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Fractal Fract
Special Issues
Fractal Systems and Associated Properties as Well as Integrable Systems
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Fractal Systems and Associated Properties as Well as Integrable Systems

A special issue of Fractal and Fractional (ISSN 2504-3110).

Deadline for manuscript submissions: 31 October 2024 | Viewed by 2409

Special Issue Editor

Prof. Dr. Yufeng Zhang

E-Mail Website
Guest Editor
School of Mathematics, University of Mining and Technology, Xuzhou 221116, China
Interests: fractal integrable systems; differential equations; mathematical symbolic computation
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Since fractional-order differential and integral equations have many extensive applications in the fields of physics, it is important to investigate their properties using mathematical methods. In this Special Issue, we focus on publishing the newest developments on fractional-order differential equations and differential-integral equations from various scientific fields. Simultaneously, we also publish some of the foremost research on soliton theory and integrable-system theory. Topics that invited for submission include (but are not limited to):

  • Fractal derivatives and integrals and their related transformations;
  • Some solutions of fractal differential and integral equations;
  • Soliton solutions of differential equations;
  • Integrable equations and some related properties;
  • Symmetries of differential equations;
  • The dbar method and its applications;
  • Riemann-Hilbert problems and asymptotic analysis.

Prof. Dr. Yufeng Zhang
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.

Published Papers (2 papers)

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Research

18 pages, 331 KiB  
Article
Exploring the Characteristics of Δh Bivariate Appell Polynomials: An In-Depth Investigation and Extension through Fractional Operators
by Musawa Yahya Almusawa
Fractal Fract. 2024, 8(1), 67; https://doi.org/10.3390/fractalfract8010067 - 18 Jan 2024
Viewed by 1015
Abstract
The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the [...] Read more.
The objective of this article is to introduce the ∆h bivariate Appell polynomials ∆hAs[r](λ,η;h) and their extended form via fractional operators. The study described in this paper follows the line of study in which the monomiality principle is used to develop new results. It is further discovered that these polynomials satisfy various well-known fundamental properties and explicit forms. The explicit series representation of ∆h bivariate Gould–Hopper polynomials is first obtained, and, using this outcome, the explicit series representation of the ∆h bivariate Appell polynomials is further given. The quasimonomial properties fulfilled by bivariate Appell polynomials ∆h are also proved by demonstrating that the ∆h bivariate Appell polynomials exhibit certain properties related to their behavior under multiplication and differentiation operators. The determinant form of ∆h bivariate Appell polynomials is provided, and symmetric identities for the ∆h bivariate Appell polynomials are also exhibited. By employing the concept of the forward difference operator, operational connections are established, and certain applications are derived. Different Appell polynomial members can be generated by using appropriate choices of functions in the generating expression obtained in this study for ∆h bivariate Appell polynomials. Additionally, generating relations for the ∆h bivariate Bernoulli and Euler polynomials, as well as for Genocchi polynomials, are established, and corresponding results are obtained for those polynomials. Full article
(This article belongs to the Special Issue Fractal Systems and Associated Properties as Well as Integrable Systems)
15 pages, 1329 KiB  
Article
A New Technique for Solving a Nonlinear Integro-Differential Equation with Fractional Order in Complex Space
by Amnah E. Shammaky, Eslam M. Youssef, Mohamed A. Abdou, Mahmoud M. ElBorai, Wagdy G. ElSayed and Mai Taha
Fractal Fract. 2023, 7(11), 796; https://doi.org/10.3390/fractalfract7110796 - 31 Oct 2023
Viewed by 972
Abstract
This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with [...] Read more.
This work aims to explore the solution of a nonlinear fractional integro-differential equation in the complex domain through the utilization of both analytical and numerical approaches. The demonstration of the existence and uniqueness of a solution is established under certain appropriate conditions with the use of Banach fixed point theorems. To date, no research effort has been undertaken to look into the solution of this integro equation, particularly due to its fractional order specification within the complex plane. The validation of the proposed methodology was performed by utilizing a novel strategy that involves implementing the Rationalized Haar wavelet numerical method with the application of the Bernoulli polynomial technique. The primary reason for choosing the proposed technique lies in its ability to transform the solution of the given nonlinear fractional integro-differential equation into a representation that corresponds to a linear system of algebraic equations. Furthermore, we conduct a comparative analysis between the outcomes obtained from the suggested method and those derived from the rationalized Haar wavelet method without employing any shared mathematical methodologies. In order to evaluate the precision and effectiveness of the proposed method, a series of numerical examples have been developed. Full article
(This article belongs to the Special Issue Fractal Systems and Associated Properties as Well as Integrable Systems)
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