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Mathematics
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Advanced Approaches to Mathematical Physics Problems
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Advanced Approaches to Mathematical Physics Problems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (30 November 2024) | Viewed by 3609

Special Issue Editor

Prof. Dr. Alexander G. Ramm

E-Mail Website
Guest Editor
Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA
Interests: differential and integral equations; operator theory; ill-posed and inverse problems; scattering theory; functional analysis; spectral theory; numerical analysis; theoretical electrical engineering; signal estimation; tomography
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue concentrates on the following three topics:

  1. Wave scattering by many small bodies and the creation of materials with a desired refraction coefficient. Many-body wave scattering is a major problem in mathematical physics. The assumption that a<<d<<λ, where a is the characteristic size of the small bodies, d is the minimal distance between the bodies and λ is the wavelength, allows one to give an asymptotically exact solution of the many-body scattering problem when multiple scattering is essential. 
  2. Analysis of the Navier–Stokes equation, which is one of the greatest problems in mathematical physics. This Special Issue addresses proof of the contradictory nature of the Navier–Stokes problem (NSP) in the three-dimensional space without boundaries and the NSP paradox, which solves one of the Millennium Prize Problems.
  3. Boundary values of analytic functions and singular integral equations in spaces of summable functions. 

Prof. Dr. Alexander G. Ramm
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

  • many-body wave scattering
  • creating materials with a desired refraction coefficient
  • Navier–Stokes problem
  • singular integral equations on L1(S)
  • summable boundary values of analytic functions

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Published Papers (4 papers)

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Research

27 pages, 417 KiB  
Article
Spatially-Periodic Solutions for Evolution Anisotropic Variable-Coefficient Navier–Stokes Equations: I. Weak Solution Existence
by Sergey E. Mikhailov
Mathematics 2024, 12(12), 1817; https://doi.org/10.3390/math12121817 - 12 Jun 2024
Viewed by 642
Abstract
We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the [...] Read more.
We consider evolution (non-stationary) spatially-periodic solutions to the n-dimensional non-linear Navier–Stokes equations of anisotropic fluids with the viscosity coefficient tensor variable in spatial coordinates and time and satisfying the relaxed ellipticity condition. Employing the Galerkin algorithm with the basis constituted by the eigenfunctions of the periodic Bessel-potential operator, we prove the existence of a global weak solution. Full article
(This article belongs to the Special Issue Advanced Approaches to Mathematical Physics Problems)
20 pages, 1772 KiB  
Article
Numerical Recovering of Space-Dependent Sources in Hyperbolic Transmission Problems
by Miglena N. Koleva and Lubin G. Vulkov
Mathematics 2024, 12(11), 1748; https://doi.org/10.3390/math12111748 - 4 Jun 2024
Viewed by 618
Abstract
A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, [...] Read more.
A body may have a structural, thermal, electromagnetic or optical role. In wave propagation, many models are described for transmission problems, whose solutions are defined in two or more domains. In this paper, we consider an inverse source hyperbolic problem on disconnected intervals, using solution point constraints. Applying a transform method, we reduce the inverse problems to direct ones, which are studied for well-posedness in special weighted Sobolev spaces. This means that the inverse problem is said to be well posed in the sense of Tikhonov (or conditionally well posed). The main aim of this study is to develop a finite difference method for solution of the transformed hyperbolic problems with a non-local differential operator and initial conditions. Numerical test examples are also analyzed. Full article
(This article belongs to the Special Issue Advanced Approaches to Mathematical Physics Problems)
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8 pages, 270 KiB  
Article
Functional Forms for Lorentz Invariant Velocities
by James M. Hill
Mathematics 2024, 12(11), 1609; https://doi.org/10.3390/math12111609 - 21 May 2024
Cited by 4 | Viewed by 910
Abstract
Lorentz invariance lies at the very heart of Einstein’s special relativity, and both the energy formula and the relative velocity formula are well-known to be invariant under a Lorentz transformation. Here, we investigate the spatial and temporal dependence of the velocity field itself [...] Read more.
Lorentz invariance lies at the very heart of Einstein’s special relativity, and both the energy formula and the relative velocity formula are well-known to be invariant under a Lorentz transformation. Here, we investigate the spatial and temporal dependence of the velocity field itself u(x,t) and we pose the problem of the determination of the functional form of those velocity fields u(x,t) which are automatically invariant under a Lorentz transformation. For a single spatial dimension, we determine a first-order partial differential equation for the velocity u(x,t), which appears to be unknown in the literature, and we investigate its main consequences, including demonstrating that it is entirely consistent with many of the familiar outcomes of special relativity and deriving two new partial differential relations connecting energy and momentum that are fully compatible with the Lorentz invariant energy–momentum relations. Full article
(This article belongs to the Special Issue Advanced Approaches to Mathematical Physics Problems)
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20 pages, 331 KiB  
Article
Regularized Asymptotics of the Solution of a Singularly Perturbed Mixed Problem on the Semiaxis for the Schrödinger Equation with the Potential Q = X2
by Alexander Yeliseev, Tatiana Ratnikova and Daria Shaposhnikova
Mathematics 2023, 11(20), 4328; https://doi.org/10.3390/math11204328 - 17 Oct 2023
Viewed by 783
Abstract
In this paper, we study the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis for the Schrödinger equation in the presence of a “strong” turning point for the limit operator on time interval that do not contain focal points. Based [...] Read more.
In this paper, we study the solution of a singularly perturbed inhomogeneous mixed problem on the half-axis for the Schrödinger equation in the presence of a “strong” turning point for the limit operator on time interval that do not contain focal points. Based on the ideas of the regularization method for asymptotic integration of problems with an unstable spectrum, it is shown how regularizing functions should be constructed for this type of singularity. The paper describes in detail the formalism of the regularization method, justifies the algorithm, constructs an asymptotic solution of any order in a small parameter, and proves a theorem on the asymptotic convergence of the resulting series. Full article
(This article belongs to the Special Issue Advanced Approaches to Mathematical Physics Problems)
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