An approximation technique for the numerical solution of a Stefan problem

Summary A nonlinear approximation technique for the numerical solution of certain free boundary problems is proposed. The method is shown for a degenerate one-dimensional Stefan problem. For this problem, an error estimate, which is independent of the used algorithm, is derived. Numerical examples are discussed.This paper was written when the author held a 1 1/2 year postdoctoral position at the Department of Mathematics and Applied Mathematics Institute of the University of Delaware     

A numerical approximation to the solution of an inverse heat conduction problem

The aim of this article is to study the parabolic inverse problem of determination of the leading coefficient in the heat equation with an extra condition at the terminal. After introducing a new variable, we reformulate the problem as a nonclassical parabolic equation along with the initial and boundary conditions. The uniqueness and continuous dependence of the solution upon the data are demonstrated, and then finite difference methods, backward Euler and Crank–Nicolson schemes are studied. The results of some numerical examples are presented to demonstrate the efficiency and the rapid convergence of the methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Analysis and numerical approximation of a parabolic-hyperbolic transmission problem

In this paper we investigate a mixed parabolic-hyperbolic initial boundary value problem in two disconnected intervals with Robin-Dirichlet conjugation conditions. A finite difference scheme approximating this problem is proposed and analyzed. An estimate of the convergence rate is obtained.     

A note on the numerical solution of a periodic parabolic problem

A direct method is presented for the numerical determination of solutions of the heat conduction equation having periodT in time. A simple error analysis which improves previous results is given.     

Approximate solution of a control problem on the basis of the nilpotent approximation

We consider the control problem for deterministic systems described by ordinary differential equations with linear controls. On the basis of the nilpotent approximation method, we construct an algorithm for finding an approximate solution of the control problem for three-dimensional nonlinear systems with two linear controls. The algorithm was implemented in Maple and tested in examples including the control of a mobile robot on a plane and the attitude control of a sphere rolling on a plane.     

On the numerical solution of the flow between a rotating and a stationary disk

The flow between two co-axial, infinite disks, one rotating with constant angular velocity and one stationary is treated in this paper. The problem is reduced to that of finding the solution of a two-point boundary value for a sixth order nonlinear ordinary differential equation and three boundary conditions at each of a finite interval. The numerical solutions are obtained by using a fourth order Runge-Kutta integration scheme in modification due to Gill and in conjunction with a modified shooting method to correct the initial guesses at one boundary. The numerical calculations for different Reynolds numbers are carried out. The results obtained by this method are compared with available results. The comparison shows excellent agreement.     

On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.     

A simplicial approximation of a solution for the nonlinear complementarity problem

For a given map f from the n-dimensional Euclidean space En into itself, we consider the complementary problem of finding a nonnegative vector x in En whose imagef(x) is also nonnegative and such that the two vectors are orthogonal. It is the unifying mathematical form for several problems arising in different fields such as mathematical programming, game theory and economics.In this paper a new algorithm is developed based on the adjacent simplex technique , which was used by Garcia, Lemke and Lüthi for approximating an equilibrium point of a noncooperative n-person game. An almost-complementary path leads to a complementary simplex, which approximates a stationary point. Because most of the existence proofs for the nonlinear complementarity use the relationship between stationary points and complementarity, the algorithm gives constructive proofs for many existence theorems. If a better approximation is desired, the algorithm may be restarted from any point. The dimension of the simplices on the path is varying, which computationally should result in some savings.     

Polynomial approximation to the solution of the cauchy problem

The Cauchy problem for a nonhomogeneous first-order differential-operator equation of parabolic type in a Hilbert space is considered. Polynomial approximations and estimates of their convergence are obtained which depend on the character of the right-hand side and the initial conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 427–429, March, 1991.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

Polynomial approximation to the solution of the Cauchy problem

The Cauchy problem for a nonhomogeneous first-order differential-operator equation of parabolic type in a Hilbert space is considered. Polynomial approximations and estimates of their convergence are obtained which depend on the character of the right-hand side and the initial conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 427–429, March, 1991.     

Structure analysis of solution to equations of quasi 3-D accretion disk model

In this paper we discuss the problems contained in the solution to the equations of quasi 3-D accretion disk model, and point out that the angular momentum equation should not be integrated directly. Finally, we develop a criterion of the existence of a disconnected solution to this model.     

The numerical solution of the inverse Stefan problem

Summary The inverse Stefan problem can be understood as a problem of nonlinear approximation theory which we solved numerically by a generalized Gauss-Newton method introduced by Osborne and Watson [19]. Under some assumptions on the parameter space we prove its quadratic convergence and demonstrate its high efficiency by three numerical examples.     

Best approximation and numerical methods in solution of differential equations

Let Q(x) bee polynomial of degree q interpolating x^m at the points $\text{x}{}_{\mathrm{<mtext>i</mtext>}}\text{,}\text{i}\text{= 0, 1, /3.,}\text{q}$, where $\text{x}{}_{\mathrm{<mtext>i</mtext>}}$ are zeros of the Tchebysheff polynomial of degree q + 1 on the interval [0, 1]. If q is of order √m, then Q(x) approximates x^m well enough. This result is used to obtain a good approximation to the solution of a system of linear differential equations.     

Existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem

In this paper two theorems are proved on the existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem, from which the solvability of this problem in the class of continuous functions follows.Translated from Matematicheskie Zametki, Vol. 10, No. 4, pp. 447–452, October, 1971.     

On the numerical solution of a quasilinear algebraic-differential system

We consider a quasilinear algebraic-differential system, suggest a spline collocation method for its solution, and prove a convergence theorem for this method. Results of numerical experiments are given.     

On the numerical solution of the linear complementarity problem

The well-known linear complementarity problem with definite matrices is considered. It is proposed to solve it using a global optimization algorithm in which one of the basic stages is a special local search. The proposed global search algorithm is tested using a variety of randomly generated problems; a detailed analysis of the computational experiment is given.