal heat release in a medium with random temperature fluctuations it is proposed. The method utilized solution of a system of ordinary stochastic differential equations describing temperature fluctuations of surrounding and granules. Autocorrelation function of temperature fluctuations has a finite decay time. The suggested method is verified by the comparison with exact analytical results. Random temperature behavior of a granule with internal heat release qualitatively differs from the results obtained in the deterministic approach. Mean first passage time of granules temperature intersects critical temperature is estimated at different regime parameters.

Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems

AbstractThis paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. The considered systems are described by second-order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit, drift-truncated Euler scheme that fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semistability and moment growth properties of the scheme.

Stochastic Bifurcations of Group-Invariant Solutions for a Generalized Stochastic Zakharov–Kuznetsov Equation

In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic bifurcations of group-invariant solutions for a generalized stochastic Zakharov–Kuznetsov equation.

A white noise approach to evolutionary ecology

Although the evolutionary response to random genetic drift is classically modelled as a sampling process for populations with fixed abundance, the abundances of populations in the wild fluctuate over time. Furthermore, since wild populations exhibit demographic stochasticity and since random genetic drift is in part due to demographic stochasticity, theoretical approaches are needed to understand the role of demographic stochasticity in eco-evolutionary dynamics. Here we close this gap for quantitative characters evolving in continuously reproducing populations by providing a framework to track the stochastic dynamics of abundance density across phenotypic space using stochastic partial differential equations. In the process we develop a set of heuristics to operationalize the powerful, but abstract theory of white noise and diffusion-limits of individual-based models. Applying these heuristics, we obtain stochastic ordinary differential equations that generalize classical expressions of ecological quantitative genetics. In particular, by supplying growth rate and reproductive variance as functions of abundance densities and trait values, these equations track population size, mean trait and additive genetic variance responding to mutation, demographic stochasticity, random genetic drift, deterministic selection and noise-induced selection. We demonstrate the utility of our approach by formulating a model of diffuse coevolution mediated by exploitative competition for a continuum of resources. In addition to trait and abundance distributions, this model predicts interaction networks defined by niche-overlap, competition coefficients, or selection gradients. Using a high-richness approximation, we find linear selection gradients and competition coefficients are uncorrelated, but magnitudes of linear selection gradients and quadratic selection gradients are both positively correlated with competition coefficients. Hence, competing species that strongly affect each other’s abundance tend to also impose selection on one another, but the directionality is not predicted. This approach contributes to the development of a synthetic theory of evolutionary ecology by formalizing first principle derivations of stochastic models tracking feedbacks of biological processes and the patterns of diversity they produce.

Tutorial: Langevin Dynamics methods for aerosol particle trajectory simulations and collision rate constant modeling

The Langevin Dynamics (LD) method (also known in the literature as Brownian Dynamics) is routinely used to simulate aerosol particle trajectories for transport rate constant calculations as well as to understand aerosol particle transport in internal and external fluid flows. This tutorial intends to explain the methodological details of setting up a LD simulation of a population of aerosol particles and to deduce rate constants from an ensemble of classical trajectories. We discuss the applicability and limitations of the translational Langevin equation to model the combined stochastic and deterministic motion of particles in fields of force or fluid flow. The drag force and stochastic “diffusion” force terms that appear in the Langevin equation are discussed elaborately, along with a summary of common forces relevant to aerosol systems (electrostatic, gravity, van der Waals, …); a commonly used first order and a fourth order Runge-Kutta time stepping schemes for linear stochastic ordinary differential equations are presented. A MATLAB® implementation of a LD computer code for simulating particle settling under gravity using the first order scheme is included for illustration. Scaling analysis of aerosol transport processes and the selection of timestep and domain size for trajectory simulations are demonstrated through two specific aerosol processes: particle diffusion charging and coagulation. Fortran® implementations of the first order and fourth order time-stepping schemes are included for simulating the 3D motion of a particle in a periodic domain. Potential applications and caveats to the usage of LD are included as a summary.

Relations between cumulants and central moments and their applications

At the beginning of this paper, we consider the solution to the problem of constructing relations between mixed central moments and cumulants (semi-invariants) of an arbitrary vector random variable. The sought relations are derived on the basis of formal operations over the Maclaurin series, which are various expansions of the characteristic function of the random vector. In the case under consideration, the coefficients of the expansions are mixed moments, cumulants, and central moments. One of the applications for the recurrence relations obtained is their usage to closure systems of ordinary differential equations (ODE) for the functions of the mathematical expectations and the functions of central moments until a given order. These functions are the main probabilistic speci cations for the state vectors of systems of stochastic ordinary differential equations (SODE) describing a behavior of stochastic dynamic systems. Therefore we have devoted the last part of the paper to derivation of the ODE system satisfied by the indicated moment functions.

An explicit order 2 scheme for the strong approximation of Stratonovich stochastic differential equations with scalar noise

AbstractA new class of stochastic Runge–Kutta (SRK) methods for the strong approximation of Stratonovich stochastic ordinary differential equations (SODEs) is presented. The proposed method is an alternative to the method of Xiao and Tang (Numer. Algor. 72: 259–296, 2016) and converges with order 2 in the strong sense. To validate the efficiency and to compare with some known methods, numerical simulations which involve generating Stratonovich stochastic integrals of level 3, are finally given.

Averaging principle on infinite intervals for stochastic ordinary differential equations

<p style='text-indent:20px;'>In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.</p>

Averaging principle on infinite intervals for stochastic ordinary differential equations with Lévy noise

<abstract> In this paper, we establish an averaging principle on the infinite time intervals for semilinear stochastic ordinary differential equations with Lévy noise. In particular, under suitable conditions we prove that if the coefficients are Poisson stable (including periodic, quasi-periodic, almost periodic, almost automorphic etc), then there exists a unique $ \mathcal L^2 $-bounded solution of the original equation, which inherits the recurrence property of the coefficients, and the recurrent solution uniformly converges to the stationary solution of the averaged equation on the whole real axis in distribution sense. </abstract>

A model for viruses propagation throughout networks

In this paper, we propose a model for virus propagation throughout computers' network. Recently interests for stochastic models have been increasing very fast where stochasticity can be implemented in at least two different ways. The first one consists of considering the same deterministic ODE system to which we add a stochastic noise, the model turns to be governed by a system of stochastic ordinary differential equations (SODEs). The second way, which we have considered in this work, consists of considering each node apart, and assume that its dynamics is governed by a given Markov chain (MC), we have called such model 'of microscopic type'. We have studied both models and pointed out significant similarities and differences. In particular, we have shown that ODE and SODE models do not consider interconnections between nodes while a modified MC model that we have considered does.

A local discontinuous Galerkin method for nonlinear parabolic SPDEs

In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove theL2-stability of the numerical scheme for fully nonlinear equations. Optimal error estimates (O(h(k+1))) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degreekare used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method.

Rough Center Manifolds

Since the breakthrough in rough paths theory for stochastic ordinary differential equations, there has been a strong interest in investigating the rough differential equation (RDE) approach and its...

Stability of delay evolution equations with fading stochastic perturbations

ABSTRACT Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity then an exponentially stable deterministic system remains to be exponentially stable (in mean square). This idea has already been checked for ordinary linear stochastic differential equations and linear stochastic difference equations. Here a similar statement is proven for a nonlinear stochastic evolution equation with quickly enough fading stochastic perturbations. More exactly: if the level of the stochastic perturbation is given by a continuous and square integrable function, then the zero solution of the considered exponentially stable deterministic system remains exponentially mean square stable independently on the maximum magnitude of the stochastic perturbations. Applications of the obtained results to stochastic reaction-diffusion equations and stochastic 2D Navier-Stokes model are shown. Consideration of other types of fading stochastic perturbations (for instance, if stochastic perturbations fade on the infinity not so quickly) is an open problem.

Renormalization of stochastic differential equations with multiplicative noise using effective potential methods

We present a method to renormalize stochastic differential equations subjected to multiplicative noise. The method is based on the widely used concept of effective potential in high-energy physics and has already been successfully applied to the renormalization of stochastic differential equations subjected to additive noise. We derive a general formula for the one-loop effective potential of a single ordinary stochastic differential equation (with arbitrary interaction terms) subjected to multiplicative Gaussian noise (provided the noise satisfies a certain normalization condition). To illustrate the usefulness (and limitations) of the method, we use the effective potential to renormalize a toy chemical model based on a simplified Gray-Scott reaction. In particular, we use it to compute the scale dependence of the toy model's parameters (in perturbation theory) when subjected to a Gaussian power-law noise with short time correlations.

Fast-diffusion limit for reaction-diffusion equations with multiplicative noise

In this paper we consider a class of a stochastic partial differential equations with multiplicative noise and consider the limit of fast diffusion or other fast elliptic operator. We show that one can approximate solutions of the stochastic partial differential equations by the solution of a suitable stochastic ordinary differential equations on the kernel of the linear operator. We focus on equations with polynomial nonlinearities and give applications to Fisher–Kolmogorov–Petrovsky–Piscounov equation and the Ginzburg–Landau equation.

Analytical and Numerical Solution of the Equation for the Probability Density Function of the Particle Velocity in a Turbulent Flow

A study has been made of the random motion of inertial particles in a homogeneous isotropic turbulent gas flow. Fluctuations of the gas velocity along the particle path were modeled by the Gaussian random process with a finite time of degeneracy of the autocorrelation function. A closed equation has been obtained for the probability density function of the particle velocity, for which two methods of numerical solution have been proposed: using the finite-difference scheme and using one based on direct numerical modeling of an empirical probability density function. The empirical probability density function was obtained as a result of the averaging of random particle paths, which are a solution of a system of ordinary stochastic differential equations. The results of numerical calculation have been compared with the analytical solution describing the dynamics of the probability density function of the particle-velocity distribution.

Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Once the existence of well-defined LEs is guaranteed, we proceed to the use of numerical simulation techniques to determine the LEs numerically. Discrete and continuous QR decomposition-based numerical methods are implemented to compute the fundamental solution matrix and use it in the computation of the LEs. Important computational features of both methods are illustrated via numerical tests. Finally, the methods are applied to two applications from power systems engineering, including the single-machine infinite-bus (SMIB) power system model.