Solvability of Nonlinear Equations with Parameters: Branching, Regularization, Group Symmetry and Solutions Blow-Up

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 14373

Special Issue Editor

Institute of Mathematics and Information Technologies, Irkutsk State University, 1 Karl Marx Str., 664003 Irkutsk, Russia
Interests: branching theory of nonlinear equations; bifurcation; singular problems; regularization; approximate methods; differential-operator equations; kinetic systems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Starting with the seminal works of A.M. Lyapunov,  A. Poincaré,  and E. Schmidt, the branching theory of nonlinear parameter-dependent equations enabled various essential applications in natural sciences and engineering over the course of the last hundred years. V.I. Yudovich  pioneered the application of symmetry methods in branching theory. A series of applications of the Lyapunov–Schmidt method, Conley index theory, and the central manifold methods in the conditions of group symmetry were reported in many seminal works during the last decades. Various critical processes in plasma physics, fluid dynamics, and thermo-dynamics are modelled using the branching theory of nonlinear differential-operator parameter-dependent equations. The objective of this Special Issue is to report on the cutting edge development of the advanced branching theory of nonlinear equations and their applications. The Special issue will bring together experts in qualitative theory of differential-operator equations, numerical analysts, and practitioners in the various applied fields of contemporary natural sciences. The results on the solvability of non-standard nonlinear equations with parameters will be reported, focusing on the analysis of the problems associated with branching, regularization, group symmetry, and solution blow-up phenomena.

Prof. Dr. Nikolai A. Sidorov
Guest Editor

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Keywords

  • nonlinear analysis
  • bifurcation
  • singular problems
  • inverse problems
  • regularization
  • approximate methods
  • differential-operator equations
  • kinetic systems
  • Vlasov-Maxwell system
  • solution blow-up
  • functional equations
  • group symmetry
  • Lyapunov-Schmidt methods
  • analytical-numerical methods

Published Papers (8 papers)

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Editorial

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4 pages, 157 KiB  
Editorial
Special Issue Editorial “Solvability of Nonlinear Equations with Parameters: Branching, Regularization, Group Symmetry and Solutions Blow-Up”
Symmetry 2022, 14(2), 226; https://doi.org/10.3390/sym14020226 - 24 Jan 2022
Cited by 3 | Viewed by 1478
Abstract
Nonlinear dynamical models with parameters are at the heart of natural science, and they serve as essential instrument to analyze and solve various appealing problems in engineering areas [...] Full article

Research

Jump to: Editorial

14 pages, 289 KiB  
Article
On Nonlinear Forced Impulsive Differential Equations under Canonical and Non-Canonical Conditions
Symmetry 2021, 13(11), 2066; https://doi.org/10.3390/sym13112066 - 02 Nov 2021
Cited by 5 | Viewed by 1150
Abstract
This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefficient. Furthermore, sufficient conditions are obtained for the existence of positive bounded [...] Read more.
This study is connected with the nonoscillatory and oscillatory behaviour to the solutions of nonlinear neutral impulsive systems with forcing term which is studied for various ranges of of the neutral coefficient. Furthermore, sufficient conditions are obtained for the existence of positive bounded solutions of the impulsive system. The mentioned example shows the feasibility and efficiency of the main results. Full article
12 pages, 909 KiB  
Article
CT Image Reconstruction via Nonlocal Low-Rank Regularization and Data-Driven Tight Frame
Symmetry 2021, 13(10), 1873; https://doi.org/10.3390/sym13101873 - 04 Oct 2021
Cited by 2 | Viewed by 1306
Abstract
X-ray computed tomography (CT) is widely used in medical applications, where many efforts have been made for decades to eliminate artifacts caused by incomplete projection. In this paper, we propose a new CT image reconstruction model based on nonlocal low-rank regularity and data-driven [...] Read more.
X-ray computed tomography (CT) is widely used in medical applications, where many efforts have been made for decades to eliminate artifacts caused by incomplete projection. In this paper, we propose a new CT image reconstruction model based on nonlocal low-rank regularity and data-driven tight frame (NLR-DDTF). Unlike the Spatial-Radon domain data-driven tight frame regularization, the proposed NLR-DDTF model uses an asymmetric treatment for image reconstruction and Radon domain inpainting, which combines the nonlocal low-rank approximation method for spatial domain CT image reconstruction and data-driven tight frame-based regularization for Radon domain image inpainting. An alternative direction minimization algorithm is designed to solve the proposed model. Several numerical experiments and comparisons are provided to illustrate the superior performance of the NLR-DDTF method. Full article
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20 pages, 514 KiB  
Article
Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions
Symmetry 2020, 12(7), 1084; https://doi.org/10.3390/sym12071084 - 01 Jul 2020
Cited by 3 | Viewed by 1532
Abstract
The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The [...] Read more.
The famous Toms effect (1948) consists of a substantial increase of the critical Reynolds number when a small amount of soluble polymer is introduced into water. The most noticeable influence of polymer additives is manifested in the boundary layer near solid surfaces. The task includes the ratio of two characteristic length scales, one of which is the Prandtl scale, and the other is defined as the square root of the normalized coefficient of relaxation viscosity (Frolovskaya and Pukhnachev, 2018) and does not depend on the characteristics of the motion. In the limit case, when the ratio of these two scales tends to zero, the equations of the boundary layer are exactly integrated. One of the goals of the present paper is group analysis of the boundary layer equations in two mathematical models of the flow of aqueous polymer solutions: the second grade fluid (Rivlin and Ericksen, 1955) and the Pavlovskii model (1971). The equations of the plane non-stationary boundary layer in the Pavlovskii model are studied in more details. The equations contain an arbitrary function depending on the longitudinal coordinate and time. This function sets the pressure gradient of the external flow. The problem of group classification with respect to this function is analyzed. All functions for which there is an extension of the kernels of admitted Lie groups are found. Among the invariant solutions of the new model of the boundary layer, a special place is taken by the solution of the stationary problem of flow around a rectilinear plate. Full article
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16 pages, 840 KiB  
Article
Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type
Symmetry 2020, 12(6), 999; https://doi.org/10.3390/sym12060999 - 11 Jun 2020
Cited by 12 | Viewed by 1739
Abstract
The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the [...] Read more.
The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya. Full article
15 pages, 1187 KiB  
Article
On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term
Symmetry 2020, 12(6), 921; https://doi.org/10.3390/sym12060921 - 02 Jun 2020
Cited by 12 | Viewed by 2257
Abstract
The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation [...] Read more.
The paper deals with two-dimensional boundary-value problems for the degenerate nonlinear parabolic equation with a source term, which describes the process of heat conduction in the case of the power-law temperature dependence of the heat conductivity coefficient. We consider a heat wave propagation problem with a specified zero front in the case of two spatial variables. The solution existence and uniqueness theorem is proved in the class of analytic functions. The solution is constructed as a power series with coefficients to be calculated by a proposed constructive recurrent procedure. An algorithm based on the boundary element method using the dual reciprocity method is developed to solve the problem numerically. The efficiency of the application of the dual reciprocity method for various systems of radial basis functions is analyzed. An approach to constructing invariant solutions of the problem in the case of central symmetry is proposed. The constructed solutions are used to verify the developed numerical algorithm. The test calculations have shown the high efficiency of the algorithm. Full article
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19 pages, 363 KiB  
Article
Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator
Symmetry 2020, 12(6), 912; https://doi.org/10.3390/sym12060912 - 01 Jun 2020
Cited by 3 | Viewed by 2021
Abstract
The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, [...] Read more.
The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations. Full article
10 pages, 714 KiB  
Article
A Boundary Value Problem for Noninsulated Magnetic Regime in a Vacuum Diode
Symmetry 2020, 12(4), 617; https://doi.org/10.3390/sym12040617 - 14 Apr 2020
Cited by 3 | Viewed by 1592
Abstract
In this paper, we study the stationary boundary value problem derived from the magnetic (non) insulated regime on a plane diode. Our main goal is to prove the existence of non-negative solutions for that nonlinear singular system of second-order ordinary differential equations. To [...] Read more.
In this paper, we study the stationary boundary value problem derived from the magnetic (non) insulated regime on a plane diode. Our main goal is to prove the existence of non-negative solutions for that nonlinear singular system of second-order ordinary differential equations. To attain such a goal, we reduce the boundary value problem to a singular system of coupled nonlinear Fredholm integral equations, then we analyze its solvability through the existence of fixed points for the related operators. This system of integral equations is studied by means of Leray-Schauder’s topological degree theory. Full article
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