Research

15 pages, 303 KiB

Open AccessArticle

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

by Gani Stamov and Ivanka Stamova

Mathematics 2021, 9(12), 1413; https://doi.org/10.3390/math9121413 - 18 Jun 2021

Viewed by 1136

In the present paper, the concept of almost periodic waves is introduced to discontinuous impulsive fractional inclusions involving Caputo fractional derivative. New results on the existence and uniqueness are established by using the theory of operator semigroups, Hausdorff measure of noncompactness, fixed point [...] Read more.

In the present paper, the concept of almost periodic waves is introduced to discontinuous impulsive fractional inclusions involving Caputo fractional derivative. New results on the existence and uniqueness are established by using the theory of operator semigroups, Hausdorff measure of noncompactness, fixed point theorems and fractional calculus techniques. Applications to a class of fractional-order impulsive gene regulatory network (GRN) models are proposed to illustrate the results. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

10 pages, 319 KiB

Open AccessArticle

The Cădariu-Radu Method for Existence, Uniqueness and Gauss Hypergeometric Stability of Ω-Hilfer Fractional Differential Equations

by Safoura Rezaei Aderyani, Reza Saadati and Michal Fečkan

Mathematics 2021, 9(12), 1408; https://doi.org/10.3390/math9121408 - 17 Jun 2021

Cited by 11 | Viewed by 1654

Abstract

Using the Cădariu–Radu method derived from the Diaz–Margolis theorem, we study the existence, uniqueness and Gauss hypergeometric stability of Ω-Hilfer fractional differential equations defined on compact domains. Next, we show the main results for unbounded domains. To illustrate the main result for a [...] Read more.

Using the Cădariu–Radu method derived from the Diaz–Margolis theorem, we study the existence, uniqueness and Gauss hypergeometric stability of Ω-Hilfer fractional differential equations defined on compact domains. Next, we show the main results for unbounded domains. To illustrate the main result for a fractional system, we present an example. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

22 pages, 313 KiB

Open AccessArticle

Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

by Shahram Rezapour, Sotiris K. Ntouyas, Abdelkader Amara, Sina Etemad and Jessada Tariboon

Mathematics 2021, 9(11), 1165; https://doi.org/10.3390/math9111165 - 21 May 2021

Cited by 8 | Viewed by 1448

Abstract

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary [...] Read more.

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary conditions, we extract an equivalent integral equation and then define new operators based on it. With the help of three distinct fixed-point theorems attributed to Krasnosel’skiĭ, Leray–Schauder, and Banach, we investigate desired uniqueness and existence results. Additionally, the dependence criterion of solutions for this CFBVP is checked via the generalized version of the Gronwall inequality. Next, three simulative examples are designed to examine our findings based on the procedures applied in the theorems. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

18 pages, 350 KiB

Open AccessArticle

Existence of Absolutely Continuous Fundamental Matrix of Linear Fractional System with Distributed Delays

by Hristo Kiskinov, Ekaterina Madamlieva, Magdalena Veselinova and Andrey Zahariev

Mathematics 2021, 9(2), 150; https://doi.org/10.3390/math9020150 - 12 Jan 2021

Cited by 7 | Viewed by 1638 | Correction

Abstract

The goal of the present paper is to obtain sufficient conditions that guaranty the existence and uniqueness of an absolutely continuous fundamental matrix for a retarded linear fractional differential system with Caputo type derivatives and distributed delays. Some applications of the obtained result [...] Read more.

The goal of the present paper is to obtain sufficient conditions that guaranty the existence and uniqueness of an absolutely continuous fundamental matrix for a retarded linear fractional differential system with Caputo type derivatives and distributed delays. Some applications of the obtained result concerning the integral representation of the solutions are given too. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

17 pages, 1138 KiB

Open AccessArticle

Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations

by Oana Brandibur, Eva Kaslik, Dorota Mozyrska and Małgorzata Wyrwas

Mathematics 2020, 8(10), 1751; https://doi.org/10.3390/math8101751 - 12 Oct 2020

Cited by 7 | Viewed by 1668

Abstract

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders [...] Read more.

Linear autonomous incommensurate systems that consist of two fractional-order difference equations of Caputo-type are studied in terms of their asymptotic stability and instability properties. More precisely, the asymptotic stability of the considered linear system is fully characterized, in terms of the fractional orders of the considered Caputo-type differences, as well as the elements of the linear system’s matrix and the discretization step size. Moreover, fractional-order-independent sufficient conditions are also derived for the instability of the system under investigation. With the aim of exemplifying the theoretical results, a fractional-order discrete version of the FitzHugh–Nagumo neuronal model is constructed and analyzed. Furthermore, numerical simulations are undertaken in order to substantiate the theoretical findings, showing that the membrane potential may exhibit complex bursting behavior for suitable choices of the model parameters and fractional orders of the Caputo-type differences. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

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10 pages, 271 KiB

Open AccessArticle

Existence of Solutions for Some Coupled Systems of Fractional Differential Inclusions

by Aurelian Cernea

Mathematics 2020, 8(5), 700; https://doi.org/10.3390/math8050700 - 02 May 2020

Cited by 1 | Viewed by 1326

Abstract

We study two coupled systems of nonconvex fractional differential inclusions with certain nonlocal boundary conditions and we prove the existence of solutions in the case when the set-valued maps are Lipschitz in the state variables. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

22 pages, 878 KiB

Open AccessArticle

An Efficient Numerical Method for Fractional SIR Epidemic Model of Infectious Disease by Using Bernstein Wavelets

by Sunil Kumar, Ali Ahmadian, Ranbir Kumar, Devendra Kumar, Jagdev Singh, Dumitru Baleanu and Mehdi Salimi

Mathematics 2020, 8(4), 558; https://doi.org/10.3390/math8040558 - 10 Apr 2020

Cited by 163 | Viewed by 6886

Abstract

In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that arises in medical science to study epidemiology and medical care for the injured. [...] Read more.

In this paper, the operational matrix based on Bernstein wavelets is presented for solving fractional SIR model with unknown parameters. The SIR model is a system of differential equations that arises in medical science to study epidemiology and medical care for the injured. Operational matrices merged with the collocation method are used to convert fractional-order problems into algebraic equations. The Adams–Bashforth–Moulton predictor correcter scheme is also discussed for solving the same. We have compared the solutions with the Adams–Bashforth predictor correcter scheme for the accuracy and applicability of the Bernstein wavelet method. The convergence analysis of the Bernstein wavelet has been also discussed for the validity of the method. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

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13 pages, 277 KiB

Open AccessArticle

On a Fractional Operator Combining Proportional and Classical Differintegrals

by Dumitru Baleanu, Arran Fernandez and Ali Akgül

Mathematics 2020, 8(3), 360; https://doi.org/10.3390/math8030360 - 06 Mar 2020

Cited by 218 | Viewed by 6314

Abstract

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function

f (t)

, by a fractional integral operator applied to the derivative [...] Read more.

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function

f (t)

, by a fractional integral operator applied to the derivative

f^{'} (t)

. We define a new fractional operator by substituting for this

f^{'} (t)

a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function. Full article

(This article belongs to the Special Issue Qualitative Theory of Fractional-Order Systems)

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