Solution of nonlinear two-point boundary-value problems using spline functions

Journal of Optimization Theory and Applications - This paper describes a technique for computing spline function approximations to the solution of two-point boundary-value problems. A performance...     

Solution of boundary-value problems for elliptic equations in the space of distributions

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order2m. The representation of a distribution from (C ^∞ (S))’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.     

Solution of boundary-value problems for xk-analytical functions in a halfplane with two cuts

We consider two boundary-value problems for x^k-analytical fonctions (k=2m+1, m=1,2,3,...) of the variable z=x+iy in a halfplane with cuts along the intervals (0, 1) and (b, ) of the real line (b > a), when the imaginary or the real part of the x-analytical function is given at the ends of the finite cut and resp. the real or the imaginary part of the function is given at the ends of the semi-infinite cut. Closed-form solutions are obtained for this problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 3–10, 1986     

Variational principles and mathematical methods of solving boundary-value problems for piecewise uniform nonlinear media

We give a variational formulation and a scheme for numerical solution of boundary-value problems for piecewise-uniform nonlinear media taking account of frictional forces. We solve the problem of determining the stressed state of a thick three-layer cylinder of elastoplastic material with linear reinforcing under the action of internal pressure and an axially symmetric temperature field.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 64–67.     

Solution of two-dimensional boundary-value problems of the theory of thin composite shells

Discrete analogues of the boundary-value problems of a two-dimensional refined theory of anisotropic shells taking into account the transverse shear deformation are presented. The systems of resolving equations in the general form are obtained for arbitrary nonshallow shells of variable curvature whose coordinate lines of the reduction surface may not coincide with the lines of principal curvatures. The algebraic problems of determining the stress-strain state in shells made of composite materials with stress concentrators under various kinds of loads are obtained as particular cases of the schemes presented. The results of calculating the stress concentration near a nonsmall circular hole in a transversely isotropic nonshallow spherical shell under internal pressure are presented. The dependences of stress concentration factors on the hole dimension and on a change in the shear stiffness of the shells are studied. A comparison between the calculation results obtained within the framework of the theories of shallow and nonshallow shells is given.Presented at the 11th International Conference on the Mechanics of Composite Materials (Riga, June 11–15, 2000).Timoshenko Institute of Mechanics, Ukranian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 465–472, July–August, 2000.     

Generalized boundary-value problems

If each of n people defines a (suitable) measure on a compact convex cake I, then there exists a division of I into n connected parts, and an assignment of these n parts to the n people, in such a way that the piece of cake assigned to each person is at least as large (in his own measure) as that assigned to anyone else. (The proof uses Brouwer's fixed-point theorem and Hall's theorem on systems of distinct representatives.) Slight progress is made towards finding algorithms for this problem.     

Carleman estimates for stratified media

We consider anisotropic elliptic and parabolic operators in a bounded stratified media in Rn characterized by discontinuities of the coefficients in one direction. The surfaces of discontinuities cross the boundary of the domain. We prove Carleman estimates for these operators with an arbitrary observation region.     

Regularization of boundary-value problems for hyperbolic equations

The problem of the stability of wave propagation in anisotropic inhomogeneous media is considered. The class of approximate solutions possessing the stability property with respect to the small deviations of the input data in the form regularizing the operators R(φ, ψ, x, t, α) is constructed. Here an important role is played by the choice of the smoothing function and by the conditions for matching the regularization parameter with the error.     

Scattering theory for perturbed stratified media

We study the operatorH = -c ² x,y)Μx,y)∇ · Μ ^-1 (x,y)∇, wherec andΜ are perturbations of functionsc ₀(y) andΜ ₀(y) which depend only on the one-dimensional variabley. In particular, we study the spatial asymptotics of lim_ε↺0(H - (λ +iε)²)^-1 applied to functions which have compact support or are otherwise well-behaved at infinity and relate the scattering matrix to the asymptotics of the generalized eigenfunctions. We then prove a trace formula for the operatorH in terms of the scattering phase, and, in a very special situation, use the trace formula to find spectral asymptotics forH. Partially supported by an NSF Postdoctoral Fellowship and the University of Missouri Research Board.     

Degenerate nonlinear boundary-value problems

We establish necessary and sufficient conditions for the existence of solutions of weakly nonlinear degenerate boundary-value problems for systems of ordinary differential equations with a Noetherian operator in the linear part. We propose a convergent iterative procedure for finding solutions and establish the relationship between necessary and sufficient conditions.     

On two-point boundary-value problems

Summary Boundary-value problems are considered for systems of m ordinary linear differential equations of order n. For such a problem L, a canonically associated first-order problem M, with mn equations, is introduced in such a may that associates of adjoint problems are adjoint first-order problems. The solution of a boundary-value problem L is given in terms of a generalized Green's function. This work has been supported by Sandia Corporation, a prime contractor to the U. S. Atomic Energy Commission.     

Overdetermined parabolic boundary-value problems

The concept of an overdetermined, linear, parabolic system of differential equations of arbitrary order is introduced, and a theory of general over-determined, parabolic boundary-value problems in compact and noncompact domains with smooth boundaries is developed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 69, pp. 240–272, 1977.     

Time-space boundary element formulation for boundary-value problems

A time-space boundary element formulation is presented for the boundary-value problems with a governing equation expressed in terms of a certain type of linear operator. The boundary as well as the time domain are divided into a finite series of time-space elements, and the interpolation functions in time and space are introduced in order to construct a final discretized equation for the assembled system. The solving scheme is discussed, and the relation to the practical engineering problems is shown.