Fractional Stochastic Systems Research Articles

Generalized Ulam-Hyer stability results for fuzzy fractional stochastic system under Caputo fractional generalized Hukuhara differentiability concept

Generalized Ulam-Hyer stability results for fuzzy fractional stochastic system under Caputo fractional generalized Hukuhara differentiability concept

A hybrid yang transform adomian decomposition method for solving time-fractional nonlinear partial differential equation

Nonlinear time-fractional partial differential equations (NTFPDEs) play a great role in the mathematical modeling of real-world phenomena like traffic models, the design of earthquakes, fractional stochastic systems, diffusion processes, and control processing. Solving such problems is reasonably challenging, and the nonlinear part and fractional operator make them more problematic. Thus, developing suitable numerical methods is an active area of research. In this paper, we develop a new numerical method called Yang transform Adomian decomposition method (YTADM) by mixing the Yang transform and the Adomian decomposition method for solving NTFPDEs. The derivative of the problem is considered in sense of Caputo fractional order. The stability and convergence of the developed method are discussed in the Banach space sense. The effectiveness, validity, and practicability of the method are demonstrated by solving four examples of NTFPEs. The findings suggest that the proposed method gives a better solution than other compared numerical methods. Additionally, the proposed scheme achieves an accurate solution with a few numbers of iteration, and thus the method is suitable for handling a wide class of NTFPDEs arising in the application of nonlinear phenomena.

The averaging principle of Atangana–Baleanu fractional stochastic integro-differential systems with delay

The averaging principle of Atangana–Baleanu fractional stochastic integro-differential systems with delay

T-controllability of higher-order fractional stochastic delay system via integral contractor

The existence, uniqueness and trajectory (T)-controllability results for mild solutions to the fractional neutral stochastic integro-delay differential system are presented in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the controllability results, and the relevant non-linear function does not have to meet the Lipschitz condition. Furthermore, T-controllability results for neutral integro-delay differential systems with non-local conditions have been established. Finally, an application to demonstrate the acquired results is discussed.

Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process

Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process

Novel Numerical Solutions to the Fractional-Stochastic Systems

Fractional-stochastic differential equations are widely used tools to simulate a wide range of engineering and scientific phenomena. In this paper, the applicability of the approach of indeterminate coefficients to various fractional-stochastic models is examined. These models have a fractional white noise term and are mostly produced by fractional-order derivative operators. We also look into the application of a polynomial chaos algorithm to stochastic Lotka-Volterra and Benney systems. Fractional-stochastic equations are entirely novel systems that have the potential to function as models for a wide range of scientific and engineering phenomena. It is noted that fractional-order systems with uncertainty or a noise term can benefit from the effective use of Galerkin type approaches used in this article. Keywords: Galerkin method, Numerical solutions, Stochastic-fractional systems

Leader-Following Formation Control for Discrete-Time Fractional Stochastic Multi-Agent Systems by Event-Triggered Strategy

Fractional differential equations, which are non-local and can better describe memory and genetic properties, are widely used to describe various physical, chemical, and biological phenomena. Therefore, the multi-agent systems based on discrete-time fractional stochastic models are established. First, some followers are selected for pinning control. In order to save resources and energy, an event-triggered based control mechanism is proposed. Second, under this control mechanism, sufficient conditions on the interaction graph and the fractional derivative order such that formation control can be achieved are given. Additionally, influenced by noise, the multi-agent system completes formation control in the mean square. In addition to that, these results are equally applicable to the discrete-time fractional formation problem without noise. Finally, the example of numerical simulation is given to prove the correctness of the results.

Relatively exact controllability of fractional stochastic neutral system with two incommensurate constant delays

This paper is devoted to analyzing a kind of fractional stochastic neutral system (FSNS). Firstly, by introducing the notion of newly defined two‐parameter Mittag–Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.

Controllability of stochastic fractional systems involving state-dependent delay and impulsive effects

In this paper, the controllability concept of a nonlinear fractional stochastic system involving state-dependent delay and impulsive effects is addressed by employing Caputo derivatives and Mittag-Leffler (ML) functions. Based on stochastic analysis theory, novel sufficient conditions are derived for the considered nonlinear system by utilizing Krasnoselkii’s fixed point theorem. Correspondingly, the applicability of the derived theoretical results is indicated by an example.

An analysis on the approximate controllability outcomes for fractional stochastic Sobolev‐type hemivariational inequalities of order 1<r<2$$ 1<r<2 $$ using sectorial operators

In this paper, we deal with the approximate controllability of fractional stochastic Sobolev‐type hemivariational differential systems of order with sectorial operators. Firstly, by using stochastic analysis, fractional calculus, sine and cosine family operators, sectorial operators, and the fixed‐point theorems of multivalued maps, we show the existence of mild solutions for the fractional stochastic evolution systems. Then, we provide a sufficient condition to guarantee the approximate controllability of the stochastic evolution systems. Next, our results cover problems involving nonlocal conditions. Finally, we present theoretical and practical applications to support the validity of the study.

Study approximate controllability and null controllability of neutral delay Hilfer fractional stochastic integrodifferential system with Rosenblatt process

In this paper, we investigate the sufficient conditions for approximate controllability and null controllability of neutral delay stochastic Hilfer fractional integrodifferential system with Rosenblatt process are established. The required results are obtained based on a fixed point technique, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.

Sobolev‐type existence results for impulsive nonlocal fractional stochastic integrodifferential inclusions of order ϱ∈(1,2) with infinite delay via sectorial operator

The impulsive effect on the fractional stochastic integrodifferential system of order with infinite delay via sectorial operator is investigated in this work. The primary findings of this study are tested by referring to multivalued functions, sectorial operators, fractional derivatives, and the fixed point approach. Initially, the presence of a mild solution for the systems class was introduced. The existence outcomes of semilinear control problems were then established. Eventually, an application is given for deriving the theory of the key findings.

Hilfer fractional neutral stochastic Sobolev-type evolution hemivariational inequality: Existence and controllability☆

This paper discusses the approximate controllability of hemivariational inequalities of the Sobolev-type Hilfer fractional neutral stochastic evolution system. Firstly, by using fixed point theorems, fractional calculus, Hilfer fractional derivatives, multivalued maps and stochastic analysis, for stochastic fractional evolution systems, we demonstrate mild solutions. Whereupon we introduce necessary conditions that ensure the approximate controllability of stochastic fractional systems. As a final step, we define our primary outcomes using an example.

Results on approximate controllability for fractional stochastic delay differential systems of order r ∈ (1,2)

In this paper, we deal with the approximate controllability of fractional stochastic delay differential inclusions of order [Formula: see text]. By using fractional calculus, stochastic analysis, the theory of cosine family and Dhage fixed point techniques, a new set of necessary and sufficient conditions are formulated which guarantees the approximate controllability of the nonlinear fractional stochastic system. In particular, the results are established with the assumption that the associated linear part of the system is approximately controllable. Further, the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions. Finally, an example is presented to illustrate the theory of the obtained result.

Fractional stochastic vibration system under recycling noise

The fractional stochastic vibration system is quite different from the traditional one, and its application potential is enormous if the noise can be deployed correctly and the connection between the fractional order and the noise property is unlocked. This article uses a fractional modification of the well-known van der Pol oscillator with multiplicative and additive recycling noises as an example to study its stationary response and its stochastic bifurcation. First, based on the principle of the minimum mean square error, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Second, the Itô differential equations and One-dimensional Markov process are obtained according to the stochastic averaging method, using Oseledec multiplicative ergodic theorem and maximal Lyapunov exponent to judge local stability, and judging global stability is done by using the singularity theory. Lastly, the stochastic D-bifurcation behavior of the model is analyzed by using the Lyapunov exponent of the dynamical system invariant measure, and the stationary probability density function of the system is solved according to the FPK equation. The results show that the fractional order and noise property can greatly affect the system’s dynamical properties. This paper offers a profound, original, and challenging window for investigating fractional stochastic vibration systems.

A note on existence and exact controllability of fractional stochastic system with finite delay

A note on existence and exact controllability of fractional stochastic system with finite delay

A study on the approximate controllability results of fractional stochastic integro-differential inclusion systems via sectorial operators

The study deals with the findings of the outcome of the approximate controllability results of inclusion type fractional stochastic system in Banach space with the order of the fractional system varrho in (1,2). At first, we implement Bohnenblust-Karlin's fixed point technique to deduce the required conditions on which the fractional system with inital conditions is approximately controllable, and there by, we postulate the sufficient conditions for extending the obtained results to the system with nonlocal conditions.

Approximate Controllability of Nonlinear Fractional Stochastic Systems Involving Impulsive Effects and State Dependent Delay

This paper studies the analysis of approximate controllability for the fractional order neutral stochastic impulsive integro-differential systems involving nonlocal condition and State Dependent Delay (SDD). Sufficient conditions are designed to illustrate the evaluation of approximate controllability. It is exhibited that the proposed protocol can explicitly drive the results by Krasnoselskii's fixed point technique and semigroup theory. As a final point, the derived scheme is validated through an example.

Approximate Controllability of Fractional Stochastic Evolution Inclusions with Non-Local Conditions

This article investigates the approximate controllability of non-linear fractional stochastic differential inclusions with non-local conditions. We establish a set of sufficient conditions for their approximate controllability and provide results in terms of controllability for the fractional stochastic control system. Our approach relies on using fractional calculus and the fixed-point theorem for multiple-valued operators. Finally, we present an illustrative example to support our findings.

Sampling‐based model order reduction for stochastic differential equations driven by fractional Brownian motion

AbstractIn this paper, we study large‐scale linear fractional stochastic systems representing, e.g., spatially discretized stochastic partial differential equations (SPDEs) driven by fractional Brownian motion (fBm) with Hurst parameter H > ½. Such equations are more realistic in modeling real‐world phenomena in comparison to frameworks not capturing memory effects. To the best of our knowledge, dimension reduction schemes for such settings have not been studied so far.In this work, we investigate empirical reduced order methods that are either based on snapshots (e.g. proper orthogonal decomposition) or on approximated Gramians. In each case, dominant subspaces are learned from data. Such model reduction techniques are introduced and analyzed for stochastic systems with fractional noise and later applied to spatially discretized SPDEs driven by fBm in order to reduce the computational cost arising from both the high dimension of the considered stochastic system and a large number of required Monte Carlo runs.We validate our proposed techniques with numerical experiments for some large‐scale stochastic differential equations driven by fBm.