Recent Advances in Fractional Differential Equations and Inclusions II

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

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 11250

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 (8 papers)

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Research

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25 pages, 375 KiB  
Article
Spatial Discretization for Stochastic Semilinear Superdiffusion Driven by Fractionally Integrated Multiplicative Space–Time White Noise
Foundations 2023, 3(4), 763-787; https://doi.org/10.3390/foundations3040043 - 06 Dec 2023
Viewed by 438
Abstract
We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in [...] Read more.
We investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative space–time white noise. The white noise is characterized by its properties of being white in both space and time, and the time fractional derivative is considered in the Caputo sense with an order α∈ (1, 2). A spatial discretization scheme is introduced by approximating the space–time white noise with the Euler method in the spatial direction and approximating the second-order space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied, and the optimal error estimates that depend on the smoothness of the initial values are established. Full article
33 pages, 481 KiB  
Article
Galerkin Finite Element Approximation of a Stochastic Semilinear Fractional Wave Equation Driven by Fractionally Integrated Additive Noise
Foundations 2023, 3(2), 290-322; https://doi.org/10.3390/foundations3020023 - 29 May 2023
Viewed by 4347
Abstract
We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α(1,2). The existence of [...] Read more.
We investigate the application of the Galerkin finite element method to approximate a stochastic semilinear space–time fractional wave equation. The equation is driven by integrated additive noise, and the time fractional order α(1,2). The existence of a unique solution of the problem is proved by using the Banach fixed point theorem, and the spatial and temporal regularities of the solution are established. The noise is approximated with the piecewise constant function in time in order to obtain a stochastic regularized semilinear space–time wave equation which is then approximated using the Galerkin finite element method. The optimal error estimates are proved based on the various smoothing properties of the Mittag–Leffler functions. Numerical examples are provided to demonstrate the consistency between the theoretical findings and the obtained numerical results. Full article
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15 pages, 309 KiB  
Article
Random Solutions for Generalized Caputo Periodic and Non-Local Boundary Value Problems
Foundations 2023, 3(2), 275-289; https://doi.org/10.3390/foundations3020022 - 29 May 2023
Cited by 1 | Viewed by 779
Abstract
In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed [...] Read more.
In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented. Full article
15 pages, 321 KiB  
Article
Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions
Foundations 2023, 3(2), 260-274; https://doi.org/10.3390/foundations3020021 - 26 May 2023
Cited by 1 | Viewed by 687
Abstract
One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that [...] Read more.
One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence. Full article
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19 pages, 355 KiB  
Article
Coupled Systems of Nonlinear Proportional Fractional Differential Equations of the Hilfer-Type with Multi-Point and Integro-Multi-Strip Boundary Conditions
Foundations 2023, 3(2), 241-259; https://doi.org/10.3390/foundations3020020 - 24 May 2023
Cited by 1 | Viewed by 2286
Abstract
In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, [...] Read more.
In this paper, we study a coupled system of nonlinear proportional fractional differential equations of the Hilfer-type with a new kind of multi-point and integro-multi-strip boundary conditions. Results on the existence and uniqueness of the solutions are achieved by using Banach’s contraction principle, the Leray–Schauder alternative and the well-known fixed-point theorem of Krasnosel’skiĭ. Finally, the main results are illustrated by constructing numerical examples. Full article
18 pages, 327 KiB  
Article
A Comparison Result for the Nabla Fractional Difference Operator
Foundations 2023, 3(2), 181-198; https://doi.org/10.3390/foundations3020016 - 12 Apr 2023
Viewed by 813
Abstract
This article establishes a comparison principle for the nabla fractional difference operator ρ(a)ν, 1<ν<2. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive [...] Read more.
This article establishes a comparison principle for the nabla fractional difference operator ρ(a)ν, 1<ν<2. For this purpose, we consider a two-point nabla fractional boundary value problem with separated boundary conditions and derive the corresponding Green’s function. I prove that this Green’s function satisfies a positivity property. Then, I deduce a relatively general comparison result for the considered boundary value problem. Full article

Review

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64 pages, 648 KiB  
Review
Ostrowski-Type Fractional Integral Inequalities: A Survey
Foundations 2023, 3(4), 660-723; https://doi.org/10.3390/foundations3040040 - 13 Nov 2023
Viewed by 434
Abstract
This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex [...] Read more.
This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions, (ζ,m)-convex functions, s-convex functions, (s,r)-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions, MT-convex functions, P-convex functions, m-convex functions, (s,m)-convex functions, exponentially s-convex functions, (β,m)-convex functions, exponential-convex functions, ζ¯,β,γ,δ-convex functions, quasi-geometrically convex functions, se-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article
40 pages, 480 KiB  
Review
A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Quantum Calculus
Foundations 2023, 3(2), 340-379; https://doi.org/10.3390/foundations3020026 - 15 Jun 2023
Cited by 1 | Viewed by 811
Abstract
A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions, (p,s) [...] Read more.
A review of results on Hermite–Hadamard (H-H) type inequalities in quantum calculus, associated with a variety of classes of convexities, is presented. In the various classes of convexities this includes classical convex functions, quasi-convex functions, p-convex functions, (p,s)-convex functions, modified (p,s)-convex functions, (p,h)-convex functions, tgs-convex functions, η-quasi-convex functions, ϕ-convex functions, (α,m)-convex functions, ϕ-quasi-convex functions, and coordinated convex functions. Quantum H-H type inequalities via preinvex functions and Green functions are also presented. Finally, H-H type inequalities for (p,q)-calculus, h-calculus, and (qh)-calculus are also included. Full article
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