scholarly journals Semilinear fractional stochastic differential equations driven by a γ-Hölder continuous signal with γ > 2/3

2020 ◽  
Vol 21 (01) ◽  
pp. 2050039
Author(s):  
Jorge A. León ◽  
David Márquez-Carreras
Keyword(s):  
Acta Math ◽  
Stochastic Calculus ◽  
Initial Condition ◽  
Hölder Continuous ◽  
Fractional Stochastic Differential Equations ◽  
Stieltjes Integration ◽  
Fractal Functions ◽  
Fractional Differential ◽  
Young Integral ◽  
Semilinear Fractional Differential Equation

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a [Formula: see text]-Hölder continuous function [Formula: see text] with [Formula: see text]. Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to [Formula: see text] is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374].

scholarly journals Uniqueness of Solutionfor Nonlinear Implicit Fractional Differential Equation

2016 ◽  
Vol 12 (11) ◽  
pp. 6807-6811
Author(s):  
Haribhau Laxman Tidke
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Initial Condition ◽  
Fractional Differential

We study the uniquenessof solutionfor nonlinear implicit fractional differential equation with initial condition involving Caputo fractional derivative. The technique used in our analysis is based on an application of Bihari and Medved inequalities.

scholarly journals Mean square calculus and random linear fractional differential equations: Theory and applications

2017 ◽  
Vol 2 (2) ◽  
pp. 317-328 ◽  
Author(s):  
C. Burgos ◽  
J.C Cortés ◽  
L. Villafuerte ◽  
R.J. Villanueva
Keyword(s):  
Stochastic Process ◽  
Second Order ◽  
Initial Condition ◽  
Mean Square ◽  
Initial Value ◽  
Mild Conditions ◽  
Fractional Differential ◽  
Linear Differential ◽  
Linear Fractional Differential Equations ◽  
Statistical Functions

AbstractThe aim of this paper is to study, in mean square sense, a class of random fractional linear differential equation where the initial condition and the forcing term are assumed to be second-order random variables. The solution stochastic process of its associated Cauchy problem is constructed combining the application of a mean square chain rule for differentiating second-order stochastic processes and the random Fröbenius method. To conduct our study, first the classical Caputo derivative is extended to the random framework, in mean square sense. Furthermore, a sufficient condition to guarantee the existence of this operator is provided. Afterwards, the solution of a random fractional initial value problem is built under mild conditions. The main statistical functions of the solution stochastic process are also computed. Finally, several examples illustrate our theoretical findings.

Initial Condition Problems of Fractional Viscoelastic Equations

2003 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Initial Condition ◽  
Initial Values ◽  
Viscoelastic Equation ◽  
Key Issues ◽  
Fractional Differential ◽  
Viscoelastic Equations ◽  
Viscoelastic Oscillator ◽  
Numerical And Analytical Methods

The 2nd order differential equation with fractional derivatives describing dynamic behavior of a single-degree-of-freedom viscoelastic oscillator, referred to as fractional viscoelastic equation (FVE), is considered. Some types of viscoelastic damped mechanical systems may be described by FVE. The differential equation with fractional derivatives is often called the fractional differential equation (FDE). FDE can be solved for zero initial values, but it can not generally be solved for non-zero initial values. How to solve the problem is one of the key issues in this field. This is called “Initial condition (value) problems” of FDE. In this paper, initial condition problems of FVE are solved by making use of the prehistory functions of unknowns which are specified before the initial instance (referred to as the initial functions) starts. Introduction of initial functions into FDE reflects the physical state in giving the initial values. In this paper, several types of initial function are used to solve unique solutions for a type of FVE (referred to as FVE-I). The solutions of FVE-I are obtained by means of both numerical and analytical methods. Implication of the solutions to viscoelastic material will also be discussed.

scholarly journals Explicit Solutions of Initial Value Problems for Linear Scalar Riemann-Liouville Fractional Differential Equations With a Constant Delay

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Initial Condition ◽  
Constant Delay ◽  
Initial Value ◽  
Fractional Differential ◽  
Set Up ◽  
Ordinary Derivative ◽  
Linear Scalar

In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.

scholarly journals Existence and stability results for Caputo fractional stochastic differential equations with Lévy noise

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1739-1751
Author(s):  
P. Umamaheswari ◽  
K. Balachandran ◽  
N. Annapoorani
Keyword(s):  
Differential Equations ◽  
Approximation Scheme ◽  
Existence Of Solution ◽  
Existence And Stability ◽  
Fractional Stochastic Differential Equations ◽  
Fractional Differential ◽  
Stochastic Fractional Differential Equations ◽  
The Stability ◽  
Fractional Dynamical System ◽  
Stability Results

In this paper, the existence of solution of stochastic fractional differential equations with L?vy noise is established by the Picard-Lindel?f successive approximation scheme. The stability of nonlinear stochastic fractional dynamical system with L?vy noise is obtained using Mittag Leffler function. Examples are provided to illustrate the theory.

scholarly journals Addendum to stability analysis of neutral linear fractional system with distributed delays (Filomat 30:3 (2016), 841-851)

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 5013-5017
Author(s):  
Magdalena Veselinova ◽  
Hristo Kiskinov ◽  
Andrey Zahariev
Keyword(s):  
Fractional Integral ◽  
Initial Conditions ◽  
Initial Condition ◽  
Initial Value ◽  
Short Article ◽  
Delayed Argument ◽  
Fractional Differential ◽  
Fractional System ◽  
The Right ◽  
Equations With Delay

In this short article we discuss the initial condition of the initial value problem for fractional differential equations with delayed argument and derivatives in Riemann-Liouville sense. We provide also a new lemma - a ?mirror? analogue of the Kilbas Lemma, concerning the right side Riemann-Liouville fractional integral, which is important for the correct setting of the initial conditions, especially in the case of equations with delay.

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