Numerical solutions of a two-phase membrane problem

In this paper different numerical methods for a two-phase free boundary problem are discussed. In the first method a novel iterative scheme for the two-phase membrane is considered. We study the regularization method and give an a posteriori error estimate which is needed for the implementation of the regularization method. Moreover, an efficient algorithm based on the finite element method is presented. It is shown that the sequence constructed by the algorithm is monotone and converges to the solution of the given free boundary problem. These methods can be applied for the one-phase obstacle problem as well.     

Numerical solutions of the Dirichlet problem via a density theorem

Under mild conditions a certain subspace M, consisting of functions which are analytic in a simply connected domain Ω and continuous on the boundary Gamma;, is shown to have real parts which are dense, in the sup norm, in the set of all solutions to the Dirichlet problem for continuous boundary data. Similar results hold for L^p boundary data. Numerical solutions of sample Dirichlet problems are computed. © 1994 John Wiley & Sons, Inc.     

Numerical estimates for stability domains of Lagrangian solutions to the restricted three-body problem

The geometric parameters of stability domains of Lagrangian solutions to the classical restricted three-body problem are quantitatively estimated. It is shown that these domains are ellipsoid-like plane figures stretched along the tangent to the circle that passes through the Lagrangian triangle solutions. A heuristic algorithm is proposed for determining the maximum size of these domains of attraction.     

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 57–62, 1990.     

Numerical studies of the stability of steady-state solutions to a contact problem in coupled thermoelasticity

A finite element approximation is used to study the stability of steady-state solutions and the erratic behavior that is present in a problem of heat conduction through an elastic rod that may come into contact with a rigid wall. The quasi-static fully coupled theory of linear thermoelasticity is assumed and a heat exchange coefficient that depends on the pressure and the gap size is imposed across the region of contact.     

Analytic solutions of the Borel problem

LetF ⊂ ℂ be a dense-in-itself set that has a nonempty connected interior and contains the origin, and let be the space of infinitely differentiable complex-valued functions onF. For some classes of such setsF, we prove that for an arbitrary sequence of complex numbers there exists a functionf ε withf ⁽ⁿ⁾(0)=d _n,n=0, 1, 2, ..., and study the analyticity properties off. The functionf is constructed in the form of various function series, namely, a power series, a series of simple fractions, and an exponential series. Analytic solutions of the multidimensional Borel problem are also considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 525–538, April, 2000.     

Rational solutions of the graphsack problem

A graphsack problem is a certain binary linear optimization problem with applications in optimal network design. From there a rational graphsack problem is derived by allowing the variables to vary continuously between 0 and 1. In this paper we deal with rational graphsack problems. First we develop the concept of compressed solutions and the concept of augmenting cuts. Making use of these concepts a very simple optimality criterion is derived. Finally an efficient algorithm solving rational graphsack problems is given which is polynomially bounded in time and which is closely related to the simplex algorithm.     

Numerical extremal solutions for a mixed problem with singular {\phi} -Laplacian

We are concerned with extremal solutions for the mixed boundary value problem $$-\left(r^{N-1}\phi(u')\right)' = r^{N-1} g(r, u), \quad u'(0) = 0 = u(R),$$ where ${g : [0, R] \times \mathbb{R} \to \mathbb{R}}$ is a continuous function and ${\phi : (-\eta, \eta) \to \mathbb{R}}$ is an increasing homeomorphism with ${\phi(0) = 0.}$ We prove the existence of minimal and maximal solutions in presence of well-ordered lower and upper solutions and develop a numerical algorithm for theirs approximation. Also, we provide numerical experiments confirming the theoretical aspects.     

Steady-state solutions of the aerotaxis problem

We study the steady-state system of aerotaxis equations in higher dimensions. It is shown that the existence and multiplicity of solutions depend on the total mass of the colony of bacteria, the energy function, and the boundary conditions.     

Numerical solutions of the poisson equation

In this article we shall give practical and numerical solutions of the Poisson equation on multidimensional spaces and show their numerical experiments by using computers.     

Numerical solutions of the thermistor equations

This paper examines heat conduction in a thermistor used as a current surge regulator. The problem consists of coupled nonlinear, nonlocal parabolic initial boundary value problems. Simplifying assumptions are made which lead to two different problems each of which consists of a one (space) dimensional nonlocal parabolic initial boundary value problem.Numerical methods for the approximate solutions of both the steady state and the transient problems are described and the results of the numerical experiments are presented.     

Wavelet solutions for the Dirichlet problem

Summary. A modified classical penalty method for solving a Dirichlet boundary value problem is presented. This new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains. It is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast, approximate calculation of boundary integral. The compact support and orthonormality of the basis are essential for representing the boundary measure numerically, and therefore for implementing this methodology. Received June 3, 1992 / Revised version received November 8, 1993     

On solutions to the Goldshtik problem

The Goldshtik model for separated flows in an incompressible fluid is considered. A solution to this two-dimensional problem of mathematical physics for a finite domain is found with a finite element method. Estimates of the differential operator are obtained. A result on the number of solutions to the Goldshtik problem is obtained using a variational method.     

Compactness of solutions to the Yamabe problem

We establish compactness of solutions to the Yamabe problem on any smooth compact connected Riemannian manifold (not conformally diffeomorphic to standard spheres) of dimension n?7 as well as on any manifold of dimension n?8 under some additional hypothesis. To cite this article: Y.Y. Li, L. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Multiplicity of solutions for the bounce problem

The bounce trajectories in a convex set which assume assigned positions in two fixed time instants are sought. Sufficient conditions in order to obtain the existence of infinitely many bounce trajectories are found.