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18 pages, 1176 KiB

Open AccessArticle

On Cauchy Problems of Caputo Fractional Differential Inclusion with an Application to Fractional Non-Smooth Systems

by Jimin Yu, Zeming Zhao and Yabin Shao

Mathematics 2023, 11(3), 653; https://doi.org/10.3390/math11030653 - 28 Jan 2023

Viewed by 1196

In this innovative study, we investigate the properties of existence and uniqueness of solutions to initial value problem of Caputo fractional differential inclusion. In the study of existence problems, we considered the case of convex and non-convex multivalued maps. We obtained the existence [...] Read more.

In this innovative study, we investigate the properties of existence and uniqueness of solutions to initial value problem of Caputo fractional differential inclusion. In the study of existence problems, we considered the case of convex and non-convex multivalued maps. We obtained the existence results for both cases by means of the appropriate fixed point theorem. Furthermore, the uniqueness corresponding to both cases was also determined. Finally, we took a non-smooth system, the modified Murali–Lakshmanan–Chua (MLC) fractional-order circuit system, as an example to verify its existence and uniqueness conditions, and through several sets of simulation results, we discuss the implications. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

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17 pages, 351 KiB

Open AccessArticle

About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays

by Hristo Kiskinov, Mariyan Milev and Andrey Zahariev

Mathematics 2022, 10(23), 4573; https://doi.org/10.3390/math10234573 - 2 Dec 2022

Cited by 3 | Viewed by 1303

Abstract

The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent [...] Read more.

The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo’s sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

13 pages, 283 KiB

Open AccessArticle

Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions

by Petar Popivanov and Angela Slavova

Mathematics 2022, 10(23), 4413; https://doi.org/10.3390/math10234413 - 23 Nov 2022

Viewed by 920

Abstract

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball

B_{1}

a Green function is constructed in the cases

c > 0

, [...] Read more.

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball

B_{1}

a Green function is constructed in the cases

c > 0

,

c \notin - N

, where c is the coefficient in front of u in the boundary condition

\frac{\partial u}{\partial n} + c u = f

. To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation

Λ u + c u = f

. Elliptic boundary value problems for

Δ^{m} u = 0

in

B_{1}

are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for

Δ u = 0

,

Δ^{2} u = 0

and

Δ^{3} u = 0

in

B_{1}

as well as some additional results from the theory of spherical functions are proposed. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

15 pages, 316 KiB

Open AccessArticle

Taylor Series for the Mittag–Leffler Functions and Their Multi-Index Analogues

by Jordanka Paneva-Konovska

Mathematics 2022, 10(22), 4305; https://doi.org/10.3390/math10224305 - 17 Nov 2022

Viewed by 1154

Abstract

It has been obtained that the n-th derivative of the 2-parametric Mittag–Leffler function is a 3-parametric Mittag–Leffler function, with exactness to a constant. Following the analogy, the author later obtained the n-th derivative of the

2 m

-parametric multi-index Mittag–Leffler function. [...] Read more.

It has been obtained that the n-th derivative of the 2-parametric Mittag–Leffler function is a 3-parametric Mittag–Leffler function, with exactness to a constant. Following the analogy, the author later obtained the n-th derivative of the

2 m

-parametric multi-index Mittag–Leffler function. It turns out that this is expressed via the

3 m

-parametric Mittag–Leffler function. In this paper, upper estimates of the remainder terms of these derivatives are found, depending on n. Some asymptotics are also found for “large” values of the parameters. Further, the Taylor series of the 2 and

2 m

-parametric Mittag–Leffler functions around a given point are obtained. Their coefficients are expressed through the values of the corresponding n-th order derivatives at this point. The convergence of the series to the represented Mittag–Leffler functions is justified. Finally, the Bessel-type functions are discussed as special cases of the multi-index (

2 m

-parametric) Mittag–Leffler functions. Their Taylor series are derived from the general case as corollaries, as well. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

13 pages, 540 KiB

Open AccessArticle

Advanced Study on the Delay Differential Equation y′(t) = ay(t) + by(ct)

by Aneefah H. S. Alenazy, Abdelhalim Ebaid, Ebrahem A. Algehyne and Hind K. Al-Jeaid

Mathematics 2022, 10(22), 4302; https://doi.org/10.3390/math10224302 - 17 Nov 2022

Cited by 9 | Viewed by 1838

Abstract

Many real-world problems have been modeled via delay differential equations. The pantograph delay differential equation

y^{'} (t) = a y (t) + b y (c t)

belongs to such a set of delay differential equations. To the authors’ [...] Read more.

Many real-world problems have been modeled via delay differential equations. The pantograph delay differential equation

y^{'} (t) = a y (t) + b y (c t)

belongs to such a set of delay differential equations. To the authors’ knowledge, there are no standard methods to solve the delay differential equations, i.e., unlike the ordinary differential equations, for which numerous and standard methods are well-known. In this paper, the Adomian decomposition method is suggested to analyze the pantograph delay differential equation utilizing two different canonical forms. A power series solution is obtained through the first canonical form, while the second canonical form leads to the exponential function solution. The obtained power series solution coincides with the corresponding ones in the literature for special cases. Moreover, several exact solutions are derived from the present power series solution at a specific restriction of the proportional delay parameter c in terms of the parameters a and b. The exponential function solution is successfully obtained in a closed form and then compared with the available exact solutions (derived from the power series solution). The obtained results reveal that the present analysis is efficient and effective in dealing with pantograph delay differential equations. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

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13 pages, 283 KiB

Open AccessArticle

Notes on Convergence Results for Parabolic Equations with Riemann–Liouville Derivatives

by Long Le Dinh and O’regan Donal

Mathematics 2022, 10(21), 4026; https://doi.org/10.3390/math10214026 - 30 Oct 2022

Cited by 1 | Viewed by 1046

Abstract

Fractional diffusion equations have applications in various fields and in this paper we consider a fractional diffusion equation with a Riemann–Liouville derivative. The main objective is to investigate the convergence of solutions of the problem when the fractional order tends to

1^{-}

[...] Read more.

Fractional diffusion equations have applications in various fields and in this paper we consider a fractional diffusion equation with a Riemann–Liouville derivative. The main objective is to investigate the convergence of solutions of the problem when the fractional order tends to

1^{-}

. Under some suitable conditions on the Cauchy data, we prove the convergence results in a reasonable sense. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

21 pages, 308 KiB

Open AccessArticle

On Some Characterizations for Uniform Dichotomy of Evolution Operators in Banach Spaces

by Rovana Boruga (Toma) and Mihail Megan

Mathematics 2022, 10(19), 3704; https://doi.org/10.3390/math10193704 - 10 Oct 2022

Cited by 7 | Viewed by 1085

Abstract

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform [...] Read more.

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform polynomial growth, we give some characterizations for the uniform exponential dichotomy, respectively for the uniform polynomial dichotomy. The proof techniques that we use for the polynomial case are new. In addition, connections between the concepts approached are established. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

26 pages, 401 KiB

Open AccessFeature PaperArticle

Generalized Proportional Caputo Fractional Differential Equations with Noninstantaneous Impulses: Concepts, Integral Representations, and Ulam-Type Stability

by Ravi Agarwal, Snezhana Hristova and Donal O’Regan

Mathematics 2022, 10(13), 2315; https://doi.org/10.3390/math10132315 - 1 Jul 2022

Cited by 7 | Viewed by 1488

Abstract

The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence [...] Read more.

The generalized proportional Caputo fractional derivative is a comparatively new type of derivative that is a generalization of the classical Caputo fractional derivative, and it gives more opportunities to adequately model complex phenomena in physics, chemistry, biology, etc. In this paper, the presence of noninstantaneous impulses in differential equations with generalized proportional Caputo fractional derivatives is discussed. Generalized proportional Caputo fractional derivatives with fixed lower limits at the initial time as well as generalized proportional Caputo fractional derivatives with changeable lower limits at each impulsive time are considered. The statements of the problems in both cases are set up and the integral representation of the solution of the defined problem in each case is presented. Ulam-type stability is also investigated and some examples are given illustrating these concepts. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

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15 pages, 411 KiB

Open AccessArticle

Generalized Proportional Caputo Fractional Differential Equations with Delay and Practical Stability by the Razumikhin Method

by Ravi Agarwal, Snezhana Hristova and Donal O’Regan

Mathematics 2022, 10(11), 1849; https://doi.org/10.3390/math10111849 - 27 May 2022

Cited by 1 | Viewed by 1329

Abstract

Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The [...] Read more.

Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The study is based on the application of Lyapunov like functions and their generalized proportional Caputo fractional derivatives among solutions of the studied system where appropriate Razumikhin like conditions are applied (appropriately modified in connection with the fractional derivative considered). The theory is illustrated with several nonlinear examples. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

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15 pages, 334 KiB

Open AccessArticle

One Sided Lipschitz Evolution Inclusions in Banach Spaces

by Ali N. A. Koam, Tzanko Donchev, Alina I. Lazu, Muhammad Rafaqat and Ali Ahmad

Mathematics 2021, 9(24), 3265; https://doi.org/10.3390/math9243265 - 16 Dec 2021

Cited by 1 | Viewed by 1765

Abstract

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular

L^{1}

. The commonly used Lipschitz condition on the right-hand side is [...] Read more.

Using the notion of limit solution, we study multivalued perturbations of m-dissipative differential inclusions with nonlocal initial conditions. These solutions enable us to work in general Banach spaces, in particular

L^{1}

. The commonly used Lipschitz condition on the right-hand side is weakened to a one-sided Lipschitz one. No compactness assumptions are required. We consider the cases of an arbitrary one-sided Lipschitz condition and the case of a negative one-sided Lipschitz constant. Illustrative examples, which can be modifications of real models, are provided. Full article

(This article belongs to the Special Issue Recent Investigations of Differential and Fractional Equations and Inclusions II)

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