Features of the block-element method in nonstationary problems

Abstract

The blockelement method developed in (1-4) is used for solving unsteady boundaryvalue problems, the timedependent coefficients of differential equa� tions. Such boundaryvalue problems are inherent for materials with timevariable properties, unsteady pro� cesses arising in the description of timevariable phys� ical fields, and various engineering applications when investigating processes proceeding in time. Examples of the important application of unsteady problems for engineering practice are given, for example, in the remarkable monograph (5). The block element is constructed by the differential factorization method (6, 7) in the fourdimensional space of three geometrical coordinates and time. It is shown that the traditional formulation of the initial boundaryvalue problems for the partial differential equations accompanied with the formulation of the boundary and initial conditions in the case of using the blockelement method undergoes a modification: the concept of initial conditions is lost, and they pass into the category of boundary conditions. When using the algorithms of investigating the boundaryvalue problems by the blockelement method, the possibility of their application to all types of equations—elliptic, parabolic, and hyperbolic—is revealed. Technical difficulties arise in the investiga� tion of hyperbolic equations for those block elements, through the carrier of which the wave front passes, and it is necessary to satisfy the causality principle (8).