On the differential factorization method in inhomogeneous problems

Variations of block-element boundaries

“…Let us construct the block element in this region for the inner boundary value problem for the Helmholtz differential equation in the form of (2) It is known that the pseudo differential equations of the block element make it possible to consider all pos sible variants of the boundary conditions for the partial differential equation [4]. Precisely, these are the Dirichlet, Neumann, or mixed boundary conditions that are considered the same as in a previous study [1].…”

“…The block ele ments have a number of advantages, whose possibili ties can considerably contribute to the investigation of solutions of the differential equations [3]. It is neces sary to note that the block element method and the differential and integral factorization methods [1,4,5] do not repeat the approach based on the automor phism and related to the Lie group analysis [6][7][8]. The purpose of the first ones is the construction of the solu tions of boundary value problems for large sets of par tial differential equations, while the last one is aimed at construction of solutions for linear and nonlinear differential equations disregarding the boundary con ditions.…”

The oscillations of layered elastic media with a wavy boundary

“…The problem of finding the classical component of the solution is described in detail in previous papers. [3][4][5][6][7][8][9] We recall that, unlike a finite element, a block element, more correctly, albeit in a more complex way, describes the local properties of the solutions of boundary value problems. 3 Using block elements and combining them, one can form complex structures or, conversely, complex structures can be separated into fragments in the form of block elements.…”

“…When using block elements, accompanying expressions, obtained using external forms, are derived, without which a block element is not completely described: functional equations, pseudodifferential equations and formulae for representing the solutions in integral form. [3][4][5][6][7][8][9] Using the above, we will consider the case when we have a boundary value problem for an elastic unbounded layer of thickness h with a wavy upper boundary. We will describe the waviness by an elastic rectangular parallelepiped lying on the upper boundary of the layer and rigidly connected to it.…”

“…Differential and integral methods of factorization, which have a topological basis, [7][8][9] are the foundation of the theory of block structures and a block element. Their application in boundary value problems of the mechanics of a deformable solid in static and dynamic cases 10 involves the factorization of matrix functions, which complicates the discussion.…”