Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (21 June 2024) | Viewed by 14881

Special Issue Editor


E-Mail Website
Guest Editor
Department of Mathematics, "Gheorghe Asachi" Technical University of Iasi, Blvd. Carol I, nr. 11, 700506 Iasi, Romania
Interests: fractional differential equations; ordinary differential equations; partial differential equations; finite difference equations; boundary value problems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional differential equations have extensive applications in the mathematical modeling of real-world phenomena which occur in scientific and engineering disciplines. This Special Issue will cover new aspects of the recent developments in the theory and applications of fractional differential equations, inclusions, inequalities, and systems of fractional differential equations with Riemann–Liouville, Caputo, and Hadamard derivatives or other generalized fractional derivatives, subject to various initial and boundary conditions. Problems such as the existence, uniqueness, multiplicity, nonexistence of solutions or positive solutions, stability of solutions, and numerical computations for these models are of great interest for readers who work in this field.

Prof. Dr. Rodica Luca
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional differential equations
  • fractional differential inclusions
  • fractional differential inequalities
  • initial value problems
  • boundary value problems
  • existence, nonexistence
  • uniqueness, multiplicity
  • stability
  • numerical computations

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here.

Published Papers (13 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

23 pages, 367 KiB  
Article
Quantum Laplace Transforms for the Ulam–Hyers Stability of Certain q-Difference Equations of the Caputo-like Type
by Sina Etemad, Ivanka Stamova, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2024, 8(8), 443; https://doi.org/10.3390/fractalfract8080443 - 28 Jul 2024
Viewed by 742
Abstract
We aim to investigate the stability property for the certain linear and nonlinear fractional q-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q-difference equations of the q-Caputo-like type [...] Read more.
We aim to investigate the stability property for the certain linear and nonlinear fractional q-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q-difference equations of the q-Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy q-difference initial value problem, we use the same quantum Laplace transform and the q-Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. Full article
22 pages, 651 KiB  
Article
Analytic Solution of the Time-Fractional Partial Differential Equation Using a Multi-G-Laplace Transform Method
by Hassan Eltayeb
Fractal Fract. 2024, 8(8), 435; https://doi.org/10.3390/fractalfract8080435 - 23 Jul 2024
Viewed by 679
Abstract
In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) [...] Read more.
In several recent studies, many researchers have shown the advantage of fractional calculus in the production of particular solutions of a huge number of linear and nonlinear partial differential equations. In this research work, different theorems related to the G-double Laplace transform (DGLT) are proved. The solution of the system of time-fractional partial differential equations is addressed using a new analytical method. This technique is a combination of the multi-G-Laplace transform and decomposition methods (MGLTDM). Moreover, we discuss the convergence of this method. Two examples are provided to check the applicability and efficiency of our technique. Full article
Show Figures

Figure 1

22 pages, 1507 KiB  
Article
Novel Approach by Shifted Fibonacci Polynomials for Solving the Fractional Burgers Equation
by Mohammed H. Alharbi, Abdullah F. Abu Sunayh, Ahmed Gamal Atta and Waleed Mohamed Abd-Elhameed
Fractal Fract. 2024, 8(7), 427; https://doi.org/10.3390/fractalfract8070427 - 20 Jul 2024
Cited by 1 | Viewed by 728
Abstract
This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas [...] Read more.
This paper analyzes a novel use of the shifted Fibonacci polynomials (SFPs) to treat the time-fractional Burgers equation (TFBE). We first develop the fundamental formulas of these polynomials, which include their power series representation and the inversion formula. We establish other new formulas for the SFPs, including integer and fractional derivatives, in order to design the collocation approach for treating the TFBE. These derivative formulas serve as tools that aid in constructing the operational metrics for the integer and fractional derivatives of the SFPs. We use these matrices to transform the problem and its underlying conditions into a system of nonlinear equations that can be treated numerically. An error analysis is analyzed in detail. We also present three illustrative numerical examples and comparisons to test our proposed algorithm. These results showed that the proposed algorithm is advantageous since highly accurate approximate solutions can be obtained by choosing a few terms of retained modes of SFPs. Full article
Show Figures

Figure 1

15 pages, 297 KiB  
Article
The Existence and Ulam Stability Analysis of a Multi-Term Implicit Fractional Differential Equation with Boundary Conditions
by Peiguang Wang, Bing Han and Junyan Bao
Fractal Fract. 2024, 8(6), 311; https://doi.org/10.3390/fractalfract8060311 - 24 May 2024
Viewed by 604
Abstract
In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the [...] Read more.
In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results. Full article
29 pages, 464 KiB  
Article
On Solutions of Two Post-Quantum Fractional Generalized Sequential Navier Problems: An Application on the Elastic Beam
by Sina Etemad, Sotiris K. Ntouyas, Ivanka Stamova and Jessada Tariboon
Fractal Fract. 2024, 8(4), 236; https://doi.org/10.3390/fractalfract8040236 - 17 Apr 2024
Cited by 1 | Viewed by 1155
Abstract
Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, [...] Read more.
Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p;q)-difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p;q)-operators in this regard. Application of the fixed points of the ρ-θ-contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p;q)-difference Navier problems. Full article
23 pages, 393 KiB  
Article
Systems of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Boundary Conditions
by Alexandru Tudorache and Rodica Luca
Fractal Fract. 2023, 7(11), 816; https://doi.org/10.3390/fractalfract7110816 - 11 Nov 2023
Cited by 3 | Viewed by 1183
Abstract
We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply [...] Read more.
We study the existence and uniqueness of solutions for a system of Hilfer–Hadamard fractional differential equations. These equations are subject to coupled nonlocal boundary conditions that incorporate Riemann–Stieltjes integrals and a range of Hadamard fractional derivatives. To establish our key findings, we apply various fixed point theorems, notably including the Banach contraction mapping principle, the Krasnosel’skii fixed point theorem applied to the sum of two operators, the Schaefer fixed point theorem, and the Leray–Schauder nonlinear alternative. Full article
12 pages, 304 KiB  
Article
Affine-Periodic Boundary Value Problem for a Fractional Differential Inclusion
by Shanshan Gao, Sen Zhao and Jing Lu
Fractal Fract. 2023, 7(9), 647; https://doi.org/10.3390/fractalfract7090647 - 25 Aug 2023
Viewed by 764
Abstract
In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values [...] Read more.
In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values and nonconvex values, respectively. Full article
15 pages, 345 KiB  
Article
Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives
by Vladimir E. Fedorov, Marina V. Plekhanova and Daria V. Melekhina
Fractal Fract. 2023, 7(6), 464; https://doi.org/10.3390/fractalfract7060464 - 7 Jun 2023
Cited by 2 | Viewed by 945
Abstract
The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order [...] Read more.
The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bounded operator, a continuous nonlinear operator that depends on lower-order Dzhrbashyan–Nersesyan derivatives, and an unknown element. The inverse problem is given by an equation, special initial value conditions for lower Dzhrbashyan–Nersesyan derivatives, and an overdetermination condition, which is defined by a linear continuous operator. Applying the fixed-point method for contraction mapping a theorem on the existence of a local unique solution is proved under the condition of local Lipschitz continuity of the nonlinear mapping. Analogous nonlocal results were obtained for the case of the nonlocally Lipschitz continuous nonlinear operator in the equation. The obtained results for the problem in arbitrary Banach spaces were used for the research of nonlinear inverse problems with time-dependent unknown coefficients at lower-order Dzhrbashyan–Nersesyan time-fractional derivatives for integro-differential equations and for a linearized system of dynamics of fractional Kelvin–Voigt viscoelastic media. Full article
24 pages, 390 KiB  
Article
On a System of Hadamard Fractional Differential Equations with Nonlocal Boundary Conditions on an Infinite Interval
by Rodica Luca and Alexandru Tudorache
Fractal Fract. 2023, 7(6), 458; https://doi.org/10.3390/fractalfract7060458 - 3 Jun 2023
Cited by 7 | Viewed by 1027
Abstract
Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes [...] Read more.
Our research focuses on investigating the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations. These equations are defined on an infinite interval and involve non-negative nonlinear terms. Additionally, they are subject to nonlocal coupled boundary conditions, incorporating Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main theorems, we employ the Guo–Krasnosel’skii fixed point theorem and the Leggett–Williams fixed point theorem. Full article
18 pages, 1009 KiB  
Article
An Implicit Numerical Method for the Riemann–Liouville Distributed-Order Space Fractional Diffusion Equation
by Mengchen Zhang, Ming Shen and Hui Chen
Fractal Fract. 2023, 7(5), 382; https://doi.org/10.3390/fractalfract7050382 - 2 May 2023
Cited by 4 | Viewed by 1487
Abstract
This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit [...] Read more.
This paper investigates a two-dimensional Riemann–Liouville distributed-order space fractional diffusion equation (RLDO-SFDE). However, many challenges exist in deriving analytical solutions for fractional dynamic systems. Efficient and reliable methods need to be explored for solving the RLDO-SFDE numerically. We develop an alternating direction implicit scheme and prove that the numerical method is unconditionally stable and convergent with an accuracy of O(σ2+ρ2+τ+hx+hy). After employing an extrapolated technique, the convergence order is improved to second order in time and space. Furthermore, a fast algorithm is constructed to reduce computational costs. Two numerical examples are presented to verify the effectiveness of the numerical methods. This study may provide more possibilities for simulating diffusion complexities by fractional calculus. Full article
Show Figures

Figure 1

22 pages, 1303 KiB  
Article
Numerical Identification of External Boundary Conditions for Time Fractional Parabolic Equations on Disjoint Domains
by Miglena N. Koleva and Lubin G. Vulkov
Fractal Fract. 2023, 7(4), 326; https://doi.org/10.3390/fractalfract7040326 - 13 Apr 2023
Cited by 8 | Viewed by 1496
Abstract
We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we [...] Read more.
We consider fractional mathematical models of fluid-porous interfaces in channel geometry. This provokes us to deal with numerical identification of the external boundary conditions for 1D and 2D time fractional parabolic problems on disjoint domains. First, we discuss the time discretization, then we decouple the full inverse problem into two Dirichlet problems at each time level. On this base, we develop decomposition techniques to obtain exact formulas for the unknown boundary conditions at point measurements. A discrete version of the analytical approach is realized on time adaptive mesh for different fractional order of the equations in each of the disjoint domains. A variety of numerical examples are discussed. Full article
Show Figures

Figure 1

20 pages, 373 KiB  
Article
Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions
by Mohammed Said Souid, Ahmed Refice and Kanokwan Sitthithakerngkiet
Fractal Fract. 2023, 7(2), 198; https://doi.org/10.3390/fractalfract7020198 - 17 Feb 2023
Cited by 3 | Viewed by 1213
Abstract
The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (Lp(·)). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers [...] Read more.
The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (Lp(·)). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example. Full article
23 pages, 390 KiB  
Article
On a System of Sequential Caputo Fractional Differential Equations with Nonlocal Boundary Conditions
by Alexandru Tudorache and Rodica Luca
Fractal Fract. 2023, 7(2), 181; https://doi.org/10.3390/fractalfract7020181 - 12 Feb 2023
Cited by 6 | Viewed by 1168
Abstract
We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on [...] Read more.
We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on the Banach fixed point theorem and the Leray–Schauder alternative. Full article
Back to TopTop