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Mathematics
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Direct and Inverse Problems for Fractional Differential Equations
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Direct and Inverse Problems for Fractional Differential Equations

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

Deadline for manuscript submissions: closed (31 July 2020) | Viewed by 20380

Special Issue Editor

Prof. Dr. Vladimir E. Fedorov

E-Mail Website
Guest Editor
Department of Mathematical Analysis, Chelyabinsk State University, Chelyabinsk, Russia
Interests: fractional derivative; distributed derivative; fractional differential equation; initial value problem; boundary value problem; inverse problem; identification problem; existence and uniqueness of solution; group analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional integro-differential calculus in general, and fractional differential equations in particular, are becoming an increasingly important tool for mathematical modeling, and, accordingly, are of increasing interest to researchers. Fractional differential equations have their own specifics, and at the same time, many of their properties are similar to the properties of differential equations of integer orders that are similar in form.

The aim of this Special Issue is to gather a collection of articles reflecting the latest developments in the theory of fractional differential equations, unique solvability issues for initial value problems to differential equations with fractional derivatives in Banach spaces, for initial boundary value problems to fractional order partial differential equations, for inverse coefficient problems to such equations, and so on. The various qualitative properties of fractional differential equations, the search for exact solutions, and group analyses for fractional differential equations are also of interest to this Special Issue.

Prof. Dr. Vladimir E. Fedorov
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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 derivative
  • Distributed derivative
  • Fractional differential equation
  • Initial value problem
  • Boundary value problem
  • Inverse problem
  • Identification problem
  • Existence and uniqueness of solution
  • Group analysis

Published Papers (10 papers)

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Research

11 pages, 248 KiB  
Article
Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order
by Murat Mamchuev
Mathematics 2020, 8(9), 1475; https://doi.org/10.3390/math8091475 - 01 Sep 2020
Cited by 6 | Viewed by 2647
Abstract
We investigate the initial problem for a linear system of ordinary differential equations with constant coefficients and with the Dzhrbashyan–Nersesyan fractional differentiation operator. The existence and uniqueness theorems of the solution of the boundary value problem under the study are proved. The solution [...] Read more.
We investigate the initial problem for a linear system of ordinary differential equations with constant coefficients and with the Dzhrbashyan–Nersesyan fractional differentiation operator. The existence and uniqueness theorems of the solution of the boundary value problem under the study are proved. The solution is constructed explicitly in terms of the Mittag–Leffler function of the matrix argument. The Dzhrbashyan–Nersesyan operator is a generalization of the Riemann–Liouville, Caputo and Miller–Ross fractional differentiation operators. The obtained results as particular cases contain the results related to the study of initial problems for the systems of ordinary differential equations with Riemann–Liouville, Caputo and Miller–Ross derivatives and the investigated initial problem that generalizes them. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
15 pages, 311 KiB  
Article
Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations
by Vladimir E. Fedorov
Mathematics 2020, 8(8), 1306; https://doi.org/10.3390/math8081306 - 06 Aug 2020
Cited by 12 | Viewed by 1330
Abstract
Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the [...] Read more.
Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the unique solvability of inhomogeneous equations. A perturbation theorem for the obtained class of generators is proved. The results of the work are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
16 pages, 305 KiB  
Article
Existence of a Unique Weak Solution to a Nonlinear Non-Autonomous Time-Fractional Wave Equation (of Distributed-Order)
by Karel Van Bockstal
Mathematics 2020, 8(8), 1283; https://doi.org/10.3390/math8081283 - 03 Aug 2020
Cited by 8 | Viewed by 1767
Abstract
We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak [...] Read more.
We study an initial-boundary value problem for a fractional wave equation of time distributed-order with a nonlinear source term. The coefficients of the second order differential operator are dependent on the spatial and time variables. We show the existence of a unique weak solution to the problem under low regularity assumptions on the data, which includes weakly singular solutions in the class of admissible problems. A similar result holds true for the fractional wave equation with Caputo fractional derivative. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
20 pages, 5550 KiB  
Article
Tube-Based Taut String Algorithms for Total Variation Regularization
by Artyom Makovetskii, Sergei Voronin, Vitaly Kober and Aleksei Voronin
Mathematics 2020, 8(7), 1141; https://doi.org/10.3390/math8071141 - 13 Jul 2020
Cited by 5 | Viewed by 2497
Abstract
Removing noise from signals using total variation regularization is a challenging signal processing problem arising in many practical applications. The taut string method is one of the most efficient approaches for solving the 1D TV regularization problem. In this paper we propose a [...] Read more.
Removing noise from signals using total variation regularization is a challenging signal processing problem arising in many practical applications. The taut string method is one of the most efficient approaches for solving the 1D TV regularization problem. In this paper we propose a geometric description of the linearized taut string method. This geometric description leads to the notion of the “tube”. We propose three tube-based taut string algorithms for total variation regularization. Different weight functionals can be used in the 1D TV regularization that lead to different types of tubes. We consider uniform, vertically nonuniform, vertically and horizontally nonuniform tubes. The proposed geometric approach is used to speed-up TV regularization processing by dividing the tubes into subtubes and using parallel processing. We introduce the concept of a relatively convex tube and describe the relationship between the geometric characteristics of tubes and exact solutions to the TV regularization. The properties of exact solutions can also be used to design efficient algorithms for solving the TV regularization problem. The performance of the proposed algorithms is discussed and illustrated by computer simulation. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
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12 pages, 266 KiB  
Article
Nonlocal Integro-Differential Equations of the Second Order with Degeneration
by Aleksandr I. Kozhanov
Mathematics 2020, 8(4), 606; https://doi.org/10.3390/math8040606 - 16 Apr 2020
Cited by 1 | Viewed by 1440
Abstract
We study the solvability for boundary value problems to some nonlocal second-order integro–differential equations that degenerate by a selected variable. The possibility of degeneration in the equations under consideration means that the statements of the corresponding boundary value problems have to change depending [...] Read more.
We study the solvability for boundary value problems to some nonlocal second-order integro–differential equations that degenerate by a selected variable. The possibility of degeneration in the equations under consideration means that the statements of the corresponding boundary value problems have to change depending on the nature of the degeneration, while the nonlocality in the equations implies that the boundary conditions will also have a nonlocal form. For the problems under study, the paper provides conditions that ensure their well-posedness. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
11 pages, 283 KiB  
Article
Inverse Problems for Degenerate Fractional Integro-Differential Equations
by Mohammed Al Horani, Mauro Fabrizio, Angelo Favini and Hiroki Tanabe
Mathematics 2020, 8(4), 532; https://doi.org/10.3390/math8040532 - 03 Apr 2020
Cited by 2 | Viewed by 1788
Abstract
This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on [...] Read more.
This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of non-fractional equations. Our method is based on transforming the inverse problem to a direct problem and identifying the conditions under which this direct problem has a unique solution. The conditions under which the unique strict solution can be compared with the case of a mild solution, obtained in previous studies under quite restrictive requirements, are on the underlying functions. Applications from partial differential equations are given to illustrate our abstract results. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
9 pages, 251 KiB  
Article
Multi-Term Fractional Degenerate Evolution Equations and Optimal Control Problems
by Marina Plekhanova and Guzel Baybulatova
Mathematics 2020, 8(4), 483; https://doi.org/10.3390/math8040483 - 01 Apr 2020
Cited by 2 | Viewed by 1384
Abstract
A theorem on unique solvability in the sense of the strong solutions is proved for a class of degenerate multi-term fractional equations in Banach spaces. It applies to the deriving of the conditions on unique solution existence for an optimal control problem to [...] Read more.
A theorem on unique solvability in the sense of the strong solutions is proved for a class of degenerate multi-term fractional equations in Banach spaces. It applies to the deriving of the conditions on unique solution existence for an optimal control problem to the corresponding equation. Obtained results are used to an optimal control problem study for a model system which is described by an initial-boundary value problem for a partial differential equation. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
15 pages, 315 KiB  
Article
Green Functions of the First Boundary-Value Problem for a Fractional Diffusion—Wave Equation in Multidimensional Domains
by Arsen Pskhu
Mathematics 2020, 8(4), 464; https://doi.org/10.3390/math8040464 - 26 Mar 2020
Cited by 8 | Viewed by 2318
Abstract
We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump [...] Read more.
We construct the Green function of the first boundary-value problem for a diffusion-wave equation with fractional derivative with respect to the time variable. The Green function is sought in terms of a double-layer potential of the equation under consideration. We prove a jump relation and solve an integral equation for an unknown density. Using the Green function, we give a solution of the first boundary-value problem in a multidimensional cylindrical domain. The fractional differentiation is given by the Dzhrbashyan–Nersesyan fractional differentiation operator. In particular, this covers the cases of equations with the Riemann–Liouville and Caputo derivatives. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
9 pages, 357 KiB  
Article
Bayesian Derivative Order Estimation for a Fractional Logistic Model
by Francisco J. Ariza-Hernandez, Martin P. Arciga-Alejandre, Jorge Sanchez-Ortiz and Alberto Fleitas-Imbert
Mathematics 2020, 8(1), 109; https://doi.org/10.3390/math8010109 - 10 Jan 2020
Cited by 5 | Viewed by 2223
Abstract
In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Grünwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the [...] Read more.
In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Grünwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the likelihood function, we propose an explicit numerical scheme based on the truncated series of the derivative definition. By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations. To evaluate the methodology, a simulation was performed using synthetic data, where the bias and mean square error are calculated, the results give evidence of the effectiveness for the method and the suitable performance of the proposed model. Moreover, an example with real data is presented as evidence of the relevance of using a fractional model. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
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21 pages, 363 KiB  
Article
A Fractional Equation with Left-Sided Fractional Bessel Derivatives of Gerasimov–Caputo Type
by Elina Shishkina and Sergey Sitnik
Mathematics 2019, 7(12), 1216; https://doi.org/10.3390/math7121216 - 10 Dec 2019
Cited by 14 | Viewed by 2287
Abstract
In this article we propose and study a method to solve ordinary differential equations with left-sided fractional Bessel derivatives on semi-axes of Gerasimov–Caputo type. We derive explicit solutions to equations with fractional powers of the Bessel operator using the Meijer integral transform. Full article
(This article belongs to the Special Issue Direct and Inverse Problems for Fractional Differential Equations)
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