Operator's Approach to the Problems with Concentrated Factors
by
Boško S. Jovanović
University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Belgrade, Yugoslavia
Coauthors: Lubin G. Vulkov (University of Rousse, Department of Applied Mathematics and Informatics, Studentska str. 8, 7017 Rousse, Bulgaria)

The finite-difference method is a basic tool for solution of partial differential equations. For the problems with discontinuous coefficients and concentrated factors (Dirac-delta functions, free boundaries, etc.) because of the low global regularity of the solution it is impossible to establish convergence of the finite difference schemes using the classical Taylor's expansion. Often, the Bramble-Hilbert lemma takes the role of the Taylor's formula for the functions from the Sobolev spaces.

One interesting class of parabolic problems models processes in heat-conduction media with concentrated capacity for which in the heat capacity coefficient is involved Dirac-delta function. In this case the jump of the heat flow in the singular point is proportional to the time derivative of the temperature. Dynamical boundary conditions cause similar effect. The current problems are non-standard and the classical analysis is difficult to apply for convergence analysis.

In the present paper finite-difference schemes approximating the one-dimensional boundary value problems for the heat equation with concentrated capacity or dynamical boundary conditions are derived. An abstract operator's method is developed for studying such problem. Special Sobolev's norms with weight operator are constructed. In these norms convergence rate estimates compatible with the smoothness of the boundary value problem data are obtained.

Date received: February 16, 2000