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Fractional Differential Equations: Theory and Application
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Fractional Differential Equations: Theory and Application

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 30 June 2025 | Viewed by 19534

Special Issue Editor

Prof. Dr. Mieczysław Cichoń

E-Mail Website
Guest Editor
Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614 Poznan, Poland
Interests: differential and integral eqations; multifunctions

Special Issue Information

Dear Colleagues,

Fractional calculus is a generalization of classical calculus, dealing with integration and differentiation operations of any positive order (and it is called "fractional" for historical reasons). It is primarily known as the basis of many applied disciplines, including fractional geometry, fractional differential equations, and fractional dynamics. However, it can be considered a branch of mathematics dealing with integral-differential equations, where integrals are of the braided type. The applications of fractional calculus today are very wide. It is true that hardly any field of modern science remains untouched by the tools and techniques of fractional calculus. Although the tools of fractional calculus are available and applicable in various fields of science, the study of the theory of fractional differential equations has started relatively recently and seems to deserve a stand-alone development of its theory in parallel with the well-known theory of ordinary differential equations.

Since efficient analytical and numerical methods still require special attention, the purpose of this Special Issue is to provide a collection of new articles that reflect recent mathematical results as well as some results in applied sciences untouched by the tools and techniques of fractional calculus.

Prof. Dr. Mieczysław Cichoń
Guest Editor

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Keywords

  • fractional calculus
  • fractional differential equations
  • generalized hypergeometric functions
  • function spaces fractional integrals
  • derivatives fractional calculus of variations

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Published Papers (13 papers)

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Research

11 pages, 260 KiB  
Article
Study of Generalized Double-Phase Problem with ς-Laplacian Operator
by Elhoussain Arhrrabi, Hamza El-Houari, Abdeljabbar Ghanmi and Khaled Kefi
Mathematics 2025, 13(5), 700; https://doi.org/10.3390/math13050700 - 21 Feb 2025
Viewed by 401
Abstract
In this paper, we explore a novel class of double-phase ς-Laplacian problems involving a ϕ-Hilfer fractional operator. Employing variational techniques and weighted Musielak space theory, we establish the existence of infinitely many positive solutions under suitable assumptions on the nonlinearities. Our [...] Read more.
In this paper, we explore a novel class of double-phase ς-Laplacian problems involving a ϕ-Hilfer fractional operator. Employing variational techniques and weighted Musielak space theory, we establish the existence of infinitely many positive solutions under suitable assumptions on the nonlinearities. Our main results are original and significantly advance the literature on problems featuring ϕ-Hilfer derivatives and the ς-Laplacian operator. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
15 pages, 291 KiB  
Article
Ulam–Hyers and Generalized Ulam–Hyers Stability of Fractional Differential Equations with Deviating Arguments
by Natalia Dilna, Gusztáv Fekete, Martina Langerová and Balázs Tóth
Mathematics 2024, 12(21), 3418; https://doi.org/10.3390/math12213418 - 31 Oct 2024
Viewed by 1030
Abstract
In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii’s fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the [...] Read more.
In this paper, we study the initial value problem for the fractional differential equation with multiple deviating arguments. By using Krasnoselskii’s fixed point theorem, the conditions of solvability of the problem are obtained. Furthermore, we establish Ulam–Hyers and generalized Ulam–Hyers stability of the fractional functional differential problem. Finally, two examples are presented to illustrate our results, one is with a pantograph-type equation and the other is numerical. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
16 pages, 591 KiB  
Article
Analysis of Fractional Order-Adaptive Systems Represented by Error Model 1 Using a Fractional-Order Gradient Approach
by Maibeth Sánchez-Rivero, Manuel A. Duarte-Mermoud, Juan Carlos Travieso-Torres, Marcos E. Orchard and Gustavo Ceballos-Benavides
Mathematics 2024, 12(20), 3212; https://doi.org/10.3390/math12203212 - 14 Oct 2024
Viewed by 909
Abstract
In adaptive control, error models use system output error and adaptive laws to update controller parameters for control or identification tasks. Fractional-order calculus, involving non-integer-order derivatives and integrals, is increasingly important for modeling, estimation, and control due to its ability to generalize classical [...] Read more.
In adaptive control, error models use system output error and adaptive laws to update controller parameters for control or identification tasks. Fractional-order calculus, involving non-integer-order derivatives and integrals, is increasingly important for modeling, estimation, and control due to its ability to generalize classical methods and offer improved robustness to disturbances. This paper addresses the gap in the literature where fractional-order gradient methods have not yet been extensively applied in identification and adaptive control schemes. We introduce a fractional-order error model with fractional-order gradient (FOEM1-FG), which integrates fractional gradient operators based on the Caputo fractional derivative. By using theoretical analysis and simulations, we confirm that FOEM1-FG maintains stability and ensures bounded output errors across a variety of input signals. Notably, the fractional gradient’s performance improves as the order, β, increases with β>1, leading to faster convergence. Compared to existing integer-order methods, the proposed approach provides a more flexible and efficient solution in adaptive identification and control schemes. Our results show that FOEM1-FG offers superior stability and convergence characteristics, contributing new insights to the field of fractional calculus in adaptive systems. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
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14 pages, 306 KiB  
Article
Applications of Extended Kummer’s Summation Theorem
by Xiaoxia Wang, Arjun K. Rathie, Eunyoung Lim and Hwajoon Kim
Mathematics 2024, 12(19), 3030; https://doi.org/10.3390/math12193030 - 27 Sep 2024
Viewed by 655
Abstract
In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series F12; those of Watson, Dixon, Whipple, and Saalschutz for the series [...] Read more.
In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series F12; those of Watson, Dixon, Whipple, and Saalschutz for the series F23; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan’s summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer’s summation theorem in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
25 pages, 7649 KiB  
Article
A New Solution to the Fractional Black–Scholes Equation Using the Daftardar-Gejji Method
by Agus Sugandha, Endang Rusyaman, Sukono and Ema Carnia
Mathematics 2023, 11(24), 4887; https://doi.org/10.3390/math11244887 - 6 Dec 2023
Cited by 2 | Viewed by 1708
Abstract
The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the [...] Read more.
The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation’s assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
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14 pages, 328 KiB  
Article
A Novel Vieta–Fibonacci Projection Method for Solving a System of Fractional Integrodifferential Equations
by Abdelkader Moumen, Abdelaziz Mennouni and Mohamed Bouye
Mathematics 2023, 11(18), 3985; https://doi.org/10.3390/math11183985 - 19 Sep 2023
Cited by 4 | Viewed by 1337
Abstract
In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient [...] Read more.
In this paper, a new approach for numerically solving the system of fractional integrodifferential equations is devised. To approximate the issue, we employ Vieta–Fibonacci polynomials as basis functions and derive the projection method for Caputo fractional order for the first time. An efficient transformation reduces the problem to a system of two independent equations. Solving two algebraic equations yields an approximate solution to the problem. The proposed method’s efficiency and accuracy are validated. We demonstrate the existence of the solution to the approximate problem and conduct an error analysis. Numerical tests reinforce the interpretations of the theory. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
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15 pages, 300 KiB  
Article
Sharp Existence of Ground States Solutions for a Class of Elliptic Equations with Mixed Local and Nonlocal Operators and General Nonlinearity
by Tingjian Luo and Qihuan Xie
Mathematics 2023, 11(16), 3464; https://doi.org/10.3390/math11163464 - 10 Aug 2023
Viewed by 1167
Abstract
In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity:  [...] Read more.
In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity: (−▵)su−▵u+λu=f(u),x∈RN, which is driven by the superposition of Brownian and Lévy processes. By considering a constrained variational problem, under suitable assumptions on f, we manage to establish a sharp existence of the ground state solutions to the equation considered. These results improve the ones in the existing reference. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
16 pages, 356 KiB  
Article
On Fractional Langevin Equations with Stieltjes Integral Conditions
by Binlin Zhang, Rafia Majeed and Mehboob Alam
Mathematics 2022, 10(20), 3877; https://doi.org/10.3390/math10203877 - 19 Oct 2022
Cited by 6 | Viewed by 1626
Abstract
In this paper, we focus on the study of the implicit FDE involving Stieltjes integral boundary conditions. We first exploit some sufficient conditions to guarantee the existence and uniqueness of solutions for the above problems based on the Banach contraction principle and Schaefer’s [...] Read more.
In this paper, we focus on the study of the implicit FDE involving Stieltjes integral boundary conditions. We first exploit some sufficient conditions to guarantee the existence and uniqueness of solutions for the above problems based on the Banach contraction principle and Schaefer’s fixed point theorem. Then, we present different kinds of stability such as UHS, GUHS, UHRS, and GUHRS by employing the classical techniques. In the end, the main results are demonstrated by two examples. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
40 pages, 783 KiB  
Article
Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρk,ϕk)-Hilfer Fractional Integro-Differential Equations
by Marisa Kaewsuwan, Rachanee Phuwapathanapun, Weerawat Sudsutad, Jehad Alzabut, Chatthai Thaiprayoon and Jutarat Kongson
Mathematics 2022, 10(20), 3874; https://doi.org/10.3390/math10203874 - 18 Oct 2022
Cited by 6 | Viewed by 1719
Abstract
In this paper, we establish the existence and stability results for the (ρk,ϕk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the [...] Read more.
In this paper, we establish the existence and stability results for the (ρk,ϕk)-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the (ρk,ϕk)-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
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15 pages, 299 KiB  
Article
Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives
by Nemat Nyamoradi, Sotiris K. Ntouyas and Jessada Tariboon
Mathematics 2022, 10(17), 3068; https://doi.org/10.3390/math10173068 - 25 Aug 2022
Cited by 1 | Viewed by 1359
Abstract
In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: [...] Read more.
In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: HDαx(t)=Af(t,x(t))+∑i=1kCiHIβigi(t,x(t)),t∈(1,e), supplemented with fractional Hadamard boundary conditions: HDξx(1)=0,HDξx(e)=aHDα−ξ−12(HDξx(t))|t=δ,δ∈(1,e), where 1<α≤2, 0<ξ≤12, a∈(0,∞), 1<α−ξ<2, 0<βi<1, A,Ci, 1≤i≤k, are real constants, HDα is the Hadamard fractional derivative of order α and HIβi is the Hadamard fractional integral of order βi. By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
17 pages, 358 KiB  
Article
Nonlocal Integro-Multi-Point (k, ψ)-Hilfer Type Fractional Boundary Value Problems
by Sotiris K. Ntouyas, Bashir Ahmad, Jessada Tariboon and Mohammad S. Alhodaly
Mathematics 2022, 10(13), 2357; https://doi.org/10.3390/math10132357 - 5 Jul 2022
Cited by 8 | Viewed by 1600
Abstract
In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving (k,ψ)-Hilfer fractional derivative operator of order in (1,2], equipped with nonlocal [...] Read more.
In this paper we investigate the criteria for the existence of solutions for single-valued as well as multi-valued boundary value problems involving (k,ψ)-Hilfer fractional derivative operator of order in (1,2], equipped with nonlocal integral multi-point boundary conditions. For the single-valued case, we rely on fixed point theorems due to Banach and Krasnosel’skiĭ, and Leray–Schauder alternative to establish the desired results. The existence results for the multi-valued problem are obtained by applying the Leray–Schauder nonlinear alternative for multi-valued maps for convex-valued case, while the nonconvex-valued case is studied with the aid of Covit–Nadler’s fixed point theorem for multi-valued contractions. Numerical examples are presented for the illustration of the obtained results. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
18 pages, 337 KiB  
Article
Integrable Solutions for Gripenberg-Type Equations with m-Product of Fractional Operators and Applications to Initial Value Problems
by Ateq Alsaadi, Mieczysław Cichoń and Mohamed M. A. Metwali
Mathematics 2022, 10(7), 1172; https://doi.org/10.3390/math10071172 - 4 Apr 2022
Cited by 9 | Viewed by 1854
Abstract
In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line R+=[0,∞). We prove the existence of solutions in some weighted spaces of [...] Read more.
In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line R+=[0,∞). We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called L1N-solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
13 pages, 271 KiB  
Article
Global Existence for an Implicit Hybrid Differential Equation of Arbitrary Orders with a Delay
by Ahmed M. A. El-Sayed, Sheren A. Abd El-Salam and Hind H. G. Hashem
Mathematics 2022, 10(6), 967; https://doi.org/10.3390/math10060967 - 17 Mar 2022
Cited by 1 | Viewed by 1529
Abstract
In this paper, we present a qualitative study of an implicit fractional differential equation involving Riemann–Liouville fractional derivative with delay and its corresponding integral equation. Under some sufficient conditions, we establish the global and local existence results for that problem by applying some [...] Read more.
In this paper, we present a qualitative study of an implicit fractional differential equation involving Riemann–Liouville fractional derivative with delay and its corresponding integral equation. Under some sufficient conditions, we establish the global and local existence results for that problem by applying some fixed point theorems. In addition, we have investigated the continuous and integrable solutions for that problem. Moreover, we discuss the continuous dependence of the solution on the delay function and on some data. Finally, further results and particular cases are presented. Full article
(This article belongs to the Special Issue Fractional Differential Equations: Theory and Application)
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