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Mathematics
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Fractional Differential Equations, Inclusions and Inequalities with Applications II
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Fractional Differential Equations, Inclusions and Inequalities with Applications II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: 25 March 2025 | Viewed by 16722

Special Issue Editor

Prof. Dr. Sotiris K. Ntouyas

grade E-Mail Website
Guest Editor
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Interests: initial value problems; boundary value problems; inequalities
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

During the last few decades, fractional differential equations, inclusions, and inequalities have been studied extensively. As a matter of fact, fractional derivatives and integrals provide a much better tool for the description of memory and hereditary properties of various materials and processes than integer derivatives. Engineers and scientists have developed new precise models that involve fractional differential equations and inequalities. These models have been applied successfully, e.g., in physics, biomathematics, blood flow phenomena, ecology, environmental issues, viscoelasticity, aerodynamics, electrodynamics of complex medium, electrical circuits, electroanalytical chemistry, control theory, etc.

The purpose of this Special Issue is to establish a collection of articles that reflect the latest mathematical developments in the field of fractional differential equations, inclusions, and inequalities with their applications.

Prof. Dr. Sotiris K. Ntouyas
Guest Editor

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Keywords

  • fractional differential equations
  • fractional differential inclusions
  • fractional inequalities
  • boundary value problem
  • existence

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Published Papers (13 papers)

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Research

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16 pages, 292 KiB  
Article
Theoretical Results on Positive Solutions in Delta Riemann–Liouville Setting
by Pshtiwan Othman Mohammed, Ravi P. Agarwal, Majeed A. Yousif, Eman Al-Sarairah, Alina Alb Lupas and Mohamed Abdelwahed
Mathematics 2024, 12(18), 2864; https://doi.org/10.3390/math12182864 - 14 Sep 2024
Viewed by 228
Abstract
This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the [...] Read more.
This article primarily focuses on examining the existence and uniqueness analysis of boundary fractional difference equations in a class of Riemann–Liouville operators. To this end, we firstly recall the general solution of the homogeneous fractional operator problem. Then, the Green function to the corresponding fractional boundary value problems will be reconstructed, and homogeneous boundary conditions are used to find the unknown constants. Next, the existence of solutions will be studied depending on the fixed-point theorems on the constructed Green’s function. The uniqueness of the problem is also derived via Lipschitz constant conditions. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
23 pages, 344 KiB  
Article
On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces
by Mieczysław Cichoń, Hussein A. H. Salem and Wafa Shammakh
Mathematics 2024, 12(17), 2631; https://doi.org/10.3390/math12172631 - 24 Aug 2024
Viewed by 553
Abstract
As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional [...] Read more.
As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation dψβ,μdtβdψα,μdtα+λx(t)=f(t,x(t)),t∈[a,b],λ∈R, for f∈C[a,b]×R and some critical orders β,α∈(0,1), combined with appropriate initial or boundary conditions, and with general classes of ψ-tempered Hilfer problems with ψ-tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
15 pages, 3018 KiB  
Article
Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations
by Agus Sugandha, Endang Rusyaman, Sukono and Ema Carnia
Mathematics 2024, 12(7), 1077; https://doi.org/10.3390/math12071077 - 3 Apr 2024
Viewed by 1012
Abstract
This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the [...] Read more.
This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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14 pages, 287 KiB  
Article
A Study of Some New Hermite–Hadamard Inequalities via Specific Convex Functions with Applications
by Moin-ud-Din Junjua, Ather Qayyum, Arslan Munir, Hüseyin Budak, Muhammad Mohsen Saleem and Siti Suzlin Supadi
Mathematics 2024, 12(3), 478; https://doi.org/10.3390/math12030478 - 2 Feb 2024
Cited by 3 | Viewed by 1080
Abstract
Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal [...] Read more.
Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal of this article is to establish a novel and generalized identity for the Caputo–Fabrizio fractional operator. With the help of this specific developed identity, we derive new fractional integral inequalities via exponential convex functions. Furthermore, we give an application to some special means. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
8 pages, 651 KiB  
Article
On Λ-Fractional Wave Propagation in Solids
by Kostantinos A. Lazopoulos and Anastasios K. Lazopoulos
Mathematics 2023, 11(19), 4183; https://doi.org/10.3390/math11194183 - 6 Oct 2023
Viewed by 846
Abstract
Wave propagation in solids is discussed, based upon inherently non-local Λ-fractional analysis. Following the governing equations of Λ-fractional continuum mechanics, the Λ-fractional wave equations are derived. Since the variational procedures are only global, in the present Λ-fractional analysis, various jumpings, either in the [...] Read more.
Wave propagation in solids is discussed, based upon inherently non-local Λ-fractional analysis. Following the governing equations of Λ-fractional continuum mechanics, the Λ-fractional wave equations are derived. Since the variational procedures are only global, in the present Λ-fractional analysis, various jumpings, either in the strain or the stress, may be shown. The proposed theory is applied to impact-induced transitions in two-phase elastic materials and viscoelastic materials. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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12 pages, 319 KiB  
Article
A Numerical Approach for Dealing with Fractional Boundary Value Problems
by Abeer A. Al-Nana, Iqbal M. Batiha and Shaher Momani
Mathematics 2023, 11(19), 4082; https://doi.org/10.3390/math11194082 - 26 Sep 2023
Cited by 11 | Viewed by 942
Abstract
This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order α and the fractional central formula for [...] Read more.
This paper proposes a novel numerical approach for handling fractional boundary value problems. Such an approach is established on the basis of two numerical formulas; the fractional central formula for approximating the Caputo differentiator of order α and the fractional central formula for approximating the Caputo differentiator of order 2α, where 0<α≤1. The first formula is recalled here, whereas the second one is derived based on the generalized Taylor theorem. The stability of the proposed approach is investigated in view of some formulated results. In addition, several numerical examples are included to illustrate the efficiency and applicability of our approach. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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12 pages, 294 KiB  
Article
Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations
by Surang Sitho, Sotiris K. Ntouyas, Chayapat Sudprasert and Jessada Tariboon
Mathematics 2023, 11(4), 867; https://doi.org/10.3390/math11040867 - 8 Feb 2023
Cited by 4 | Viewed by 1652
Abstract
In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ1-Hilfer and ψ2-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the [...] Read more.
In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ1-Hilfer and ψ2-Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
29 pages, 4859 KiB  
Article
An Investigation on Existence, Uniqueness, and Approximate Solutions for Two-Dimensional Nonlinear Fractional Integro-Differential Equations
by Tahereh Eftekhari and Jalil Rashidinia
Mathematics 2023, 11(4), 824; https://doi.org/10.3390/math11040824 - 6 Feb 2023
Cited by 3 | Viewed by 1178
Abstract
In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials [...] Read more.
In this research, we provide sufficient conditions to prove the existence of local and global solutions for the general two-dimensional nonlinear fractional integro-differential equations. Furthermore, we prove that these solutions are unique. In addition, we use operational matrices of two-variable shifted Jacobi polynomials via the collocation method to reduce the equations into a system of equations. Error bounds of the presented method are obtained. Five test problems are solved. The obtained numerical results show the accuracy, efficiency, and applicability of the proposed approach. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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9 pages, 273 KiB  
Article
Topological Structure and Existence of Solutions Set for q-Fractional Differential Inclusion in Banach Space
by Ali Rezaiguia and Taher S. Hassan
Mathematics 2023, 11(3), 683; https://doi.org/10.3390/math11030683 - 29 Jan 2023
Viewed by 1054
Abstract
In this work, we concentrate on the existence of the solutions set of the following problem [...] Read more.
In this work, we concentrate on the existence of the solutions set of the following problem cDqασ(t)∈F(t,σ(t),cDqασ(t)),t∈I=[0,T]σ0=σ0∈E, as well as its topological structure in Banach space E. By transforming the problem posed into a fixed point problem, we provide the necessary conditions for the existence and compactness of solutions set. Finally, we present an example as an illustration of main results. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
24 pages, 491 KiB  
Article
New Hermite–Hadamard and Ostrowski-Type Inequalities for Newly Introduced Co-Ordinated Convexity with Respect to a Pair of Functions
by Muhammad Aamir Ali, Fongchan Wannalookkhee, Hüseyin Budak, Sina Etemad and Shahram Rezapour
Mathematics 2022, 10(19), 3469; https://doi.org/10.3390/math10193469 - 23 Sep 2022
Cited by 2 | Viewed by 1029
Abstract
In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls [...] Read more.
In both pure and applied mathematics, convex functions are used in many different problems. They are crucial to investigate both linear and non-linear programming issues. Since a convex function is one whose epigraph is a convex set, the theory of convex functions falls under the umbrella of convexity. However, it is a significant theory that affects practically all areas of mathematics. In this paper, we introduce the notions of g,h-convexity or convexity with respect to a pair of functions on co-ordinates and discuss its fundamental properties. Moreover, we establish some novel Hermite–Hadamard- and Ostrowski-type inequalities for newly introduced co-ordinated convexity. Additionally, it is presented that the newly introduced notion of the convexity and given inequalities are generalizations of existing studies in the literature. Lastly, we look at various mathematical examples and graphs to confirm the validity of the newly found inequalities. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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27 pages, 471 KiB  
Article
Sequential Fractional Hybrid Inclusions: A Theoretical Study via Dhage’s Technique and Special Contractions
by Sina Etemad, Sotiris K. Ntouyas, Bashir Ahmad, Shahram Rezapour and Jessada Tariboon
Mathematics 2022, 10(12), 2090; https://doi.org/10.3390/math10122090 - 16 Jun 2022
Cited by 2 | Viewed by 1126
Abstract
The most important objective of the present research is to establish some theoretical existence results on a novel combined configuration of a Caputo sequential inclusion problem and the hybrid integro-differential one in which the boundary conditions are also formulated as the hybrid multi-order [...] Read more.
The most important objective of the present research is to establish some theoretical existence results on a novel combined configuration of a Caputo sequential inclusion problem and the hybrid integro-differential one in which the boundary conditions are also formulated as the hybrid multi-order integro-differential conditions. In this respect, firstly, some inequalities are proven in relation to the corresponding integral equation. Then, we employ some newly defined theoretical techniques with the help of the product operators on a Banach algebra and also with the aid of some special functions including α-ψ-contractions and α-admissible mappings to extract the existence criteria corresponding to the given mixed sequential hybrid BVPs. Some important useful properties such as the approximate endpoint property, (Cα)-property, and the compactness play a key role in this regard. The final part of the manuscript is devoted to formulating and computing two applicable examples to guarantee the correctness of the obtained results. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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15 pages, 818 KiB  
Article
Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions
by Pheak Neang, Kamsing Nonlaopon, Jessada Tariboon, Sotiris K. Ntouyas and Bashir Ahmad
Mathematics 2022, 10(5), 767; https://doi.org/10.3390/math10050767 - 28 Feb 2022
Cited by 7 | Viewed by 2042
Abstract
In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence [...] Read more.
In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)

Review

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28 pages, 370 KiB  
Review
A Survey on the Oscillation of Solutions for Fractional Difference Equations
by Jehad Alzabut, Ravi P. Agarwal, Said R. Grace, Jagan M. Jonnalagadda, A. George Maria Selvam and Chao Wang
Mathematics 2022, 10(6), 894; https://doi.org/10.3390/math10060894 - 11 Mar 2022
Cited by 5 | Viewed by 2000
Abstract
In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and [...] Read more.
In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and for different types of equations. The investigation was carried out within the delta and nabla operators. Full article
(This article belongs to the Special Issue Fractional Differential Equations, Inclusions and Inequalities with Applications II)
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