Random Fractional Differential Equations Research Articles

On the convergence of the Galerkin method for random fractional differential equations

On the convergence of the Galerkin method for random fractional differential equations

Khasminskii Approach for ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo Fractional Stochastic Pantograph Problem

In this manuscript, we study an averaging principle for fractional stochastic pantograph differential equations (FSDPEs) in the ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-sense accompanied by Brownian movement. Under certain assumptions, we are able to approximate solutions for FSPEs by solutions to averaged stochastic systems in the sense of mean square. Analysis of system solutions before and after the average allows extending the classical Khasminskii approach to random fractional differential equations in the sense of ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo. For clarity, we present at the end an applied example to facilitate the clarification of the theoretical results obtained

A survey on random fractional differential equations involving the generalized Caputo fractional-order derivative

This article discusses the preliminary results of a general form of random fractional differential equations (RFDEs) involving the generalized Caputo fractional derivative. The problem proposed here allows us to interpolate various types of RFDEs. By applying the fixed point theorem and the generalized Gronwall fractional integral inequality, the existence of a unique solution and the Hyers–Ulam stability of RFDEs are presented. Furthermore, a new technique called the Hybrid Laplace-like transform method (HLTM) is proposed to seek the solutions of RFDEs in the nonlinear form. Numerical examples are provided to verify the effectiveness of the proposed method.

Existence of general mean square solutions for a sequential random fractional differential system with nonlocal conditions

In this study, we discuss the existence of solutions for a coupled system of two- sequential random fractional differential equations involving nonlocal periodic boundary conditions. Using a version of Leray-Schauder alternative theorem, we present recent results on the existence of coupled solutions for the problem. The obtained solutions as well as their Caputo derivatives satisfy some stochastic processes properties. An illustrative example is also discussed at the end of this paper.

Solvability for a sequential system of random fractional differential equations of Hermite type

In this work, we study the existence and uniqueness of solutions for a coupled problem of two-sequential random fractional differential equations. The considered problem allows us to obtain the classical Hermite random differential equation as a special case. Using the Banach contraction principle, we prove new existence and uniqueness results. New results on the Ulam-Hyers stability are also discussed.

Nonlinear Random Differential Equations with n Sequential Fractional Derivatives

Abstract This article deals with a solvability for a problem of random fractional differential equations with n sequential derivatives and nonlocal conditions. The existence and uniqueness of solutions for the problem is obtained by using Banach contraction principle. New random data concepts for the considered problem are introduced and some related definitions are given. Also, some results related to the dependance on the introduced data are established for both random and deterministic cases.

Solving random fractional second-order linear equations via the mean square Laplace transform: Theory and statistical computing

This paper deals with random fractional differential equations of the form, CD0+αX(t)+AX˙(t)+BX(t)=0, t>0, with initial conditions, X(0)=C0 and X˙(0)=C1, where CD0+αX(t) stands for the Caputo fractional derivative of X(t). We consider the case that the fractional differentiation order is 1<α<2. For the sake of generality, we further assume that C0, C1, A and B are random variables satisfying certain mild hypotheses. Then, we first construct a solution stochastic process, via a generalized power series, which is mean square convergent for all t>0. Secondly, we provide explicit approximations of the expectation and variance functions of the solution. To complete the random analysis and from this latter key information, we take advantage of the Principle of Maximum Entropy to calculate approximations of the first probability density function of the solution. All the theoretical findings are illustrated via numerical experiments.

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.

System of boundary random fractional differential equations via Hadamard derivative

Abstract We study the existence of solutions for random system of fractional differential equations with boundary nonlocal initial conditions. Our approach is based on random fixed point principles of Schaefer and Perov, combined with a vector approach that uses matrices that converge to zero. We prove existence and uniqueness results for these systems. Some examples are presented to illustrate the theory.

Dynamics and stability for Katugampola random fractional differential equations

<abstract> This paper deals with some existence of random solutions and the Ulam stability for a class of Katugampola random fractional differential equations in Banach spaces. A random fixed point theorem is used for the existence of random solutions, and we prove that our problem is generalized Ulam-Hyers-Rassias stable. An illustrative example is presented in the last section. </abstract>

Boundary Value Problem for Caputo–Fabrizio Random Fractional Differential Equations

Abstract This article deals with some existence of random solutions and Ulam stability results for a class of Caputo-Fabrizio random fractional differential equations with boundary conditions in Banach spaces. Our results are based on the fixed point theory and random operators. Two illustrative examples are presented in the last section.

Mean square convergent numerical solutions of random fractional differential equations: Approximations of moments and density

A fractional forward Euler-like method is developed to solve initial value problems with uncertainties formulated via the Caputo fractional derivative. The analysis is conducted by using the so-called random mean square calculus. Under mild conditions on the data, the mean square convergence of the numerical method is proved. This type of stochastic convergence guarantees the approximations of the mean and the variance of the solution stochastic process, computed via the aforementioned numerical scheme, will converge to their corresponding exact values. Furthermore, from this probability information, we calculate reliable approximations to the first probability density function of the solution by taking advantage of the Maximum Entropy Principle. The theoretical analysis is illustrated by two examples.

On initial value problem of random fractional differential equation with impulses

In this paper, we prove the existence and uniqueness of solution for random fractional differential equation with impulses via Banach fixed point theorem and Schauder fixed point theorem. Moreover, the continuous dependence of the solution on the initial data is investigated.

Study of a boundary value problem for fractional order psi -Hilfer fractional derivative

This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in psi -Hilfer sense. By utilizing classical fixed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justification of our analysis we provide two examples.

Random fractional differential equations with Riemann-Liouville-type fuzzy differentiability concept

Random fractional differential equations with Riemann-Liouville-type fuzzy differentiability concept

Hilfer and Hadamard random fractional differential equations in Fréchet spaces

Hilfer and Hadamard random fractional differential equations in Fréchet spaces

Coupled Hilfer fractional differential systems with random effects

This paper deals with some existence results for two classes of coupled systems of Hilfer and Hilfer–Hadamard random fractional differential equations. The main tool used to carry out our results is Itoh’s random fixed point theorem.

Extending the deterministic Riemann–Liouville and Caputo operators to the random framework: A mean square approach with applications to solve random fractional differential equations

This paper extends both the deterministic fractional Riemann–Liouville integral and the Caputo fractional derivative to the random framework using the mean square random calculus. Characterizations and sufficient conditions to guarantee the existence of both fractional random operators are given. Assuming mild conditions on the random input parameters (initial condition, forcing term and diffusion coefficient), the solution of the general random fractional linear differential equation, whose fractional order of the derivative is α ∈ [0, 1], is constructed. The approach is based on a mean square chain rule, recently established, together with the random Fröbenius method. Closed formulae to construct reliable approximations for the mean and the covariance of the solution stochastic process are also given. Several examples illustrating the theoretical results are included.

Existence results for random fractional differential equations

In this paper, an existence result for a random fractional differential equation is established under a Caratheodory condition. Existence results for extremal random solutions are also proved. Finally, an existence and uniqueness result is given

Polynomial Chaos for random fractional order differential equations

Fractional order models provide a powerful instrument for description of memory and hereditary properties of systems in comparison to integer order models, where such effects are difficult to incorporate and often neglected. Also, many physical real world problems that involve uncertainties and errors can be best modeled with random differential equations. Thus, to be able to deal with many real life problems, it is important to develop mathematical methodologies to solve systems that include both memory effects and uncertainty. The aim of this paper is to study the application of the generalized Polynomial Chaos (gPC) to random fractional ordinary differential equations. The method of Polynomial Chaos has played an increasingly important role when dealing with uncertainties. The main idea of the method is the projection of the random parameters and stochastic processes in the system onto the space of polynomial chaoses. However, to apply Polynomial Chaos to random fractional differential equations requires careful attention due to memory effects and the increasing of the computation time in respect to the classic random differential equations. In order to avoid more complex numerical computations and obtain accurate solutions we rely on Richardson extrapolation. It is shown that the application of generalized Polynomial Chaos method in conjunction with Richardson extrapolation is a reliable and accurate method to numerically solve random fractional ordinary differential equations.