Recent Advances in Fractional Differential Equations and Inclusions

A special issue of Foundations (ISSN 2673-9321). This special issue belongs to the section "Mathematical Sciences".

Deadline for manuscript submissions: closed (30 September 2022) | Viewed by 30828

Special Issue Editor

Special Issue Information

Dear Colleagues,

Fractional calculus is a generalization of classical calculus to an arbitrary real order and has evolved as an interesting and important area of research. Fractional differential equations have attracted much attention in literature because some real-world problems in physics, mechanics, engineering, game theory, stability, optimal control, and other fields can be described better with the help of fractional differential equations. Fractional differential equations and inclusions constitute a significant branch of nonlinear analysis.

This Special Issue invites papers that focus on recent and original research results of fractional differential equations and inclusions.

Prof. Dr. Sotiris K. Ntouyas
Guest Editor

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Keywords

  • Fractional calculus
  • Fractional derivatives
  • Fractional integrals
  • Fractional differential equations
  • Fractional differential inclusions
  • Fractional boundary value problems
  • Fractional order nonlinear systems
  • Fractional integral equations
  • Fractional differences
  • Fractional inequalities

Published Papers (19 papers)

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Research

14 pages, 313 KiB  
Article
New Generalized Hermite–Hadamard–Mercer’s Type Inequalities Using (k, ψ)-Proportional Fractional Integral Operator
by Henok Desalegn Desta, Eze R. Nwaeze, Tadesse Abdi and Jebessa B. Mijena
Foundations 2023, 3(1), 49-62; https://doi.org/10.3390/foundations3010005 - 11 Jan 2023
Cited by 2 | Viewed by 980
Abstract
In this paper, by using Jensen–Mercer’s inequality we obtain Hermite–Hadamard–Mercer’s type inequalities for a convex function employing left-sided (k, ψ)-proportional fractional integral operators involving continuous strictly increasing function. Our findings are a generalization of some results that existed in the literature. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
14 pages, 300 KiB  
Article
Analysis of Sequential Caputo Fractional Differential Equations versus Non-Sequential Caputo Fractional Differential Equations with Applications
by Aghalaya S. Vatsala, Govinda Pageni and V. Anthony Vijesh
Foundations 2022, 2(4), 1129-1142; https://doi.org/10.3390/foundations2040074 - 14 Dec 2022
Cited by 5 | Viewed by 1446
Abstract
It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is [...] Read more.
It is known that, from a modeling point of view, fractional dynamic equations are more suitable compared to integer derivative models. In fact, a fractional dynamic equation is referred to as an equation with memory. To demonstrate that the fractional dynamic model is better than the corresponding integer model, we need to compute the solutions of the fractional differential equations and compare them with an integer model relative to the data available. In this work, we will illustrate that the linear nq-order sequential Caputo fractional differential equations, which are sequential of order q where q<1 with fractional initial conditions and/or boundary conditions, can be solved. The reason for choosing sequential fractional dynamic equations is that linear non-sequential Caputo fractional dynamic equations with constant coefficients cannot be solved in general. We used the Laplace transform method to solve sequential Caputo fractional initial value problems. We used fractional boundary conditions to compute Green’s function for sequential boundary value problems. In addition, the solution of the sequential dynamic equations yields the solution of the corresponding integer-order differential equations as a special case as q1. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
16 pages, 346 KiB  
Article
Investigation of a Nonlinear Coupled (k, ψ)–Hilfer Fractional Differential System with Coupled (k, ψ)–Riemann–Liouville Fractional Integral Boundary Conditions
by Ayub Samadi, Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon
Foundations 2022, 2(4), 918-933; https://doi.org/10.3390/foundations2040063 - 18 Oct 2022
Cited by 7 | Viewed by 1288
Abstract
This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled (k,ψ)–Hilfer fractional differential equations subject to coupled (k,ψ)–Riemann–Liouville fractional integral boundary conditions. We prove two [...] Read more.
This paper is concerned with the existence of solutions for a new boundary value problem of nonlinear coupled (k,ψ)–Hilfer fractional differential equations subject to coupled (k,ψ)–Riemann–Liouville fractional integral boundary conditions. We prove two existence results by applying the Leray–Schauder alternative, and Krasnosel’skiĭ’s fixed-point theorem under different criteria, while the third result, concerning the uniqueness of solutions for the given problem, relies on the Banach’s contraction mapping principle. Examples are included for illustrating the abstract results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
13 pages, 283 KiB  
Article
Green’s Functions for a Fractional Boundary Value Problem with Three Terms
by Paul W. Eloe and Jeffrey T. Neugebauer
Foundations 2022, 2(4), 885-897; https://doi.org/10.3390/foundations2040060 - 12 Oct 2022
Cited by 1 | Viewed by 1054
Abstract
We construct a Green’s function for the three-term fractional differential equation D0+αu+aD0+μu+f(t)u=h(t), 0<t<b, where [...] Read more.
We construct a Green’s function for the three-term fractional differential equation D0+αu+aD0+μu+f(t)u=h(t), 0<t<b, where α(2,3], μ(1,2], and f is continuous, satisfying the boundary conditions u(0)=u(0)=0, D0+βu(b)=0, where β[0,2]. To accomplish this, we first construct a Green’s function for the two-term problem D0+αu+aD0+μu=h(t), 0<t<b, satisfying the same boundary conditions. A lemma from spectral theory is integral to our construction. Some limiting properties of the Green’s function for the two-term problem are also studied. Finally, existence results are given for a nonlinear problem. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
13 pages, 340 KiB  
Article
Caputo Fractional Evolution Equations in Discrete Sequences Spaces
by Alejandro Mahillo and Pedro J. Miana
Foundations 2022, 2(4), 872-884; https://doi.org/10.3390/foundations2040059 - 11 Oct 2022
Cited by 1 | Viewed by 1139
Abstract
In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces p(N0) with p1. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by [...] Read more.
In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces p(N0) with p1. The Caputo fractional calculus extends the usual derivation. The operator, associated to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra 1(Z). We treat in detail some of these compact support sequences. We use techniques from Banach algebras and a Functional Analysis to explicity check the solution of the problem. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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23 pages, 421 KiB  
Article
Detailed Error Analysis for a Fractional Adams Method on Caputo–Hadamard Fractional Differential Equations
by Charles Wing Ho Green and Yubin Yan
Foundations 2022, 2(4), 839-861; https://doi.org/10.3390/foundations2040057 - 22 Sep 2022
Cited by 3 | Viewed by 1506
Abstract
We consider a predictor–corrector numerical method for solving Caputo–Hadamard fractional differential equation over the uniform mesh logtj=loga+logtNajN,j=0,1,2,,N with [...] Read more.
We consider a predictor–corrector numerical method for solving Caputo–Hadamard fractional differential equation over the uniform mesh logtj=loga+logtNajN,j=0,1,2,,N with a1, where loga=logt0<logt1<<logtN=logT is a partition of [loga,logT]. The error estimates under the different smoothness properties of the solution y and the nonlinear function f are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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20 pages, 4757 KiB  
Article
On Λ-Fractional Differential Equations
by Konstantinos A. Lazopoulos
Foundations 2022, 2(3), 726-745; https://doi.org/10.3390/foundations2030050 - 05 Sep 2022
Cited by 2 | Viewed by 1308
Abstract
Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of [...] Read more.
Λ-fractional differential equations are discussed since they exhibit non-locality and accuracy. Fractional derivatives form fractional differential equations, considered as describing better various physical phenomena. Nevertheless, fractional derivatives fail to satisfy the prerequisites of differential topology for generating differentials. Hence, all the sources of generating fractional differential equations, such as fractional differential geometry, the fractional calculus of variations, and the fractional field theory, are not mathematically accurate. Nevertheless, the Λ-fractional derivative conforms to all prerequisites demanded by differential topology. Hence, the various mathematical forms, including those derivatives, do not lack the mathematical accuracy or defects of the well-known fractional derivatives. A summary of the Λ-fractional analysis is presented with its influence on the sources of differential equations, such as fractional differential geometry, field theorems, and calculus of variations. Λ-fractional ordinary and partial differential equations will be discussed. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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12 pages, 277 KiB  
Article
Multiple Positive Solutions for Fractional Boundary Value Problems with Integral Boundary Conditions and Parameter Dependence
by Hamza Tabti and Mohammed Belmekki
Foundations 2022, 2(3), 714-725; https://doi.org/10.3390/foundations2030049 - 29 Aug 2022
Cited by 1 | Viewed by 1116
Abstract
In this paper, we consider the existence of multiple positive solutions to boundary value problems of nonlinear fractional differential equation with integral boundary conditions and parameter dependence. To obtain our results, we used a functional-type cone expansion-compression fixed point theorem and the Leggett–Williams [...] Read more.
In this paper, we consider the existence of multiple positive solutions to boundary value problems of nonlinear fractional differential equation with integral boundary conditions and parameter dependence. To obtain our results, we used a functional-type cone expansion-compression fixed point theorem and the Leggett–Williams fixed point theorem. Examples are included to illustrate the main results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
10 pages, 283 KiB  
Article
Existence and Stability Results for Fractional Hybrid q-Difference Equations with q-Integro-Initial Condition
by Ravi P. Agarwal, Hana Al-Hutami, Bashir Ahmad and Boshra Alharbi
Foundations 2022, 2(3), 704-713; https://doi.org/10.3390/foundations2030048 - 23 Aug 2022
Cited by 1 | Viewed by 1391
Abstract
This article is concerned with the study of a new class of hybrid fractional q-integro-difference equations involving Caputo type q-derivatives and Riemann-Liouville q-integrals of different orders with a nonlocal q-integro-initial condition. An existence result for the given problem is [...] Read more.
This article is concerned with the study of a new class of hybrid fractional q-integro-difference equations involving Caputo type q-derivatives and Riemann-Liouville q-integrals of different orders with a nonlocal q-integro-initial condition. An existence result for the given problem is obtained by means of Krasnoselskii’s fixed point theorem, whereas the uniqueness of its solutions is shown by applying the Banach contraction mapping principle. We also discuss the stability of solutions of the problem at hand and find that it depends on the nonlocal parameter in contrast to the initial position of the domain. To demonstrate the application of the obtained results, examples are constructed. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
16 pages, 355 KiB  
Article
Nonlocal Boundary Value Problems for (k,ψ)-Hilfer Fractional Differential Equations and Inclusions
by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon
Foundations 2022, 2(3), 681-696; https://doi.org/10.3390/foundations2030046 - 19 Aug 2022
Cited by 3 | Viewed by 1145
Abstract
In the present research, single and multi-valued (k,ψ)-Hilfer type fractional boundary value problems of order in (1,2] involving nonlocal integral boundary conditions were studied. In the single-valued case, the Banach and Krasnosel’skiĭ fixed point theorems as well as the [...] Read more.
In the present research, single and multi-valued (k,ψ)-Hilfer type fractional boundary value problems of order in (1,2] involving nonlocal integral boundary conditions were studied. In the single-valued case, the Banach and Krasnosel’skiĭ fixed point theorems as well as the Leray–Schauder nonlinear alternative were used to establish the existence and uniqueness results. In the multi-valued case, when the right-hand side of the inclusion has convex values, we established an existence result via the Leray–Schauder nonlinear alternative method for multi-valued maps, while the second existence result, dealing with the non-convex valued right-hand side of the inclusion, was obtained by applying Covitz-Nadler fixed point theorem for multi-valued contractions. The obtained theoretical results are well illustrated by the numerical examples provided. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
10 pages, 280 KiB  
Article
Simpson’s Type Inequalities for s-Convex Functions via a Generalized Proportional Fractional Integral
by Henok Desalegn, Jebessa B. Mijena, Eze R. Nwaeze and Tadesse Abdi
Foundations 2022, 2(3), 607-616; https://doi.org/10.3390/foundations2030041 - 25 Jul 2022
Cited by 2 | Viewed by 1175
Abstract
In this paper, we give new Simpson’s type integral inequalities for the class of functions whose derivatives of absolute values are s-convex via generalized proportional fractional integrals. Some results in the literature are particular cases of our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
22 pages, 375 KiB  
Article
Nonlocal ψ-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions
by Sotiris K. Ntouyas, Bashir Ahmad and Jessada Tariboon
Foundations 2022, 2(2), 377-398; https://doi.org/10.3390/foundations2020026 - 22 Apr 2022
Cited by 4 | Viewed by 1513
Abstract
In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the ψ-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given [...] Read more.
In this paper, we establish existence and uniqueness results for a new class of boundary value problems involving the ψ-Hilfer generalized proportional fractional derivative operator, supplemented with mixed nonlocal boundary conditions including multipoint, fractional integral multiorder and derivative multiorder operators. The given problem is first converted into an equivalent fixed point problem, which is then solved by means of the standard fixed point theorems. The Banach contraction mapping principle is used to establish the existence of a unique solution, while the Krasnosel’skiĭ and Schaefer fixed point theorems as well as the Leray–Schauder nonlinear alternative are applied for obtaining the existence results. We also discuss the multivalued analogue of the problem at hand. The existence results for convex- and nonconvex-valued multifunctions are respectively proved by means of the Leray–Schauder nonlinear alternative for multivalued maps and Covitz–Nadler’s fixed point theorem for contractive multivalued maps. Numerical examples illustrating the obtained results are also presented. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
10 pages, 296 KiB  
Article
Generalized Fractional Integrals Involving Product of a Generalized Mittag–Leffler Function and Two H-Functions
by Prakash Singh, Shilpi Jain and Praveen Agarwal
Foundations 2022, 2(1), 298-307; https://doi.org/10.3390/foundations2010021 - 11 Mar 2022
Cited by 1 | Viewed by 1710
Abstract
The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative [...] Read more.
The objective of this research is to obtain some fractional integral formulas concerning products of the generalized Mittag–Leffler function and two H-functions. The resulting integral formulas are described in terms of the H-function of several variables. Moreover, we give some illustrative examples for the efficiency of the general approach of our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
8 pages, 259 KiB  
Article
A Note on a Coupled System of Hilfer Fractional Differential Inclusions
by Aurelian Cernea
Foundations 2022, 2(1), 290-297; https://doi.org/10.3390/foundations2010020 - 03 Mar 2022
Cited by 2 | Viewed by 1562
Abstract
A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we [...] Read more.
A coupled system of Hilfer fractional differential inclusions with nonlocal integral boundary conditions is considered. An existence result is established when the set-valued maps have non-convex values. We treat the case when the set-valued maps are Lipschitz in the state variables and we avoid the applications of fixed point theorems as usual. An illustration of the results is given by a suitable example. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
9 pages, 306 KiB  
Article
On Fractional Lyapunov Functions of Nonlinear Dynamic Systems and Mittag-Leffler Stability Thereof
by Attiq ul Rehman, Ram Singh and Praveen Agarwal
Foundations 2022, 2(1), 209-217; https://doi.org/10.3390/foundations2010013 - 07 Feb 2022
Cited by 3 | Viewed by 1875
Abstract
In this paper, fractional Lyapunov functions for epidemic models are introduced and the concept of Mittag-Leffler stability is applied. The global stability of the epidemic model at an equilibrium state is established. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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17 pages, 294 KiB  
Article
Analytical Study of a ϕ− Fractional Order Quadratic Functional Integral Equation
by Ahmed M. A. El-Sayed, Hind H. G. Hashem and Shorouk M. Al-Issa
Foundations 2022, 2(1), 167-183; https://doi.org/10.3390/foundations2010010 - 25 Jan 2022
Cited by 4 | Viewed by 2076
Abstract
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and [...] Read more.
Quadratic integral equations of fractional order have been studied from different views. Here we shall study the existence of continuous solutions of a ϕ fractional-orders quadratic functional integral equation, establish some properties of these solutions and prove the existence of maximal and minimal solutions of that quadratic integral equation. Moreover, we introduce some particular cases to illustrate our results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
16 pages, 293 KiB  
Article
Existence and Uniqueness of Solutions to a Nabla Fractional Difference Equation with Dual Nonlocal Boundary Conditions
by Nandhihalli Srinivas Gopal and Jagan Mohan Jonnalagadda
Foundations 2022, 2(1), 151-166; https://doi.org/10.3390/foundations2010009 - 21 Jan 2022
Cited by 4 | Viewed by 2307
Abstract
In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a [...] Read more.
In this paper, we look at the two-point boundary value problem for a finite nabla fractional difference equation with dual non-local boundary conditions. First, we derive the associated Green’s function and some of its properties. Using the Guo–Krasnoselkii fixed point theorem on a suitable cone and under appropriate conditions on the non-linear part of the difference equation, we establish sufficient requirements for at least one and at least two positive solutions of the boundary value problem. Next, we discuss the existence and uniqueness of solutions to the considered problem. For this purpose, we use Brouwer and Banach fixed point theorem, respectively. Finally, we provide a few examples to illustrate the applicability of established results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
14 pages, 1984 KiB  
Article
Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method
by Neelma, Eiman and Kamal Shah
Foundations 2022, 2(1), 6-19; https://doi.org/10.3390/foundations2010002 - 28 Dec 2021
Viewed by 1709
Abstract
This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also [...] Read more.
This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also compute analytical or semi-analytical results to the proposed problems. Also by some proper examples we demonstrate the results. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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18 pages, 1020 KiB  
Article
Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem
by Vishal Nikam, Dhananjay Gopal and Rabha W. Ibrahim
Foundations 2021, 1(2), 286-303; https://doi.org/10.3390/foundations1020021 - 03 Dec 2021
Cited by 1 | Viewed by 2082
Abstract
The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems [...] Read more.
The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the field of applied mathematics. For this purpose, we generalize a special type of fixed-point theorems. The intention of this work is to prove fixed-point theorems for the class of βG, ψG contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of fixed points. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions)
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