Problems For Systems Of Equations Research Articles

Solvability of initial boundary value problems for a quasihydrodynamic system of equations in the case of a weakly compressible fluid

Subject of research: solvability of the first initial-boundary value problem for a quasi-hydrodynamic system of equations in the case of a weakly compressible fluid. Method of research: the proof is based on the Galerkin method using a priori estimates. Main results of research: the existence theorem for generalized solutions is proven.

Vector Fixed Point Approach to Control of Kolmogorov Differential Systems

The paper presents a vector approach to control problems for systems of equations. The method is described in the case of Kolmogorov systems which arise frequently in the dynamics of populations. Three types of problems are discussed: problems with control of both per capita growth rates, problems with control parameters acting on the growth rates, and problems which combine the first two types. The controllability is obtained via a vector approach based on the Perov fixed point theorem and matrices which are convergent to zero. Four concrete illustrative examples are added.

Core competency oriented problem-solving teaching ----Taking "Solving Application Problems with a System of Equations" as an Example

The teaching of Mathematical problem-solving is one of the three key teaching points in junior high school mathematics. Solving applied problems through equations (systems) is an important component of the first module of teaching content, "Numbers and Algebra", and holds a very important position. It is a key focus in junior high school mathematics problem-solving teaching. At present, under the premise of the implementation of the new curriculum reform in the field of mathematics in China, based on the recent publication of compulsory education curriculum standards,the teachers of mathematics should focus on cultivating excellent mathematical core literacy among junior high school students, and ensuring that they can independently cope with and solve problems in the math teaching of junior high school. Based on the above background, the author focuses on the newly published mathematics curriculum standards in 2022, and studies the teaching design of "solving application problems for systems of equations". In this paper, we explore the mathematical core literacy elements contained in the teaching of solving applied problems in equations, study the improvement path of mathematical problem-solving teaching, share the teaching design practice and thinking based on core literacy oriented problem-solving teaching, and the important role that this teaching design can play in cultivating students' core literacy .

Criterion for equational noetherianity and complexity of the solvability problem for systems of equations over partially ordered sets

Criterion for equational noetherianity and complexity of the solvability problem for systems of equations over partially ordered sets

Blow up of solutions of systems of nonlinear equations of thermoelasticity

Sufficient conditions of blow up in finite time of solutions to initial boundary value problems for nonlinear systems of equations of thermoelasticity type are obtained. It is shown that solutions even with large enough initial energies of the considered problems may blow up in a finite time.

On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order.

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.

SPECIAL SPLINE APPROXIMATION FOR THE SOLUTION OF THE NON-STATIONARY 3-D MASS TRANSFER PROBLEM

In this paper we consider the conservative averaging method (CAM) with special spline approximation for solving the non-stationary 3-D mass transfer problem. The special hyperbolic type spline, which interpolates the middle integral values of piece-wise smooth function is used. With the help of these splines the initial-boundary value problem (IBVP) of mathematical physics in 3-D domain with respect to one coordinate is reduced to problems for system of equations in 2-D domain. This procedure allows reduce also the 2-D problem to a 1-D problem and thus the solution of the approximated problem can be obtained analytically. The accuracy of the approximated solution for the special 1-D IBVP is compared with the exact solution of the studied problem obtained with the Fourier series method. The numerical solution is compared with the spline solution. The above-mentioned method has extensive physical applications, related to mass and heat transfer problems in 3-D domains.

Язык моделирования гетерогенных динамических систем LISMA_HDS

Heterogeneous dynamic systems (HDS) simultaneously describe processes of different physical nature. Systems of this kind are typical for numerous applications. HDSs are characterized by the following features. They are often multimode or hybrid systems. In general, their modes are defined as initial value problems (Cauchy problems) for implicit differential-algebraic systems of equations. Due to the presence of heterogeneous dynamic components or processes evolving in both time and space, the dimension of the complete system of equations may be pretty high. In some cases, the system of equations has an internal structure, for instance, the differential-algebraic system of equations approximating a partial differential equation by the method of lines. An original huge system of equations can then be algorithmically rewritten in a compact form. Moreover, heterogeneous hybrid dynamical systems can generate events of qualitatively different types. Therefore one has to use different numerical event detection algorithms. Nowadays, HDSs are modeled and simulated in computer environments. The modeling languages widely used by engineers do not allow them to fully specify all the properties of the systems of this class. For instance, they do not include event typing constructs. That is why a declarative general-purpose modeling language named LISMA_HDS has been developed for the computer-aided modeling and ISMA simulation environment. The language takes into account all of the characteristic features of HDSs. It includes constructs for plain or algorithmic declaration of model constants, initial value problems for explicit differential-algebraic systems of equations, and initial guesses for variables. It also allows researchers to define explicit time events, modes and transitions between them upon the occurrence of events of different types, to use macros and implement event control. LISMA_HDS is defined by a generative grammar in an extended Backus-Naur form and semantic constraints. It is proved that the grammar belongs to the LL(2) subclass of context-free grammars.

The Kato Square Root Problem for Divergence Form Operators with Potential

The Kato square root problem for divergence form elliptic operators with potential V : {mathbb {R}}^{n} rightarrow {mathbb {C}} is the equivalence statement left| left( L + V right) ^{frac{1}{2}} u right| _{2} simeq left| nabla u right| _{2} + left| V^{frac{1}{2}} u right| _{2}, where L + V := - mathrm {div} (A nabla ) + V and the perturbation A is an L^{infty } complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying left| left| V right| ^{frac{alpha }{2}} u right| _{2} + left| (-Delta )^{frac{alpha }{2}}u right| _{2} lesssim left| left( left| V right| - Delta right) ^{frac{alpha }{2}}u right| _{2} for all u in D(left| V right| - Delta ) and some alpha in (1,2]. The class of potentials that will satisfy such a condition is known to contain the reverse Hölder class RH_{2} and L^{frac{n}{2}}left( {mathbb {R}}^{n} right) in dimension n > 4. To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by Axelsson, Keith and McIntosh for proving quadratic estimates is developed. In addition to applying this non-homogeneous framework to the scalar Kato problem with zero-order potential, it will also be applied to the Kato problem for systems of equations with zero-order potential.

The Variational Factors Problem for Systems of Equations Written in an Extended Kovalevskaya Form

In the paper, a solution to the variational factors problem for systems of equations written in an extended Kovalevskaya form is given. The solution is presented locally in a coordinate form.

Computational algorithms based on the fundamental solution of task operator

An approximation of matrix Green’s function for a hyperbolic system of linear differential equations of the first order has been created. An algorithm for an approximate solution of the generalized Riemann problem of discontinuity decay in the presence of additional conditions at the boundaries has been proposed. A computational algorithm for the approximate solution of boundary value problems for hyperbolic systems of equations has been set up on the basis of these solutions. The algorithm is implemented for a system of equations describing the propagation of elastic waves in a block-fractured medium. The results of the numerical experiments aimed at studying the possible mechanisms of propagation of elastic waves to a considerable depth without scattering have been presented.

Inverse problems for a class of linear Sobolev type equations with overdetermination on the kernel of operator at the derivative

Abstract The purpose of this work is to obtain sufficient conditions of a solution existence and uniqueness for a class of inverse problems for linear evolution equations with a degenerate operator at the derivative and with an unknown element in the right-hand side of the equation, which depends on the time variable. The overdetermination condition is given on the kernel of the operator at the derivative, the initial condition have the Cauchy form or the Showalter–Sidorov form. The obtained abstract results are applied to the investigation of linear inverse problems for the Sobolev system of equations and for the linearized Oskolkov system with overdetermination on the pressure gradient function.

SPECIAL HPERBOLIC TYPE APPROXIMATION FOR SOLVING OF 3-D TWO LAYER STATIONARY DIFFUSION PROBLEM

In this paper we examine the conservative averaging method (CAM) along the vertical z-coordinate for solving the 3-D boundary-value 2 layers diffusion problem. The special parabolic and hyperbolic type approximation (splines), that interpolate the middle integral values of piece-wise smooth function, is investigated. With the help of these splines the problems of mathematical physics in 3-D with respect to one coordinate are reduced to problems for system of equations in 2-D in every layer. This procedure allows reduce also the 2-D problem to a 1-D problem and the solution of the approximated problem can be obtained analytically. As the practical application of the created mathematical model, we are studying the calculation of the concentration of heavy metal calcium (Ca) in a two-layer peat block.

Asymptotic Stability of a Stationary Solution of a Multidimensional Reaction–Diffusion Equation with a Discontinuous Source

A two-dimensional reaction–diffusion equation in a medium with discontinuous characteristics is considered; the existence, local uniqueness, and asymptotic stability of its stationary solution, which has a large gradient at the interface, is proved. This paper continues the authors’ works concerning the existence and stability of solutions with internal transition layers of boundary value problems with discontinuous terms to multidimensional problems. The proof of the existence and stability of a solution is based on the method of upper and lower solutions. The methods of analysis proposed in this paper can be generalized to equations of arbitrary dimension of the spatial variables, as well as to more complex problems, e.g., problems for systems of equations. The results of this work can be used to develop numerical algorithms for solving stiff problems with discontinuous coefficients.

On Boundary-Value Problems for Second-Order Elliptic and Parabolic Systems

For nonstandard boundary-value problems for systems of equations of elliptic and parabolic types with vector boundary conditions, the well-posedness is proved. Typical examples are provided.

Determination of thermophysical characteristics of vulcanizable rubber products by the mathematical modeling method

The studies of the vulcanization kinetics of elastomers were carried out using a Truck tyre tread rubber compound. The formal kinetic scheme of vulcanization of rubbers sulfur-accelerator curing system was used which generalizes the set of reactions occurring in the curing process. A mathematical model is developed for determining the thermal parameters vulcanizable mixture comprising algorithms for solving direct and inverse problems for system of equations of heat conduction and kinetics of the curing process. The performance of the model is confirmed by the results of numerical experiments on model examples.

Scalarization of stationary semiclassical problems for systems of equations and its application in plasma physics

We propose a method for determining asymptotic solutions of stationary problems for pencils of differential (and pseudodifferential) operators whose symbol is a self-adjoint matrix. We show that in the case of constant multiplicity, the problem of constructing asymptotic solutions corresponding to a distinguished eigenvalue (called an effective Hamiltonian, term, or mode) reduces to studying objects related only to the determinant of the principal matrix symbol and the eigenvector corresponding to a given (numerical) value of this effective Hamiltonian. As an example, we show that stationary solutions can be effectively calculated in the problem of plasma motion in a tokamak.

Некоторые нестандартные задачи трехмерной теории поля

The article considers certain boundary-value problems for the Poisson system of equations in the 3D space the boundary conditions of which contain, apart from the function values, the values of gradient, divergence and curl. The statement of such problems is based on an identity containing unconditional link between the boundary values of any pair of functions and the values of vectors of their normal derivatives and curls. Such identity follows from well-known formula representing the Laplace operator in the curl-divergence form. Such conditions have a clear hydrodynamic sense and generate the corresponding subspaces of the Sobolev space. In addition, the obtained identity can be considered as the necessary condition for the theorem about the smoothness of generalized or weak solutions of the considered problems. The corresponding boundary-value problems are correctly solvable in the weak sense; that is, they have a unique weak solution. Some specific examples are given. For the proof, the author used the Galerkin method, which is based on the equality of the bilinear forms consistent with the standard formulation of the Laplace operator and its curl-divergence form. It is exactly the equality of these forms that determines different types of boundary conditions containing the basic operations of the field theory, namely, gradient, divergence and curl of the sought solution. It has been shown that the sequence of approximate solutions is compact in nature, and passage to the limit has been done. By using the proposed approach it becomes possible to formulate some nonstandard problems for the Stokes and Navier-Stokes systems of equations.

A Parametric Dirichlet Problem for Systems of Quasilinear Elliptic Equations With Gradient Dependence

ABSTRACTThe aim of this article is to study the Dirichlet boundary value problem for systems of equations involving the -Laplacian operators and parameters μi≥0 (i = 1,2) in the principal part. Another main point is that the nonlinearities in the reaction terms are allowed to depend on both the solution and its gradient. We prove results ensuring existence, uniqueness, and asymptotic behavior with respect to the parameters.

Regularity of weak solutions to the model Venttsel problem for linear parabolic systems with nonsmooth in time principal matrix: A(t)-caloric approximation method

We consider the model Venttsel type problem for linear parabolic systems of equations. The Venttsel type boundary condition is fixed on the flat part of the lateral surface of a given cylinder. It is defined by a parabolic operator (with respect to the tangential derivatives) and the conormal derivative. The Holder continuity of every weak solution of the problem is proved under optimal assumptions on the data. In particular, only boundedness in the time variable of the principal matrices of the system and the boundary operator is assumed. All results are obtained by so-called A(t)-caloric approximation method (Arkhipova et al. in Nonlinear Anal 95:421–435, 2014).