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.

A Universal Predictor-corrector Algorithm for Numerical Simulation of Generalized Fractional Differential Equations

2021 ◽  
Author(s):  
Zaid Odibat
Keyword(s):  
Numerical Simulation ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Fractional Operators ◽  
Initial Value ◽  
Generalized Fractional Integral ◽  
Fractional Differential ◽  
Predictor Corrector

Abstract This study introduces some remarks on generalized fractional integral and differential operators, that generalize some familiar fractional integral and derivative operators, with respect to a given function. We briefly explain how to formulate representations of generalized fractional operators. Then, mainly, we propose a predictor-corrector algorithm for the numerical simulation of initial value problems involving generalized Caputo-type fractional derivatives with respect to another function. Numerical solutions of some generalized Caputo-type fractional derivative models have been introduced to demonstrate the applicability and efficiency of the presented algorithm. The proposed algorithm is expected to be widely used and utilized in the field of simulating fractional-order models.

scholarly journals Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1093
Author(s):  
Daniel Cao Labora
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Initial Value ◽  
Major Question ◽  
Initial Values ◽  
Fractional Differential

One major question in Fractional Calculus is to better understand the role of the initial values in fractional differential equations. In this sense, there is no consensus about what is the reasonable fractional abstraction of the idea of “initial value problem”. This work provides an answer to this question. The techniques that are used involve known results concerning Volterra integral equations, and the spaces of summable fractional differentiability introduced by Samko et al. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann–Liouville fractional integral. In particular, we show that a fractional differential equation of a certain order with Riemann–Liouville derivatives demands, in principle, less initial values than the ceiling of the order to have a uniquely determined solution, in contrast to a widely extended opinion. We compute explicitly the amount of necessary initial values and the orders of differentiability where these conditions need to be imposed.

scholarly journals Finding symmetries by incorporating initial conditions as side conditions

2008 ◽  
Vol 19 (6) ◽  
pp. 701-715 ◽  
Author(s):  
JOANNA GOARD
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Invariant Solution ◽  
Lie Symmetry ◽  
Initial Condition ◽  
Initial Value ◽  
Side Condition ◽  
Symmetry Generator ◽  
Infinitesimal Symmetry

It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

scholarly journals Initial Value Problem for Stochastic Hyprid Hadamard Fractional Differential Equation

2019 ◽  
Vol 16 ◽  
pp. 8288-8296
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
Wagdy G. El-sayed ◽  
A. A. Badr ◽  
Ahmed Tarek Sayed
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Initial Value Problem ◽  
Stochastic Analysis ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Initial Value ◽  
Analysis Techniques ◽  
Fractional Differential ◽  
Hadamard Fractional Integral

In this paper, we discuss the existence of solutions for a stochastic initial value problem of Hyprid fractional dierential equations of Hadamard type given by                            where HD is the Hadamard fractional derivative, and is the Hadamard fractional integral and be such that are investigated. The fractional calculus and stochastic analysis techniques are used to obtain the required results. 

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.

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 Results for Langevin Fractional Differential Inclusions Involving Two Fractional Orders with Four-Point Multiterm Fractional Integral Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas ◽  
Bashir Ahmad
Keyword(s):  
Boundary Conditions ◽  
Differential Inclusions ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Integral Boundary Conditions ◽  
Fractional Differential Inclusions ◽  
Integral Boundary ◽  
Fractional Differential ◽  
The Right ◽  
Fractional Orders

We discuss the existence of solutions for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions. Our study relies on standard fixed point theorems for multivalued maps and covers the cases when the right-hand side of the inclusion has convex as well as nonconvex values. Illustrative examples are also presented.

scholarly journals Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1979
Author(s):  
Ravi P. Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan ◽  
Ricardo Almeida
Keyword(s):  
Fractional Derivatives ◽  
Finite Interval ◽  
Iterative Scheme ◽  
Mild Solutions ◽  
Initial Value ◽  
Monotone Iterative ◽  
Fractional Differential ◽  
The One ◽  
The Right ◽  
The Given

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 174-189 ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Source Term ◽  
Initial Conditions ◽  
Standard Procedure ◽  
Relaxation Equation ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

scholarly journals Hybrid approximations for fractional calculus

2019 ◽  
Vol 29 ◽  
pp. 01001
Author(s):  
Mohsen Razzaghi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Fractional Integral ◽  
Algebraic Equations ◽  
Initial Value ◽  
Hybrid Functions ◽  
Fractional Differential ◽  
Taylor Polynomials ◽  
Hybrid Approximations

In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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