Solution of Boundary Problems with Regard for Inherent Error on the Basis of the Lagrange Method

Abstract

A two-sided method for calculation of the dynamic distributed-parameter systems with regard for the error of the reference data was proposed and substantiated using the variational principle of ‘Lagrange. Calculation of the temperature fields with due regard for the phase transformations and errors in the parameters and characteristics was considered by way of an example. A relevant finite-difference method and computer programs for modeling the thermal physical processes of substance melting and crystallization in the case of imprecisely defined parameters and characteristics were developed. The direct Stefan problem was solved using a variant of the through “enthalpy” method. The conjugate problem was solved by smoothing the lumped heat capacity and other parameters and characteristics with a singularity of the delta-function type. Determination of the maximum/minimum of the temperature field, as well as the two-sided estimate of the solution gradient at the given point of the domain were considered as examples. In both cases, the specific power of the spatial data whose values lie within a certain band was regarded as given approximately. The present paper also cites examples of solving the similar stationary boundary problems. The results of work can be used in the practice of research and design in the areas of metallurgy, electrical apparatuses, criogenic engineering, and so on.