On the differential factorization method in inhomogeneous problems

Abstract

In this paper, we develop a method that applies to inhomogeneous and nonstationary boundary value problems without preliminary variations in the boundary conditions. The method is described as applied to deformable materials and complex block structures that may be under the action of physical fields [3]. Below, we prove that boundary value problems for inhomogeneous differential equations in block structures can be solved by the differential factorization method. Due to this, new opportunities open up for applying this method to the study and solution of boundary value problems for systems of differential equations with variable coefficients and to nonlinear cases with the use of the well-known Newton‐Kantorovich method [4]. 1. We state a boundary value problem for a block structure assuming that the underlying systems of differential equations are inhomogeneous. Suppose that the block structure domain Ω consists of subdomains Ω b ( b = 1, 2, …, B ) with boundaries ∂Ω b . It may happen that part of the block’s boundary is shared with another block, in which case this part is called a contact boundary. The remaining noncontact boundary can be free or subject to external forces. It is assumed that a boundary value problem for systems of partial differential equations with their own constant coefficients is set in each domain Ω b . For each block, the boundary value problem for the system of P partial differential equations in the threedimensional block domain Ω can be written as