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Fractal Fract
Special Issues
Contemporary Methods of Fractional Order Differential and Differential-Operator Equations
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Contemporary Methods of Fractional Order Differential and Differential-Operator Equations

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

Deadline for manuscript submissions: 30 November 2024 | Viewed by 449

Special Issue Editors

Prof. Dr. Sabir Umarov

E-Mail Website
Guest Editor
Department of Mathematics, University of New Haven, 300 Boston Post Road, West Haven, CT 06516, USA
Interests: fractional calculus; fractional differential equations; fractional derivative; differential operators and equations; stochastic differential equations; random walk models; stochastic processes; applied mathematics
Prof. Dr. Yuri Luchko

E-Mail Website
Guest Editor
Department of Mathematics-Physics-Chemistry, Berlin University of Applied Sciences and Technology, 13353 Berlin, Germany
Interests: fractional calculus; ordinary and partial fractional differential equations; mathematical modelling with fractional calculus models; fractional anomalous diffusion and wave propagation; integral transforms and special functions in fractional calculus
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yangquan Chen

grade E-Mail Website
Guest Editor
Department of Mechanical Engineering (ME), University of California, Merced, CA 95343, USA
Interests: drone-based remote sensing; 3D photogrammetry; optimal sensing in cyber-physical systems; distributed collaborative sensing in distributed parameter systems; mobile sensor networks; virtual metrology
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to new developments in the theory and applications of the fractional-order differential, partial differential, differential operator equations and systems, and stochastic differential equations. The focus is on novel effective analytic or semi-analytic methods for the solution of linear and nonlinear equations and systems, inverse problems including the determination of order(s), stability problems, and problems related to the asymptotic behavior of solutions. The novel methods for the analysis of the existence and uniqueness of a solution, continuous dependence of a solution on data, representations of solutions, fundamental solutions, properties of solutions, descriptions of solution spaces, etc., also fall within the scope of this Special Issue. Furthermore, submissions focusing on new effective applications of the fractional-order partial differential and differential operator equations and systems, which generalize famous classical equations like those of Schrödinger, Navier–Stokes, and Fokker–Planck, and other equations, in modern science (quantum physics, hydrodynamics, statistical mechanics, etc.) are also invited.

Prof. Dr. Sabir Umarov
Prof. Dr. Yuri Luchko
Prof. Dr. Yangquan Chen
Guest Editors

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-order PDEs
  • fractional-order differential operator equations
  • fractional-order SDEs
  • fractional fokker–planck equation
  • fractional-order systems
  • inverse problems for fractional-order equations
  • stability of fractional-order systems
  • applied fractional-order equations
  • applied fractional-order systems
  • determination of order(s)
  • determination of coefficients
  • solution spaces
  • representation of solution
  • fundamental solution
  • mittag–leffler- and wright-type functions
  • matrix-valued mittag–leffler and Wright functions

Published Papers (1 paper)

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Research

14 pages, 307 KiB  
Article
An Inverse Problem for the Subdiffusion Equation with a Non-Local in Time Condition
by Ravshan Ashurov, Marjona Shakarova and Sabir Umarov
Fractal Fract. 2024, 8(7), 378; https://doi.org/10.3390/fractalfract8070378 - 27 Jun 2024
Viewed by 181
Abstract
An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, instead of the Cauchy condition, the non-local in time condition [...] Read more.
An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, instead of the Cauchy condition, the non-local in time condition u(0)=u(T) is taken. The right-hand side of the equation has the form g(t)f with a given function g(t) and an unknown element f∈H. If the function g(t) preserves its sign, then under a over-determined condition u(t0)=ψ, t0∈(0,T), it is proved that the solution of the inverse problem exists and is unique. An example is given showing the violation of the uniqueness of the solution for some sign-changing functions g(t). For such functions g(t), under certain conditions on this function, one can achieve the well-posedness of the problem by choosing properly t0. Moreover, we show that for some g(t), for the existence of a solution to the inverse problem, certain orthogonality conditions must be satisfied, but in this case there is no uniqueness. To the best knowledge of authors, the inverse problem with the non-local condition u(0)=u(T) has been considered for the first time. Moreover, all the results obtained are new not only for the subdiffusion equation, but also for the classical diffusion equation. Full article
(This article belongs to the Special Issue Contemporary Methods of Fractional Order Differential and Differential-Operator Equations)
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