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Mathematics
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Nonlinear Boundary Value Problems and Their Applications
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Nonlinear Boundary Value Problems and Their Applications

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

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 10186

Special Issue Editor

Prof. Dr. Andrey Amosov

E-Mail Website
Guest Editor
Department of Mathematical and Computer Modelling, National Research University "Moscow Power Engineering Institute", 111250 Moscow, Russia
Interests: mathematical physics equations; nonlinear boundary value problems; weak solutions; mathematical theory of radiative-conductive heat exchange problems; numerical analysis; homogenization; projection methods

Special Issue Information

Dear Colleagues,

Currently, there are many scientific challenges in the fascinating field of nonlinear differential equations and their applications. An important direction is the development of new, and improvement of known, methods of research and solving nonlinear problems, but no less important is the study of the properties and modelling of nonlinear mathematical problems arising in various applied areas.

The purpose of this Special Issue is to gather the latest contributions with recent advances in the mathematical theory of nonlinear boundary value problems and applications of nonlinear mathematical models to the broad fields of science and engineering.

We invite the authors to submit original research articles and high-quality review articles in the field of nonlinear boundary value problems obtained from development of methods of mathematical analysis and simulation of nonlinear models based on ordinary differential equations, partial differential equations, systems of partial differential equations and integro-differential equations, variational methods, computational methods, numerical analysis and others.

Prof. Dr. Andrey Amosov
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

  • nonlinear analysis
  • nonlinear mathematical models in applied sciences and engineering
  • nonlinear and quasilinear partial differential equations
  • weak solutions
  • systems of partial differential equations and integro-differential equations
  • variational methods for nonlinear problems
  • blow-up phenomena
  • asymptotic behavior and stabilization of solutions
  • homogenization
  • computational methods for nonlinear boundary value problems

Published Papers (7 papers)

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Research

19 pages, 316 KiB  
Article
Existence and Multiplicity of Solutions for a Class of Particular Boundary Value Poisson Equations
by Songyue Yu and Baoqiang Yan
Mathematics 2022, 10(12), 2070; https://0-doi-org.brum.beds.ac.uk/10.3390/math10122070 - 15 Jun 2022
Viewed by 900
Abstract
In this paper, a special class of boundary value problems, −▵u=λuq+ur,inΩ,u>0, inΩ, [...] Read more.
In this paper, a special class of boundary value problems, −▵u=λuq+ur,inΩ,u>0, inΩ,n·∇u+g(u)u=0,on∂Ω, where 0<q<1<r<N+2N−2 and g:[0,∞)→(0,∞) is a nondecreasing C1 function. Here, Ω⊂RN(N≥3) is a bounded domain with smooth boundary ∂Ω and λ>0 is a parameter. The existence of the solution is verified via sub- and super-solutions method. In addition, the influences of parameters on the minimum solution are also discussed. The second positive solution is obtained by using the variational method. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
20 pages, 782 KiB  
Article
Steady-State Navier–Stokes Equations in Thin Tube Structure with the Bernoulli Pressure Inflow Boundary Conditions: Asymptotic Analysis
by Rita Juodagalvytė, Grigory Panasenko and Konstantinas Pileckas
Mathematics 2021, 9(19), 2433; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192433 - 30 Sep 2021
Cited by 1 | Viewed by 1486
Abstract
Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic [...] Read more.
Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
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17 pages, 305 KiB  
Article
Positive Solutions for a Singular Elliptic Equation Arising in a Theory of Thermal Explosion
by Song-Yue Yu and Baoqiang Yan
Mathematics 2021, 9(17), 2173; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172173 - 06 Sep 2021
Cited by 2 | Viewed by 1259
Abstract
In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case [...] Read more.
In this paper, the thermal explosion model described by a nonlinear boundary value problem is studied. Firstly, we prove the comparison principle under nonlinear boundary conditions. Secondly, using the sub-super solution theorem, we prove the existence of a positive solution for the case K(x)>0, as well as the monotonicity of the maximal solution on parameter λ. Thirdly, the uniqueness of the solution for K(x)<0 is proved, as well as the monotonicity of the solutions on parameter λ. Finally, we obtain some new results for the existence of solutions, and the dependence on the λ for the case K(x) is sign-changing. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
16 pages, 696 KiB  
Article
A Nonhomogeneous Boundary Value Problem for Steady State Navier-Stokes Equations in a Multiply-Connected Cusp Domain
by Kristina Kaulakytė and Konstantinas Pileckas
Mathematics 2021, 9(17), 2022; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172022 - 24 Aug 2021
Viewed by 1431
Abstract
The boundary value problem for the steady Navier–Stokes system is considered in a 2D multiply-connected bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with nonzero flow rates over connected [...] Read more.
The boundary value problem for the steady Navier–Stokes system is considered in a 2D multiply-connected bounded domain with the boundary having a power cusp singularity at the point O. The case of a boundary value with nonzero flow rates over connected components of the boundary is studied. It is also supposed that there is a source/sink in O. In this case the solution necessarily has an infinite Dirichlet integral. The existence of a solution to this problem is proved assuming that the flow rates are “sufficiently small”. This condition does not require the norm of the boundary data to be small. The solution is constructed as the sum of a function with the finite Dirichlet integral and a singular part coinciding with the asymptotic decomposition near the cusp point. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
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30 pages, 445 KiB  
Article
Nonstationary Radiative–Conductive Heat Transfer Problem in a Semitransparent Body with Absolutely Black Inclusions
by Andrey Amosov
Mathematics 2021, 9(13), 1471; https://0-doi-org.brum.beds.ac.uk/10.3390/math9131471 - 23 Jun 2021
Cited by 4 | Viewed by 1163
Abstract
The paper is devoted to a nonstationary initial–boundary value problem governing complex heat exchange in a convex semitransparent body containing several absolutely black inclusions. The existence and uniqueness of a weak solution to this problem are proven herein. In addition, the stability of [...] Read more.
The paper is devoted to a nonstationary initial–boundary value problem governing complex heat exchange in a convex semitransparent body containing several absolutely black inclusions. The existence and uniqueness of a weak solution to this problem are proven herein. In addition, the stability of solutions with respect to data, a comparison theorem and the results of improving the properties of solutions with an increase in the summability of the data were established. All results are global in terms of time and data. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
18 pages, 329 KiB  
Article
Boundary Value Problems for ψ-Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions
by Surang Sitho, Sotiris K. Ntouyas, Ayub Samadi and Jessada Tariboon
Mathematics 2021, 9(9), 1001; https://0-doi-org.brum.beds.ac.uk/10.3390/math9091001 - 28 Apr 2021
Cited by 17 | Viewed by 1342
Abstract
In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point [...] Read more.
In the present article, we study a new class of sequential boundary value problems of fractional order differential equations and inclusions involving ψ-Hilfer fractional derivatives, supplemented with integral multi-point boundary conditions. The main results are obtained by employing tools from fixed point theory. Thus, in the single-valued case, the existence of a unique solution is proved by using the classical Banach fixed point theorem while an existence result is established via Krasnosel’skiĭ’s fixed point theorem. The Leray–Schauder nonlinear alternative for multi-valued maps is the basic tool to prove an existence result in the multi-valued case. Finally, our results are well illustrated by numerical examples. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
10 pages, 272 KiB  
Article
Non-Stationary Model of Cerebral Oxygen Transport with Unknown Sources
by Andrey Kovtanyuk, Alexander Chebotarev, Varvara Turova, Irina Sidorenko and Renée Lampe
Mathematics 2021, 9(8), 910; https://0-doi-org.brum.beds.ac.uk/10.3390/math9080910 - 20 Apr 2021
Cited by 2 | Viewed by 1392
Abstract
An inverse problem for a system of equations modeling oxygen transport in the brain is studied. The problem consists of finding the right-hand side of the equation for the blood oxygen transport, which is a linear combination of given functionals describing the average [...] Read more.
An inverse problem for a system of equations modeling oxygen transport in the brain is studied. The problem consists of finding the right-hand side of the equation for the blood oxygen transport, which is a linear combination of given functionals describing the average oxygen concentration in the neighborhoods of the ends of arterioles and venules. The overdetermination condition is determined by the values of these functionals evaluated on the solution. The unique solvability of the problem is proven without any smallness assumptions on the model parameters. Full article
(This article belongs to the Special Issue Nonlinear Boundary Value Problems and Their Applications)
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