Solution Of Mixed Problem Research Articles

THE EXISTENCE OF GENERALIZED SOLUTION TO ONEDIMENSIONAL MIXED PROBLEM FOR ONE CLASS OF THIRD ORDER NONILINEAR PSEUDOPARABOLIC EQUATIONS

Abstract. This work is dedicated to the study of existence in small of generalized solution of onedimensional mixed problem for one class of third order nonlinear pseudoparabolic equations. Conception of generalized solution for mixed problem under consideration is introduced. After applying Fourier method, the solution of original problem is reduced to the solution of some countable system of nonlinear inteqro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence theorem in small of generalized solution of the mixed problem is proved by contracted mappings principle.

Classical solution of the mixed problem for a nonlinear equation

The first mixed problem for a nonlinear equation is considered in the quarter plane. The Cauchy conditions are set at the bottom of the boundary. The Dirichlet condition is set on the left part of the boundary. The solution is constructed using the method of characteristics in an implicit analytical form as a solution of the integral equation. The solvability of these integral equations, the smoothness of the solutions, and their dependence on the initial data are investigated. The uniqueness is proved and the conditions are established, under which there exists a piecewise smooth and classical solution of the first mixed problem.

On the evolution of solutions of mixed problems in thermoelasticity of porous bodies with dipolar structure

On the evolution of solutions of mixed problems in thermoelasticity of porous bodies with dipolar structure

Propagation of electric field generated by periodic pumping in a stable medium of two-level atoms of the Maxwell–Bloch model

We consider the problem of the propagation of an electric field generated by periodic pumping in a stable medium of two-level atoms as the mixed problem for the Maxwell–Bloch equations without spectrum broadening. An approach to the study of such a problem is proposed. We use the inverse scattering transform method in the form of the matrix Riemann–Hilbert (RH) problem, using simultaneous spectral analysis of both the Lax equations. The proposed matrix RH problem solves the problem of the propagation of a sinusoidal signal in an unperturbed stable medium (attenuator). It is proved that this RH problem provides the causality principle for the region t < x, and for the region of the light cone, 0 < x < t allows us to find the asymptotics of the transmitted signal. First, we study the asymptotics of the RH problem for large times, and then, we obtain asymptotic formulas for the mixed problem solution of the Maxwell–Bloch equations when the attenuator is long enough. Three sectors are obtained in the light cone where the asymptotics have essentially different behaviors.

One method to prove of existence weak solution of a mixed problem for 2D parabolic equations

Abstract In this study using the residue method the solution of the first type mixed problem for 2D linear parabolic equation in the bounded cylinder of the Euclidean space R 3 ( x , y , t ) is obtained in explicit form. When the smoothness of initial data does not permit to construct of the classical solution then it is necessary to extend the concept of classical solution. Based on the examined problem is proved that the obtained solution is a weak solution too. The use of this proof method to prove the existence of a weak solution can be applied to prove the existence of weak solutions for the more general problem with non-self-adjoint and even that contains higher-order derivative with respect to t in the boundary condition.

Well-Posedness of a Mixed Problem in a Cylindrical Domain for One Class of Multidimensional Hyperbolic-Parabolic Equations

We prove the unique solvability and obtain the explicit expression for the classical solution of mixed problem in a cylindrical domain for one class of multidimensional hyperbolic-parabolic equations.

Brezzi--Douglas--Marini Interpolation of Any Order on Anisotropic Triangles and Tetrahedra

Recently, the $\vec{H}(\operatorname{div})$-conforming finite element families for second order elliptic problems have come more into focus, since due to hybridization and subsequent advances in computational efficiency their use is no longer mainly theoretical. Their property of yielding exactly divergence-free solutions for mixed problems makes them interesting for a variety of applications, including incompressible fluids. In this area, boundary and interior layers are present, which demand the use of anisotropic elements. While for the Raviart-Thomas interpolation of any order on anisotropic tetrahedra optimal error estimates are known, this contribution extends these results to the Brezzi-Douglas-Marini finite elements. Optimal interpolation error estimates are proved under two different regularity conditions on the elements, which both relax the standard minimal angle condition. Additionally a numerical application on the Stokes equations is presented to illustrate the findings.

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

Absence of global solutions of a mixed problem for a Ginzburg–Landau type nonline-ar evolution equation

We study the problem of the absence of global solutions of the first mixed problem for one nonlinear evolution equation of Ginzburg–Landau type.We prove that global solutions of the studied problem are absent for “sufficiently large” values of the initial data.

Existence of solution to some mixed problems for linear differential-algebraic systems of partial differential equations

We consider a linear differential-algebraic system of PDEs with special matrix coefficients. Two cases are investigated. In the first one, the system has a small index, and the matrix at unknown vector-function, while the system written in the canonical form, is arbitrary. In the second case, the system has an arbitrary index, while a matrix at the small term is triangular. In both the cases, using the methods of characteristics and successive approximations, we prove the existence of a unique classical solution of mixed problems for the considered differential-algebraic systems of PDEs.

Some estimates on solutions of mixed problems for mixtures

Our study is dedicated to a composite, which, in fact, is a mixture of two thermoelastic micropolar bodies. We formulate the mixed initial boundary value problem in this context and define the domain of influence for given data. For any solution of the mixed problem we associate a measure and prove a second-order differential inequality for it. Based on the maximum principle for the heat equation and on the second-order differential inequality, we establish an estimate which proves that the thermal and the mechanical effects, at large distance from the domain of influence, are dominated by an exponential decay.

The study of a mixed problem for one class of third order differential equations

This work is dedicated to the study of existence and uniqueness of almost everywhere solution of a multidimensional mixed problem for one class of third order differential equations with nonlinear operator on the right-hand side. The conception of almost everywhere solution for the mixed problem under consideration is introduced. After applying the Fourier method, the solution of the original problem is reduced to the solution of some countable system of nonlinear integro-differential equations in unknown Fourier coefficients of the sought solution. Besides, existence and uniqueness theorems of almost everywhere solution of the mixed problem under consideration are also proved in this work.

Mixed problems for degenerate abstract parabolic equations and applications

Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that, these problems arise in fluid mechanics and environmental engineering.

Solvability of Mixed Problems With Integral Condition for Singular Parabolic Equations

In this paper we proved the existance and uniqness of strong generalized solution of mixed problems wih integral condition for singular parabolic equaions depending on a theorem proved in [1] in which a priori estimaion of the solution for such problems was derived.

УСТОЙЧИВОСТЬ НЕЗАКРЕПЛЕННОЙ КРУГЛОЙ ПОДЗЕМНОЙ ВЫРАБОТКИ В АКТИВНОЙ ЗОНЕ ЗАГЛУБЛЕННОГО ЛЕНТОЧНОГО ФУНДАМЕНТА

The paper presents the basic correlations of the approximate analytical solution of mixed problem of soil elasticity and plasticity theory as well as the examples of the comparison of numeric values of the experiments’ results aimed at determining the load-bearing capacity of foundation beds and the form and dimensions of plastic strain ranges with the results obtained on its basis.

Explicit Estimates for Solutions of Mixed Elliptic Problems

We deal with the existence of quantitative estimates for solutions of mixed problems to an elliptic second-order equation in divergence form with discontinuous coefficients. Our concern is to estimate the solutions with explicit constants, for domains in ℝn (n≥2) of class C0,1. The existence of L∞ and W1,q estimates is assured for q=2 and any q<n/(n-1) (depending on the data), whenever the coefficient is only measurable and bounded. The proof method of the quantitative L∞ estimates is based on the De Giorgi technique developed by Stampacchia. By using the potential theory, we derive W1,p estimates for different ranges of the exponent p depending on the fact that the coefficient is either Dini-continuous or only measurable and bounded. In this process, we establish new existences of Green functions on such domains. The last but not least concern is to unify (whenever possible) the proofs of the estimates to the extreme Dirichlet and Neumann cases of the mixed problem.

A Priori Estimation of the Solution for Mixed Problems with Integral Condition for Singular Parabolic Equations

We prove a theorem in which we get a priori estimation of the solution for mixed problems with integral condition for singular parabolic equations. Mixed problems with nonlocal boundary conditions or with nonlocal initial conditions were studied in many works lately. Our result plays an important role in the theory of heat transmission, thermoelasticity, chemical engineering, underground water flow, and plasma physics.

Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions

We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional dierential equations. Dierence methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite dierence systems. Methods of second type are truncation of the infinite dierence system, so that the resulting dierence problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-dierenti al problems can be obtained from the general model by specifying the given operators.

Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain

We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain \(\Omega\) with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of \(\partial\Omega\) with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on \((0,c)\times\Omega\) to the initial boundary value problem, for small \(c\). Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations.

Solution of Mixed Problem Including a First Order Three Dimensional P.D.E with Nonlocal and Global Boundary Conditions

In this paper solution of mixed complex boundary value problem of first order is considered. The basic term in the problem with respect to space variables, has Cauchy-Riemann operator. We first use Laplace transformation to introduce spectral problem. Then we investigate for corresponding Fredholm's type. The spectral problem here is different from classical boundary value problems. Here boundary conditions are nonlocal and global and in general linear.At the end we find asymptotic expansionfor the solution of spectral problemwhich depends on unknown complex parameter. With the help of this asymptotic expansion we prove existence and uniqueness of mixed problem.