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Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations...
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Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 31 May 2024 | Viewed by 5187

Special Issue Editor

Dr. Sergei Sitnik

E-Mail Website
Guest Editor
Institute of Engineering and Digital Technologies, Belgorod State National Research University, 308015 Belgorod, Russia
Interests: differential equations; transmutation theory; integral transforms; special functions; inequalities; numerical methods; approximation theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to include research papers and surveys which cover a wide range of topics in computational and applied mathematics. High-quality papers on the following topics are welcome: differential equations, especially concerning singular solutions and coefficients; transmutation theory, with examples of its applications; integral transform and special function theory; classical and advanced inequalities; and numerical methods for all of the abovementioned problems.

Dr. Sergei Sitnik
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

  • differential equations
  • transmutation theory
  • integral transforms
  • special functions
  • inequalities
  • numerical methods
  • approximation theory

Published Papers (7 papers)

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Research

17 pages, 709 KiB  
Article
Two-Variable q-Hermite-Based Appell Polynomials and Their Applications
by Mohammed Fadel, Maryam Salem Alatawi and Waseem Ahmad Khan
Mathematics 2024, 12(9), 1358; https://doi.org/10.3390/math12091358 (registering DOI) - 29 Apr 2024
Abstract
A noteworthy advancement within the discipline of q-special function analysis involves the extension of the concept of the monomiality principle to q-special polynomials. This extension helps analyze the quasi-monomiality of many q-special polynomials. This extension is a helpful tool for [...] Read more.
A noteworthy advancement within the discipline of q-special function analysis involves the extension of the concept of the monomiality principle to q-special polynomials. This extension helps analyze the quasi-monomiality of many q-special polynomials. This extension is a helpful tool for considering the quasi-monomiality of several q-special polynomials. This study aims to identify and establish the characteristics of the 2-variable q-Hermite–Appell polynomials via an extension of the concept of monomiality. Also, we present some applications that are taken into account. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
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22 pages, 704 KiB  
Article
A Semiclassical Approach to the Nonlocal Nonlinear Schrödinger Equation with a Non-Hermitian Term
by Anton E. Kulagin and Alexander V. Shapovalov
Mathematics 2024, 12(4), 580; https://doi.org/10.3390/math12040580 - 15 Feb 2024
Cited by 2 | Viewed by 486
Abstract
The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov [...] Read more.
The nonlinear Schrödinger equation (NLSE) with a non-Hermitian term is the model for various phenomena in nonlinear open quantum systems. We deal with the Cauchy problem for the nonlocal generalization of multidimensional NLSE with a non-Hermitian term. Using the ideas of the Maslov method, we propose the method of constructing asymptotic solutions to this equation within the framework of semiclassically concentrated states. The semiclassical nonlinear evolution operator and symmetry operators for the leading term of asymptotics are derived. Our approach is based on the solutions of the auxiliary dynamical system that effectively linearizes the problem under certain algebraic conditions. The formalism proposed is illustrated with the specific example of the NLSE with a non-Hermitian term that is the model of an atom laser. The analytical asymptotic solution to the Cauchy problem is obtained explicitly for this example. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
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14 pages, 292 KiB  
Article
Evolution Equations with Liouville Derivative on R without Initial Conditions
by Vladimir E. Fedorov and Nadezhda M. Skripka
Mathematics 2024, 12(4), 572; https://doi.org/10.3390/math12040572 - 14 Feb 2024
Viewed by 489
Abstract
New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique [...] Read more.
New classes of evolution differential equations with the Liouville derivative in Banach spaces are studied. Equations are considered on the whole real line and are not endowed by the initial conditions. Using the methods of the Fourier transform theory, we prove the unique solvability in the sense of classical solutions for the equation solved with respect to the Liouville fractional derivative with a bounded operator at the unknown function. This allows us to obtain the analogous result for the equation with a linear degenerate operator at the fractional derivative and with a spectrally bounded pair of operators. Abstract results are applied to obtain new results on the unique solvability of systems of ordinary differential equations, boundary problems to partial differential equations, and systems of equations. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
9 pages, 253 KiB  
Article
An Approach to Multidimensional Discrete Generating Series
by Svetlana S. Akhtamova, Tom Cuchta and Alexander P. Lyapin
Mathematics 2024, 12(1), 143; https://doi.org/10.3390/math12010143 - 01 Jan 2024
Viewed by 1603
Abstract
We extend existing functional relationships for the discrete generating series associated with a single-variable linear polynomial coefficient difference equation to the multivariable case. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
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27 pages, 446 KiB  
Article
Necessary and Sufficient Conditions for Solvability of an Inverse Problem for Higher-Order Differential Operators
by Natalia P. Bondarenko
Mathematics 2024, 12(1), 61; https://doi.org/10.3390/math12010061 - 24 Dec 2023
Cited by 1 | Viewed by 502
Abstract
We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, [...] Read more.
We consider an inverse spectral problem that consists in the recovery of the differential expression coefficients for higher-order operators with separate boundary conditions from the spectral data (eigenvalues and weight numbers). This paper is focused on the principal issue of inverse spectral theory, namely, on the necessary and sufficient conditions for the solvability of the inverse problem. In the framework of the method of the spectral mappings, we consider the linear main equation of the inverse problem and prove the unique solvability of this equation in the self-adjoint case. The main result is obtained for the first-order system of the general form, which can be applied to higher-order differential operators with regular and distribution coefficients. From the theorem on the main equation’s solvability, we deduce the necessary and sufficient conditions for the spectral data for a class of arbitrary order differential operators with distribution coefficients. As a corollary of our general results, we obtain the characterization of the spectral data for the fourth-order differential equation in terms of asymptotics and simple structural properties. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
17 pages, 317 KiB  
Article
Almost Automorphic Solutions to Nonlinear Difference Equations
by Marko Kostić, Halis Can Koyuncuoğlu and Vladimir E. Fedorov
Mathematics 2023, 11(23), 4824; https://doi.org/10.3390/math11234824 - 29 Nov 2023
Viewed by 661
Abstract
In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the [...] Read more.
In the present work, we concentrate on a certain class of nonlinear difference equations and propose sufficient conditions for the existence of their almost automorphic solutions. In our analysis, we invert an appropriate mapping and obtain the main existence outcomes by utilizing the contraction mapping principle. As the second objective of the manuscript, we reconsider one of the landmark results, namely the Bohr–Neugebauer theorem, in the qualitative theory of dynamical equations, and we investigate the relationship between the existence of almost automorphic solutions and the existence of solutions with a relatively compact range for the proposed difference equation type. Thus, we provide a discrete counterpart of the Bohr–Neugebauer theorem due to the almost automorphy notion under some technical conditions. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
13 pages, 321 KiB  
Article
A Family of Holomorphic and m-Fold Symmetric Bi-Univalent Functions Endowed with Coefficient Estimate Problems
by Pishtiwan Othman Sabir, Hari Mohan Srivastava, Waggas Galib Atshan, Pshtiwan Othman Mohammed, Nejmeddine Chorfi and Miguel Vivas-Cortez
Mathematics 2023, 11(18), 3970; https://doi.org/10.3390/math11183970 - 19 Sep 2023
Cited by 4 | Viewed by 786
Abstract
This paper presents a new general subfamily NΣmu,v(η,μ,γ,ℓ) of the family Σm that contains holomorphic normalized m-fold symmetric bi-univalent functions in the open unit disk D associated [...] Read more.
This paper presents a new general subfamily NΣmu,v(η,μ,γ,ℓ) of the family Σm that contains holomorphic normalized m-fold symmetric bi-univalent functions in the open unit disk D associated with the Ruscheweyh derivative operator. For functions belonging to the family introduced here, we find estimates of the Taylor–Maclaurin coefficients am+1 and a2m+1, and the consequences of the results are discussed. The current findings both extend and enhance certain recent studies in this field, and in specific scenarios, they also establish several connections with known results. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
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