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Mathematics
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Contemporary Generalizations of Differential Equations and Related Enhancements in Analysis
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Contemporary Generalizations of Differential Equations and Related Enhancements in Analysis

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

Deadline for manuscript submissions: closed (1 February 2024) | Viewed by 3429

Special Issue Editor

Dr. Andrey Borisovich Muravnik

E-Mail Website
Guest Editor
Nikol’skii Mathematical Institute, Peoples' Friendship University of Russia, 117198 Moscow, Russia
Interests: partial differential equations and related areas of analysis

Special Issue Information

Dear Colleagues,

Differential equations have been one of the most important and influential fields of mathematics for centuries, with wide-ranging applications in physics, engineering, finance, and many other fields. The study of differential equations has led to many important discoveries, and researchers have developed a variety of techniques to solve them. In recent years, the study of differential equations has expanded to include various generalizations.

This Special Issue aims to include research papers and surveys covering a wide range of topics in pure and applied mathematics. High-quality papers on the following topics are welcome: functional-differential equations, fractional-order differential equations, degenerate and singular differential equations, and nonlinear differential equations. Related areas of analysis within the framework of the Special Issue include: spectral theory, weight functions spaces, special functions, integral transformations, nonlocal aspects of functional analysis, nonlinear analysis, asymptotical analysis, and applications of all the above-mentioned theoretical directions.

Dr. Andrey Borisovich Muravnik
Guest Editor

Manuscript Submission Information

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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

  • estimates of solutions
  • nonlocal properties
  • qualitative behavior
  • spectral problems
  • boundary-value problems
  • nonlocal problems
  • Fourier analysis
  • integrals and derivatives of fractional orders
  • nonclassical integral transformations

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Published Papers (4 papers)

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Research

9 pages, 240 KiB  
Article
On Global Solutions of Two-Dimensional Hyperbolic Equations with General-Kind Nonlocal Potentials
by Andrey B. Muravnik
Mathematics 2024, 12(12), 1811; https://doi.org/10.3390/math12121811 - 11 Jun 2024
Viewed by 701
Abstract
In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not [...] Read more.
In the case of one spatial independent variable, we study hyperbolic differential-difference equations with potentials represented as linear combinations of translations of the desired function along the spatial variable. The qualitative novelty of this investigation is that, unlike previous research, it is not assumed that the real part of the symbol of the differential-difference operator contained in the equation has a constant sign. Previously, it was possible to remove that substantial restriction (i.e., the specified sign constancy) only for the case where the nonlocal term (i.e., the translated potential) is unique. In the present paper, we consider the case of the general-kind one-variable nonlocal potential, i.e., equations with an arbitrary amount of translated terms. No commensurability assumptions are imposed on the translation lengths. The following results are presented: We find a condition relating the coefficients at the nonlocal terms of the investigated equation and the length of the translations, providing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation. Full article
(This article belongs to the Special Issue Contemporary Generalizations of Differential Equations and Related Enhancements in Analysis)
33 pages, 500 KiB  
Article
Resolvent Convergence for Differential–Difference Operators with Small Variable Translations
by Denis Ivanovich Borisov and Dmitry Mikhailovich Polyakov
Mathematics 2023, 11(20), 4260; https://doi.org/10.3390/math11204260 - 12 Oct 2023
Cited by 2 | Viewed by 751
Abstract
We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations [...] Read more.
We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations are governed by a small multi-dimensional parameter. The operators are considered either on the entire space or an arbitrary multi-dimensional domain with a regular boundary. The boundary conditions are also arbitrary and general and involve small variable translations. Our main results state that the considered operators converge in the norm resolvent sense to ones with zero translations in the best possible operator norm. Estimates for the convergence rates are established as well. We also prove the convergence of the spectra and pseudospectra. Full article
(This article belongs to the Special Issue Contemporary Generalizations of Differential Equations and Related Enhancements in Analysis)
22 pages, 415 KiB  
Article
Local Solvability and Stability of an Inverse Spectral Problem for Higher-Order Differential Operators
by Natalia P. Bondarenko
Mathematics 2023, 11(18), 3818; https://doi.org/10.3390/math11183818 - 5 Sep 2023
Cited by 3 | Viewed by 817
Abstract
In this paper, we, for the first time, prove the local solvability and stability of an inverse spectral problem for higher-order (n>3) differential operators with distribution coefficients. The inverse problem consists of the recovery of differential equation coefficients from [...] Read more.
In this paper, we, for the first time, prove the local solvability and stability of an inverse spectral problem for higher-order (n>3) differential operators with distribution coefficients. The inverse problem consists of the recovery of differential equation coefficients from (n−1) spectra and the corresponding weight numbers. The proof method is constructive. It is based on the reduction of the nonlinear inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation remains uniquely solvable. Furthermore, we estimate the differences of the coefficients in the corresponding functional spaces. Full article
(This article belongs to the Special Issue Contemporary Generalizations of Differential Equations and Related Enhancements in Analysis)
17 pages, 370 KiB  
Article
(F, G, C)-Resolvent Operator Families and Applications
by Vladimir E. Fedorov and Marko Kostić
Mathematics 2023, 11(16), 3505; https://doi.org/10.3390/math11163505 - 14 Aug 2023
Cited by 1 | Viewed by 781
Abstract
In this paper, we introduce and investigate several new classes of (F,G,C)-regularized resolvent operator families subgenerated by multivalued linear operators in locally convex spaces. The known classes of (a,k)-regularized C-resolvent [...] Read more.
In this paper, we introduce and investigate several new classes of (F,G,C)-regularized resolvent operator families subgenerated by multivalued linear operators in locally convex spaces. The known classes of (a,k)-regularized C-resolvent operator-type families are special cases of the classes introduced in this paper. We provide certain applications of (F,G,C)-regularized resolvent operator families and (a,k)-regularized C-resolvent families to abstract fractional differential–difference inclusions and abstract Volterra integro-difference inclusions. Full article
(This article belongs to the Special Issue Contemporary Generalizations of Differential Equations and Related Enhancements in Analysis)
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