Random Differential Equations Research Articles

An adaptive wavelet optimized finite difference B-spline polynomial chaos method for random partial differential equations

In this paper, wavelet optimized finite difference B-spline polynomial chaos method is proposed for solving stochastic partial differential equations. The generalized polynomial chaos is applied by considering linear B-spline wavelet basis. Then, stochastic Galerkin projection is executed for evaluating the deterministic coefficients of the generalized polynomial chaos. In the next step, the system of equations is discretized by using Crank-–Nicolson scheme for time integration and for approximating the differential operators, central finite difference matrices are considered. An adaptive grid is generated using the linear B-spline generalized polynomial chaos for optimizing the numerical solution. The method is then tested on three problems namely heat equation with uncertain initial conditions, Burger’s equation with random initial conditions and Burger’s equation with random viscosity as well as uncertain initial conditions. For the three test problems, grid modifications are displayed by taking periodic boundary conditions into consideration. Mean and standard deviation are plotted for each test problem. Moreover, for the third test problem, computational time comparison is performed by computing CPU time taken by the proposed method and the CPU time taken by the finite difference method on a uniform grid.

Linear Fractional Galton–Watson Processes in Random Environment and Perpetuities

Abstract Linear fractional Galton–Watson branching processes in i.i.d. random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals. On the other hand, any random difference equation defines an autoregressive Markov chain (a random affine recursion) which can be positive recurrent, null recurrent and transient and which, as the forward iterations of an iterated function system, has an a.s. convergent counterpart in the positive recurrent case given by the corresponding backward iterations. The present expository article aims to provide an explicit view at how these aspects of random difference equations and their stationary limits, called perpetuities, enter into the results and the analysis, especially in quenched regime. Although most of the results presented here are known, we hope that the offered perspective will be welcomed by some readers.

On a Coupled System of Stochastic Ito^-Differential and the Arbitrary (Fractional) Order Differential Equations with Nonlocal Random and Stochastic Integral Conditions

The fractional stochastic differential equations had many applications in interpreting many events and phenomena of life, and the nonlocal conditions describe numerous problems in physics and finance. Here, we are concerned with the combination between the three senses of derivatives, the stochastic Ito^-differential and the fractional and integer orders derivative for the second order stochastic process in two nonlocal problems of a coupled system of two random and stochastic differential equations with two nonlocal stochastic and random integral conditions and a coupled system of two stochastic and random integral conditions. We study the existence of mean square continuous solutions of these two nonlocal problems by using the Schauder fixed point theorem. We discuss the sufficient conditions and the continuous dependence for the unique solution.

A Novel Averaging Principle Provides Insights in the Impact of Intratumoral Heterogeneity on Tumor Progression

Typically stochastic differential equations (SDEs) involve an additive or multiplicative noise term. Here, we are interested in stochastic differential equations for which the white noise is nonlinearly integrated into the corresponding evolution term, typically termed as random ordinary differential equations (RODEs). The classical averaging methods fail to treat such RODEs. Therefore, we introduce a novel averaging method appropriate to be applied to a specific class of RODEs. To exemplify the importance of our method, we apply it to an important biomedical problem, in particular, we implement the method to the assessment of intratumoral heterogeneity impact on tumor dynamics. Precisely, we model gliomas according to a well-known Go or Grow (GoG) model, and tumor heterogeneity is modeled as a stochastic process. It has been shown that the corresponding deterministic GoG model exhibits an emerging Allee effect (bistability). In contrast, we analytically and computationally show that the introduction of white noise, as a model of intratumoral heterogeneity, leads to monostable tumor growth. This monostability behavior is also derived even when spatial cell diffusion is taken into account.

Adaptive neural tracking control for stochastic nonlinear multi-agent periodic time-varying systems

This paper investigates the design of adaptive tracking neural controllers for first- and second-order nonlinear multi-agent periodic time-varying systems described based on random differential equations, respectively. First, the Fourier series expansion and neural networks are used as the main tools to describe the nonlinear periodic disturbance dynamics of the follower, which transforms the completely nonlinear problem into a partially linearizable problem. Then, the fully distributed adaptive neural controllers are developed based on adaptive estimation, in which new estimation methods are designed to address periodic time-varying perturbations. Furthermore, based on the stochastic Lyapunov stability theory, it is proved that the tracking errors converge asymptotically to a small neighborhood of zero in the sense of mean square. Finally, the efficicency of the proposed controllers is verified using simulations.

A Sequential Random Airy Type Problem of Fractional Order: Existence, Uniqueness and ß-Differential Dependance

In this work, a new class of sequential random differential equations of Airy type is introduced. An existence and uniqueness criteria for stochastic process solutions for the introduced class is discussed. Some notions on β−differential dependance are also introduced. Then, new results on the β−dependance are discussed. At the end, some illustrative examples are discussed.

Learning delay dynamics for multivariate stochastic processes, with application to the prediction of the growth rate of COVID-19 cases in the United States

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show the existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.

Liouville’s equations for random systems

Given a random system, a Liouville’s equation is an exact partial differential equation that describes the evolution of the probability density function of the solution. In this article, we derive Liouville’s equations for the first-order homogeneous semilinear random partial differential equation. This is done for all finite-dimensional distributions of the random field solution, starting with dimension one, then dimension two, and finally generalizing to any dimension. Several examples, including the linear advection equation with random coefficients, are treated. As a corollary, we deduce Liouville’s equations for path-wise stochastic integrals and nonlinear random ordinary differential equations.

Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions

It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, finance and engineering. Here we are concerned with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufficient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.

Two Simulation Methods of Brownian Motion

Einstein-Smoluchowski’s theory and Langevin’s theory are two main theories to describe Brownian motion. The Einstein-Smoluchowski theory regards Brownian motion as random walking of Brownian particles, while the Langevin theory establishes a random differential equation describing the motion of Brownian particles. Both theories involve random numbers, i.e., only the statistical results can be discussed. Based on the above theories, this paper presents the corresponding simulations, which verifies the statistical conclusions as well as compares their similarities, differences, and applicable conditions. These results offer a guideline for further studies on Brownian motion.

Large deviations of branching process in a random environment

Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Y n+1=A n Y n + B n , where A 1, A 2, … are independent identically distributed random variables and B n may depend on { ( A k , B k ) , 0 ⩽ k < n } $ \{(A_k,B_k),0\leqslant k \lt n\} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.

Interacting particles systems with delay and random delay differential equations

In this work we study a kinetic model of active particles with delayed dynamics, and its limit when the number of particles goes to infinity. This limit turns out to be related to delayed differential equations with random initial conditions. We analyze two different dynamics, one based on the full knowledge of the individual trajectories of each particle, and another one based only on the trace of the particle cloud, loosing track of the individual trajectories. Notice that in the first dynamic the state of a particles is its path, whereas it is simply a point in Rd in the second case. We analyze in both cases the corresponding mean-field dynamic obtaining an equation for the time evolution of the distribution of the particles states. Well-posedness of the equation is proved by a fixed-point argument. We conclude the paper with some possible future research directions and modeling applications.

Dynamics of stochastic nonlocal partial differential equations

This paper is concerned with the asymptotic behavior of solutions to nonlocal stochastic partial differential equations with multiplicative and additive noise driven by a standard Brownian motion, respectively. First of all, the stochastic nonlocal differential equations are transformed into their associated conjugated random differential equations, we then construct the dynamical systems to the original problems via the properties of conjugation. Next, in the case of multiplicative noise, we establish the existence of the random attractor when it absorbs every bounded deterministic set. Particularly, it is shown the pullback random attractor, which is also forward attracting, becomes a singleton when the external forcing term vanishes at zero. Eventually, in the case of additive noise, two approaches are applied to prove the existence of pullback random attractors with the help of energy estimations. Actually, these two attractors turn out to be the same one.

A Spatiotemporal Prediction Model for Regional Scheduling of Shared Bicycles Based on the INLA Method

Dock-less bicycle-sharing programs have been widely accepted as an efficient mode to benefit health and reduce congestions. And modeling and prediction has always been a core proposition in the field of transportation. Most of the existing demand prediction models for shared bikes take regions as research objects; therefore, a POI-based method can be a beneficial complement to existing research, including zone-level, OD-level, and station-level techniques. Point of interest (POI) is the location description of spatial entities, which can reflect the cycling route characteristics for both commuting and noncommuting trips to a certain extent, and is also the main generating point and attraction point of shared-bike travel flow. In this study, we make an effort to model a POI-level cycling demand with a Bayesian hierarchical method. The proposed model combines the integrated nested Laplace approximation (INLA) and random partial differential equation (SPDE) to cope with the huge computation in the modeling process. In particular, we have adopted the dock-less bicycle-sharing rental records of Mobike as a case study to validate our method; the study area was one of the fastest growing urban districts in Shanghai in August 2016. The operation results show that the method can help better understand, measure, and characterize spatiotemporal patterns of bike-share ridership at the POI level and quantify the impact of the spatiotemporal effect on bicycle-sharing use.

Wong–Zakai approximations for quasilinear systems of Itô’s type stochastic differential equations

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô’s SDEs, we consider for any finite partition of the time interval [0,T] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero.

Probabilistic analysis of a class of impulsive linear random differential equations via density functions

An important class of non-homogeneous first-order linear random differential equations subject to an infinite sequence of square impulses with random intensity is studied. In applications, these equations are useful to model the dynamics of a population with periodic harvesting and migration under uncertainties. The solution is explicitly obtained via the first probability density function assuming an arbitrary joint density for all model parameters. Probabilistic stability analysis is carried out through the densities of the random sequences of minima and maxima. All the theoretical results are fully illustrated through two numerical examples.

Study on fractional random differential equations with not instantaneous impulses

In this paper, we investigate the existence of solution for not instantaneous impulsive fractional differential equations with random parameter. By using fixed point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, generalized Ulam Hyers Rassias stable solution is also obtained.

High order random fractional differential equations: Existence, uniqueness and data dependence

In this work, we are concerned with a high order random fractional differential equation with nonlocal conditions in a suitable Banach space. An existence and uniqueness of solutions for the given problem are obtained. New concepts on the continuous and high order fractional derivative dependence are introduced and some stability results on random as well for deterministic data dependence are discussed.

Stability criteria of random delay differential systems subject to random impulses

AbstractThis article investigates the stability of random impulsive delay differential systems with the kind of random impulsive intensity determined by an arbitrary random sequence or an irreducible aperiodic Markov chain. Moreover, the continuous dynamics are described by random delay differential equations whose random disturbances are driven by second‐order moment processes. Based on the approaches of average impulsive interval and mode‐dependent average impulsive interval, the criteria of global asymptotic stability in probability and exponential stability in th moment are constructed, respectively. In the end, the feasibility of the proposed theoretical results is verified by an example.