Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I

Abstract

In recent times, in connection with the development of the theory of generalized functions and the theory of generalized solutions of boundary value problems for differential equations, the approach to the construction of difference operators, which approximate the differential problem, has essentially changed. First of all we have in mind the following: 1. (1) the rejection, because of results obtained, of the attempt to approximate (in more or less explicit form) a differential equation, with continuity conditions and boundary conditions, which must be done locally and independently of each other, by systems of finite difference equations; 2. (2) the development of numerical methods for a group of non-stationary problems, in which the form of the difference schemes is determined, not by the requirements of the best local approximation, but by conditions of stability and simplicity of calculation; 3. (3) the construction and basis of difference operators which possess the integral properties of differential operators of boundary value problems. We shall dwell only on the last group of methods. Difference equations for whose investigation integral characteristics were used, have been considered, for example, in works [1]–[13]. For the study of the properties of difference operators, which keep the integral properties of the differential problems, we can successfully use functional methods of investigation. It is well known with what success functional methods are applied to the study of problems of mathematical physics, in particular the method of orthogonal projections, which form a considerable part of the theory of generalized solutions (see [14]–[17]). A significant advance in the theory of difference equations has been marked by the fact that functional methods of directly investigating their inner properties (see [18]–[20]) have been applied to equations on a network. In the present work we put forward and systematically apply a method of orthogonal projections for network vector-functions. It permits us 1. (1)to investigate the structural properties of subspaces of a space of network functions, 2. (2) to compose difference equations for some problems of mathematical physics, 3. (3) to solve the question of difference analogues of boundary conditions by using the fact that the solution is a member of a definite subspace of functions, 4. (4) to investigate the properties of the solution of difference equations, 5. (5) to give optimal evaluations for the solutions of difference equations by means of projectional operators on the right-hand sides of these equations. The difference equations obtained for boundary value problems of types I, II and III, for self-conjugate elliptic equations of the second order, possess a regular closure of the computational algorithm; with symmetrization of these systems of equations they are converted into known systems of equations of a divergent type. We have called the difference equations obtained “difference analogues” of the problems considered, wishing thereby to stress the fact that in conclusions obtained from them we did not make use of the idea of local approximation of the differential operator by a difference operator, but obtained these equations by using functional (integral) methods. Local approximation was used only in replacing the first derivative of a function by a centrally divided difference. With the selected approach to the construction of difference operators we shall consider as a natural form of approximation an approximation, in the weak sense, which characterizes the nearness of two functionals, with which we may meet not only the case of local smallness of the discrepancy in the approximation of the differential operator by a difference, but also the case of its rapid oscillation. Both cases are equally well represented in the chosen method of evaluation of the errors of solutions. For clearness and simplicity of exposition we have limited ourselves to the case of three-dimensional space and have considered problems involving the Laplace operator.