On Some Integral Transformations and Their Application to the Solution of Boundary-Value Problems in Mathematical Physics

We obtain a formula for the expansion of an arbitrary function in a series in the eigenfunctions of the Sturm–Liouville boundary-value problem for the differential equation of cone functions. On the basis of this result, we derive a series of integral transformations (including well-known ones) and inversion formulas for them. We apply these formulas to the solution of initial boundary-value problems in the theory of heat conduction for circular hollow cones truncated by spherical surfaces.     

The stress state of an elastic composite cone with centre of rotation at the apex

The torsion of a composite cone that has a centre of rotation at its apex is investigated in a spherical system of coordinates. A composite cone is a cone with one shear modulus, inserted into a conical funnel having another shear modulus and with ideal mechanical contact between its surface and the inner surface of the conical funnel. The auxiliary problem of a composite cone with its apex truncated by a spherical surface is considered first. The outer surface of such a conical body is not loaded, but a load that reduces to a torque is applied to its spherical surface. The auxiliary problem is reduced to a one-dimensional discontinuous boundary-value problem using a specially constructed integral transformation. The exact solution of this boundary-value problem is constructed. The limit is then taken in the solution obtained as the radius of the spherical surface tends to zero for the purpose of obtaining an exact solution of the problem of the torsion of a composite cone that has a centre of rotation at the apex.     

Heat Conduction of Structures with a System of Capillary Tubes

We obtain equations that describe heat conduction of bodies spatially reinforced by a system of capillary tubes filled with a moving heat-tapping fluid. The corresponding initial boundary-value problem of heat conduction is formulated and qualitatively analyzed. For thin-walled structures, we present an algorithm for reduction of the three-dimensional problem to a two-dimensional one. Specific calculations in solving the formulated problem are performed in the stationary and axisymmetric cases for a conic shell of revolution. The influence of the reinforcement parameters (intensity and direction), velocity of fluid motion in thermal tubes, dimensions of the inner sections of the tubes, and cone angle of a shell on the temperature field is investigated.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

Solvability of Linear Matrix Boundary-Value Problem

We find solvability conditions and give a construction of generalized Green operator for a linear matrix boundary-value problem. We suggest an operator which reduces a linear matrix equation to a standard linearNoetherian boundary-value problem. To solve a linearmatrix systemwe use an operatorwhich reduces a linear matrix equation to a linear algebraic equation with rectangular matrix.     

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.     

Local convexity conditions for reachability tubes of distributed control systems

For a nonlinear functional operator equation describing a wide class of controlled initial boundary-value problems we introduce the notion of an abstract reachability set analogous to the notion of a reachability tube. We obtain local sufficient conditions for the convexity of such a set. We consider a mixed boundary-value problem associated with a semilinear hyperbolic equation of the second order in a rather general form as an example illustrating the reduction of a controlled initial boundary-value problem to the studied equation, as well as the verification of the stated assumptions.     

Determining the boundary of a two-dimensional region from the solution of the external initial boundary-value problem for the heat equation

We determine the boundary of a two-dimensional region using the solution of the external initial boundary-value problem for the nonhomogeneous heat equation. The initial values for the boundary determination include the right-hand side of the equation and the solution of the initial boundary-value problem given for finitely many points outside the region. The inverse problem is reduced to solving a system of two integral equations nonlinear in the function defining the sought boundary. An iterative procedure is proposed for numerical solution of the problem involving linearization of integral equations. The efficiency of the proposed procedure is investigated by a computer experiment.     

旋转锥面$l_\infty$拟合的截断光滑化牛顿法

In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use l∞ norm than 12 norm. l∞ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large amount of cones repeatedly, moreover, when large amount of measured data are to be fitted to one rotated cone, the number of components in the maximum function is large. So it is necessary to develop efficient solution methods. To solve such optimization problems efficiently, a truncated smoothing Newton method is presented. At first, combining aggregate smoothing technique to the maximum function as well as the absolute value function and a smoothing function to the square root function, a monotonic and uniform smooth approximation to the objective function is constructed. Using the smooth approximation, a smoothing Newton method can be used to solve the problem. Then, to reduce the computation cost, a truncated aggregate smoothing technique is applied to give the truncated smoothing Newton method, such that only a small subset of component functions are aggregated in each iteration point and hence the computation cost is considerably reduced.     

Truncated Smoothing Newton Method for l_∞ Fitting Rotated Cones

In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use l∞ norm than l2 norm. l∞ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large...     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

Continuous approximations of discontinuous nonlinearities of semilinear elliptic-type equations

We obtain new variational principles of the existence of strong and semiregular solutions of principal boundary-value problems for elliptic-type second-order equations with discontinuous nonlinearity. We study a problem of proximity between the sets of solutions of an approximating problem with nonlinearity continuous in phase variable and solutions of the initial boundary-value problem with discontinuous nonlinearity. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 224–233, February, 1999.     

Solvability of thermoviscoelastic problem for Leray alpha-model

We investigate solvability problem (in a weak sense) for one initial boundary-value problem describing alpha-Leray model with viscosity depending on a temperature.     

Positive solutions for a class of boundary-value problem with integral boundary conditions in Banach spaces

This paper investigates the existence and multiplicity of positive solutions for a class of nonlinear boundary-value problem of second-order differential equations with integral boundary conditions in ordered Banach spaces. The arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators. The nonexistence of a positive solution is also studied.     

关于耦合型对流-扩散方程组的奇异摄动边值问题

本文考虑下述耦合型对流-扩散方程组的奇异摄动边值问题:本文提出两种方法:一种是初值化解法,用这种方法,原始问题转化成一系列没有扰动的一阶微分方程或方程组的初值问题,从而得到一个渐近展开式;第二种是边值化解法,用这种方法,原始问题转化成一组没有边界层现象的边值问题,从而可以得到精确解和使用经典的数值方法去得到具有关于摄动参数ε一致的高精度数值解.     

To Boundary-Value Problems for Degenerating Pseudoparabolic Equations With Gerasimov–Caputo Fractional Derivative

In this paper we consider a boundary-value problems for degenerating pseudoparabolic equation with variable coefficients and with Gerasimov–Caputo fractional derivative. To solve the problem we obtain a priori estimates in differential and difference settings. These a priori estimates imply uniqueness and stability of the solution with respect to the initial data and the right-hand side on the layer, as well as the convergence of the solution of each of the difference problem to the solution of the corresponding differential problem.     

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

Localization principle and an estimate of the rate of damping of oscillations in a viscous compressible stratified liquid

Mathematical Notes - We consider the initial boundary-value problem describing small acoustic-gravitational oscillations of a viscous stratified liquid in the linear approximation. A distinctive...     

The finite-time blowup of the solution of an initial boundary-value problem for the nonlinear equation of ion sound waves

We obtain blowup conditions for the solutions of initial boundary-value problems for the nonlinear equation of ion sound waves in a hydrogen plasma in the approximation of “hot” electrons and “heavy” ions. A specific characteristic of this nonlinear equation is the noncoercive nonlinearity of the form ?_t|?u|², which complicates its study by any energy method. We solve this problem by the Mitidieri–Pohozaev method of nonlinear capacity.