Axis-symmetric boundary-value problem describing distribution of charges in semiconductors

Under certain assumptions, the existence of two families of solutions to the stationary problem is proved and asymptotic expansions with respect to a small parameter are established. The stability of these families of solutions is shown. The problem under consideration has a solution connected with the so-called interior transition layer. In the case of a nonstationary problem, the existence and uniqueness of a solution are established for any t>0. Under certain assumptions, the nonstationary problem defines a dynamical system in some compact set. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 56–70.     

On a two-point boundary-value problem in geophysics

We present some results on the existence and uniqueness of solutions to a two-point boundary-value problem that models gyre flows in rotating spherical coordinates.     

On a boundary-value problem solvability condition

We prove a sufficient condition for the solvability of a two-point problem.Translated from Matematicheskie Zametki, Vol. 13, No. 2, pp. 247–250, February, 1973.     

An application of a multipoint differential-difference scheme to a boundary-value problem

For the boundary-value problem (2) we construct a scheme of the method of lines with a central-difference approximation of the derivative for any odd pattern. In particular cases we investigate the behavior at the net refinement of the direct solution of the boundary-value problem for the determination of the difference between the approximate solution obtained by the method of lines and the exact solution of the problem (1), (2). We also consider some modifications of the method of lines: the number of the lines of the net is taken to be equal to that of the pattern. We give an estimate for the norm of the difference between the approximate solution obtained by this method and the exact solution of the problem (1), (2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Instituta im. V. A. Steklova AN SSSR, Vol. 70, pp. 76–88, 1977.     

On a smooth solution of a boundary-value problem

We study a periodic boundary-value problem for the quasilinear equation u _tt −u _xx =F[u, u _t , u _x ], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998.     

The solvability of a boundary-value periodic problem

In the space of functions B _a³⁺={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T₃/2)=g(x, −t)}, we establish that if the condition aT ₃ (2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u _u −a ² u _xx =g(x, t), u(0, t)=u(π, t)=0, u(x, t+T ₃ )=u(x, t), ℝ², is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997 Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997     

Representation of the solution to a boundary-value problem as a continuum integral

A Feynman integral representation is obtained for the solution of the Dirichlet and Neumann problems for a second-order parabolic equation in a Riemannian space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 6–11, 1974.In conclusion, the author wishes to thank V. S. Buslaev for his attention to this work.     

Existence of three solutions to a second-order nonhomogeneous multipoint boundary-value problem

This paper is concerned with a nonhomogeneous multipoint boundary-value problem (BVP) of a second-order differential equation with one-dimensional p-Laplacian. Using multiple fixed-point theorems, new sufficient conditions to guarantee the existence of at least three solutions of this BVP are established. An example is presented to illustrate the main results. The first emphasis of this paper is to show that the approach to get three positive solutions of a BVP by using multiple fixed-point theorems can be extended to treat nonhomogeneous BVPs. The second emphasis is put on the nonlinear term f involved with the first-order delta operator.     

Collocation method for a multipoint boundary-value problem

Ukrainian Mathematical Journal -     

Multiple solutions of a boundary-value problem in enzyme kinetics

In some immobilized enzyme systems the steady state of substrate concentration may suddenly change from a low profile to a high profile or vice versa when the physical parameters of the systems pass through certain critical values. This phenomenon is due to the transition from a unique solution to multiple solutions (or vice versa) of the enzyme reaction equation. This problem is studied by considering two physical parameters which represent the internal reaction mechanism and the external influence on the boundary of the reaction-diffusion medium. Both analytical and numerical results for the problem are presented. The analytical results include some sufficient conditions for the existence of multiple steady-state solutions as well as a unique solution. Various numerical results of the problem including time-dependent solutions and their convergence to steady-state solutions are given.     

Asymptotic properties of a solution of a boundary-value problem

The asymptotic behavior of the solution of a boundary-value problem for the equation u_txx+ u_x =f when the time tends to infinity is investigated. It is proved that the time mean of the solution tends to a stationary solution everywhere except in a boundary region at the left end of the interval.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 273–284, September, 1970.     

Application of the boundary-value technique to singular perturbation problems at high Reynolds numbers

The boundary-value technique, advanced by Roberts for the solution of singular pertubation problems of ordinary differential equations where the small parameter multiples the highest derivative, is extended to the solution of the Navier-Stokes equation at high Reynolds numbers. Three standard flows—uniform flow past a plate, flow with a linearly adverse external velocity, and shear flow past a flat plate—have been chosen as test problems with a view to evaluating some of the features of the boundary-value technique, particularly in comparison with coefficient matching techniques as examplified by the method of matcher asymptotic expansions.     

Numerical treatment of a parabolic boundary-value control problem

The purpose of this paper is to present a method to compute optimal controls for a class of one-dimensional heat-diffusion processes. The approach used is in the spirit of the Ritz method and approximates the given problem with simpler tasks which are solved by means of algorithms based on the principles of semi-infinite programming. General convergence properties of the procedures are shown. Some illustrative numerical examples are also given.This research was supported by NSF Grant No. GK-31833 and by The Swedish Institute of Applied Mathematics, Stockholm, Sweden.     

Green function of a parabolic boundary-value problem and the optimization problem

We establish necessary and sufficient conditions for the choice of optimal control over systems de-scribed by a general parabolic problem with restricted internal control.