On solving the second basic three-element boundary-value Carleman-type problem in classes of bianalytic functions in a disk

In this paper, we investigate one of the basic Carleman-type boundary-value problems for bianalytic functions. We obtain a constructive method for solving the problem in the case of circular domain. We establish that solving the problem can be reduced to a sequential solving two generalized Carleman-type problems for analytic functions in a disk. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 413–426, July–September, 2006.     

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 boundary-value problem for hyperbolic-type equation of the third order

We formulate some boundary-value problems for a linear third-order equation with hyperbolic operator in the main part and study the unique solvability. Under certain conditions to given functions, using the Riemann method, we obtain an integral representation of solutions. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 591–603, October–December, 2007.     

The impedance boundary-value problem of diffraction by a strip

We consider the impedance boundary-value problem for the Helmholtz equation originated by the problem of wave diffraction by an infinite strip with imperfect conductivity. The two possible different situations of real and complex wave numbers are considered. Bessel potential spaces are used to deal with the problem, and the identification of corresponding operators of single and double layer potentials allow a reformulation of the problem into a system of integral equations. The well-posedness of the problem is obtained for a set of impedance parameters (and wave numbers), after the incorporation of some compatibility conditions on the data. At the end, an improvement of the regularity of the solution is derived for the same set of parameters previously considered.     

Integrable boundary-value problem for the Volterra chain on the half-axis

We study quasiperiodic (finite-gap) solutions of the Volterra chain satisfying an integrable boundary condition on the semiaxis. From the set of general finite-gap solutions, only those corresponding to the boundary-value problem are singled out, the relevant condition being expressed as a system of algebraic equations.     

Singularly perturbed Dirichlet boundary-value problem for a stationary system in the linear elasticity theory

We consider a singularly perturbed Dirichlet boundary-value problem for an elliptic operator of the linear elasticity theory in a bounded domain with a small cavity. The main result is the proof of the theorem about the convergence of eigenelements of the perturbed boundary-value problem to eigenelements of the corresponding limiting boundary-value problem, when the parameter ? which defines the diameter of the small cavity tends to zero.     

A nonlocal boundary-value problem for the gellerstedt equation

We consider the boundary-value problem for the Gellerstedt equation

On a nonlinear boundary value problem modeling corneal shape

In this paper we present some results concerning a boundary value problem for a nonlinear ordinary differential equation that was used before in modeling the topography of human cornea. These results generalize previously obtained theorems on existence and uniqueness. We show that our equation has a unique solution for all parameters and conditions that can arise in physical situation. In the second part of the article we derive some new estimates and approximate solutions. Numerical calculations verify that these approximations are very accurate.     

On the first stationary boundary-value problem of elasticity in weighted Sobolev spaces in exterior domains of R3

This paper solves the first boundary-value problem of elasticity both in interior and exterior domains ofR ³. The equations are set in weighted Sobolev spaces for exterior domains that describe the decay of the functions at infinity. The results established include existence, regularity, and convergence of iterations of the solution.This research was supported by the Rashi Foundation.     

On Hilbert-type boundary-value problem of poly-Hardy class on the unit disc

In this article, the poly-Hardy class on the unit disc is introduced and the boundary behaviour of the function in this class is discussed. Then the method used in Wang (Y.F. Wang, On modified Hilbert boundary-value problems of polyanalytic functions, Math. Methods Appl. Sci. 32 (2009), pp. 1415–1427) is applied to Hilbert-type boundary-value problems for the poly-Hardy class on the unit disc, and the expression of solution and the condition of solvability are explicitly obtained.     

Solvability of the boundary-value problem for a variable-order differential equation on a geometric graph

The solvability of the boundary-value problem for a string-beam model is studied. The model is described by an equation of orders 2 or 4 on dirrerent edges of an arbitrary graph. Criteria for the problem to be degenerate and nondegenerate are obtained; in particular, it is proved that the nondegeneracy of the problem is equivalent to the maximum principle.     

On solution strategies to Saint-Venant problem

Different solution strategies to the relaxed Saint-Venant problem are presented and comparatively discussed from a mechanical and computational point of view. Three approaches are considered; namely, the displacement approach, the mixed approach, and the modified potential stress approach. The different solution strategies lead to the formulation of two-dimensional Neumann and Dirichlet boundary-value problems. Several solution strategies are discussed in general, namely, the series approach, the reformulation of the boundary-value problems for the Laplace's equations as integral boundary equations, and the finite-element approach. In particular, the signatures of the finite-element weak solutions—the computational costs, the convergence, the accuracy—are discussed considering elastic cylinders whose cross sections are represented by piece-wise smooth domains.     

On boundary-value problems for second order perturbed differential inclusions

This article studies the existence of solutions to boundary-value problems for second order multi-valued perturbed differential inclusions under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions and the weaker nonconvexity conditions for multi-valued functions.     

Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

In this paper, a new difference scheme based on quartic splines is derived for solving linear and nonlinear second-order ordinary differential equations subject to Neumann-type boundary conditions. The scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary, which is superior to the well-known Numerov’s scheme with the accuracy being fourth order. Convergence analysis of the present method for linear cases is discussed. Finally, numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method.     

On the eigenvalues of second-order boundary-value problems

In this paper we investigate the properties of eigenvalues of some boundary-value problems generated by second-order Sturm-Liouville equation with distributional potentials and suitable boundary conditions. Moreover, we share a necessary condition for the problem to have an infinitely many eigenvalues. Finally, we introduce some ordinary and Frechet derivatives of the eigenvalues with respect to some elements of the data.     

Numerical solution of a time-optimal parabolic boundary-value control problem

A special time-optimal parabolic boundary-value control problem describing a one-dimensional heat-diffusion process is solved numerically. Using a bang-bang principle recently proved by Lempio, this problem can be transformed in such a way that the variables are jumps of bang-bang controls. A discretization is performed in two steps, and the convergence of the approximate solutions is proved. Finally, an algorithm to solve the discrete problem is developed and some numerical results are discussed.The author would like to thank Prof. F. Lempio, who pointed out this problem to him, and Prof. K. Glashoff for many helpful comments and suggestions.     

Global bifurcation of a class of periodic boundary-value problems

We prove that λ=0 is a global bifurcation point of the second-order periodic boundary-value problem (p(t)x^′(t))^′−λx(t)−λ²x^′(t)−f(t,x(t),x^′(t),x^″(t));x(0)=x(1),x^′(0)=x^′(1). We study this equation under hypotheses for which it may be solved explicitly for x^″(t). However, it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation. Thus, we need to study the implicit equation with regard to global bifurcation.     

On the second boundary-value problem of magnetization by the Landau-Lifschitz equation

Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 34, No. 1, pp. 86–93, January–March, 1994. Original article submitted.