Properties of the solutions of a certain class of variational inequalities

In a curvilinear quadrangle one considers an elliptic operator with linear principal terms and discontinuous leading coefficients. One investigates the solution of a variational inequality with a constraint on the derivatives, tangent to the boundary and to the discontinuity lines of the coefficients. On certain parts of the boundary one imposes the first boundary condition and on others a condition on a directional derivative. One proves the existence of a solution with square summable second derivatives at each point of the subdomains where the leading coefficients are smooth.Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 83–92, 1986.     

On the nonuniqueness of the solution of a mixed boundary value problem for the Laplace equation

We study the solvability of the mixed boundary value problem for the Laplace equation with three distinct boundary conditions, two of which include two directional derivatives with distinct tilt angles and the remaining one is the first boundary condition. An example of a nontrivial solution of the homogeneous problem is given, and conditions under which the problem has a unique solution are established. The solvability of the problem with a nonhomogeneous first boundary condition is studied.     

Solutions of the main boundary value problems for a loaded second-order parabolic equation with constant coefficients

We give well-posed statements of the main initial–boundary value problems in a rectangular domain and in a half-strip for a second-order parabolic equation that contains partial Riemann–Liouville fractional derivatives with respect to one of the two independent variables. We construct Green functions and representations of solutions of these problems. We prove existence and uniqueness theorems for the first boundary value problem and the problem in the half-strip with the boundary condition of the first kind.     

A problem with dynamic nonlocal condition for pseudohyperbolic equation

We consider an initial-boundary problem with dynamic nonlocal boundary condition for a pseudohyperbolic fourth-order equation in a cylinder. Dynamic nonlocal boundary condition represents a relation between values of a required solution, its derivatives with respect to spatial variables, second-order derivatives with respect to time variable and an integral term. The main result lies in substantiation of solvability of this problem. We prove the existence and uniqueness of a generalized solution. The proof is based on the a priori estimates obtained in this paper, Galyorkin’s procedure and the properties of the Sobolev spaces.     

Asymptotic Expansions for Two Singularly Perturbed Convection–Diffusion Problems with Discontinuous Data: The Quarter Plane and the Infinite Strip

We consider a singularly perturbed convection–diffusion equation,     , defined on two domains: a quarter plane,  ( x , y ) ∈ (0, ∞) × (0, ∞)  , and an infinite strip,  ( x , y ) ∈ (−∞, ∞) × (0, 1)  . We consider for both problems discontinuous Dirichlet boundary conditions:   u ( x , 0) = 0  and   u (0, y ) = 1  for the first one and   u ( x , 0) =χ_{[ a , b ]}( x )  and   u ( x , 1) = 0  for the second. For each problem, asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) when the singular parameter  ε→ 0⁺  (with fixed distance r to the discontinuity points of the boundary condition) and (b) when that distance   r → 0⁺  (with fixed ε). It is shown that in both problems, the first term of the expansion at  ε= 0  is an error function or a combination of error functions. This term characterizes the effect of the discontinuities on the ε-behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the discontinuities of the boundary condition, the solution u ( x , y ) of both problems is approximated by a linear function of the polar angle at the discontinuity points.     

Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition

For an equation of the parabolic-hyperbolic type, we consider an inverse problem with a nonlocal condition relating solution derivatives that belong to different types of the equation in question. We justify a uniqueness criterion and prove the existence of a solution of the problem by the spectral analysis method. We prove the stability of the solution with respect to the nonlocal boundary condition.     

一类非线性三阶三点边值问题的解和正解

考察了一类含有一阶和二阶导数的非线性三阶三点边值问题的解和正解.通过构造适当的Banach空间并且利用相应的积分方程建立了两个存在定理.主要结论表明,只要非线性项在其定义域的某个子集上的"高度"是适当的,该问题存在一个解或者正解.     

On the convergence of the Dirichlet grid problem with a singularity for a singularly perturbed convection–diffusion equation

The Dirichlet problem for a singulary perturbed convection–diffusion equation in a rectangle when a discontinuity at the flow exit the first derivative of the boundary condition gives rise to an inner layer for the solution. On piecewise-uniform Shishkin grids that condense near regular and characteristic layers, the solution obtained using the classical five-point difference scheme with a directed difference is shown to converge with respect to the small parameter to solve the original problem in the grid norm L _∞ ^h almost with the first order. This theoretical result is confirmed via numerical analysis.     

Application of a 14-point averaging operator in the grid method

The Dirichlet problem for Laplace’s equation in a rectangular parallelepiped is solved by applying the grid method. A 14-point averaging operator is used to specify the grid equations on the entire grid introduced in the parallelepiped. Given boundary values that are continuous on the parallelepiped edges and have first derivatives satisfying the Lipschitz condition on each parallelepiped face, the resulting discrete solution of the Dirichlet problem converges uniformly and quadratically with respect to the mesh size. Assuming that the boundary values on the faces have fourth derivatives satisfying the Hölder condition and the second derivatives on the edges obey an additional compatibility condition implied by Laplace’s equation, the discrete solution has uniform and quartic convergence with respect to the mesh size. The convergence of the method is also analyzed in certain cases when the boundary values are of intermediate smoothness.     

A problem with nonlocal conditions for mixed-type equations

We study a problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of degeneration line and with the displacement condition on parts of boundary characteristics of the Gellerstedt equation with singular coefficient. With the help of the maximum principle we prove the uniqueness of a solution to the problem, and with the help of the method of integral equations we prove the existence of a solution to the problem.     

Inverse problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator and with a nonlocal boundary condition

For an equation of the mixed elliptic-hyperbolic type, we study the inverse problem with a nonlocal condition relating the derivatives of the solution on the elliptic and hyperbolic parts of the boundary. We prove a uniqueness criterion and construct the solution in the form of a Fourier series.     

A non‐linear and non‐local boundary condition for a diffusion equation in petroleum engineering

This article deals with a boundary value problem for Laplace equation with a non‐linear and non‐local boundary condition. This problem comes from petroleum engineering and is used to obtain an estimation of well productivity. The non‐linear and non‐local boundary condition is written on the well boundary. On the outer reservoir boundaries, we have both Dirichlet and Neumann conditions. In this paper, we prove the existence and uniqueness of a solution to this problem. The existence is proved by Schauder theorem and the uniqueness is obtained under more restricted conditions, when the involved operator is a contraction. Copyright © 2005 John Wiley & Sons, Ltd.     

Bitsadze-Samarskii problem with a lacking shift condition for the Gellerstedt equation with a singular coefficient

For the Gellerstedt equation with a singular coefficient, we study the well-posedness of the problem with the Bitsadze-Samarskii conditions on the ellipticity boundary and on a segment of the degeneration line and with a shift condition on parts of boundary characteristics. We use the maximum principle to prove the uniqueness of the solution of the problem in the class of Hölder functions and the method of integral equations to prove its existence.     

Global existence of solutions to a coupled coagulation system

We consider a model equations describing the coagulation process of a gas on a surface. The problem is modeled by two coupled equations. The first one is a nonlinear transport equation with bilinear coagulation operator while the second one is a nonlinear ordinary differential equation. The velocity and the boundary condition of the transport equation depend on the supersaturation function satisfying the nonlinear ode. We first prove global existence and uniqueness of solution to the nonlinear transport equation then, we consider the coupled problem and prove existence in the large of solutions to the full coagulation system.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

The Impedance Boundary Value Problem for the Helmholtz Equation in a Half-Plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L_∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5]. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Local Existence of Contact Discontinuities in Relativistic Magnetohydrodynamics

We study the free boundary problem for a contact discontinuity for the system of relativistic magnetohydrodynamics. A surface of contact discontinuity is a characteristic of this system with no flow across the discontinuity for which the pressure, the velocity and the magnetic field are continuous whereas the density, the entropy and the temperature may have a jump. For the two-dimensional case, we prove the local-in-time existence in Sobolev spaces of a unique solution of the free boundary problem provided that the Rayleigh-Taylor sign condition on the jump of the normal derivative of the pressure is satisfied at each point of the initial discontinuity.     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

Smoothness of solutions of a special one-sided problem

We consider a planar domain, namely a curvilinear quadrilateral. We study a variational inequality of special form on the set of functions that are monotonically increasing on part of the boundary. This problem corresponds to a one-sided problem for an elliptic equation. A boundary condition of first kind is prescribed on part of the boundary, while on the other part of the boundary the tangential derivative is nonnegative and the product of the tangential and oblique derivatives is zero. We establish that the first derivatives of the solution satisfy a Hölder condition. Bibliography: 5 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 173–186.     

Unique Solvability of the Inverse Problem with a Time-Dependent Boundary Condition for a Sorption Model

A mixed initial boundary-value problem is considered for nonequilibrium sorption dynamics with inner-diffusion kinetics. The problem allows for convection and longitudinal diffusion and has a time-dependent boundary condition. This condition contains the time derivative of a solution component and constitutes the balance equation for the absorbed mixture near the output boundary of the sorption region—inside the diffusion barrier. Bounds on the solution of the direct problem are obtained: nonnegativity of the solution and its first time derivatives, and boundedness of the solution by known functions. The inverse problem of estimating the nonlinear system parameter—the sorption isotherm—is considered and a uniqueness theorem is proved.