Algebraic grids and their application to the numerical solution of linear integral equations — II

Abstract

The paper is a new edition of the authors’ previous work on this topic. A significant improvement in the results of the previous article is associated with the use of weight functions for the transition from an integral of lower to higher dimension.
Such a transition turned out to be necessary to obtain new estimates of the error of the approximate solution of the Fredholm integral equation of the second kind by the iteration method using algebraic grids.
The essence of this approach is that in the approximate calculation of the solution of the Fredholm integral equation of the second kind, a partial sum of the Neumann series consisting of integrals of different multiplicities is used. When using different algebraic grids corresponding to different purely real fields and one stretching parameter, it turns out that for a lower dimension, a smaller number of nodes of the algebraic grid will be used, and therefore the accuracy of the calculation will be lower. In order not to solve the complex problem of optimizing the number of nodes for different dimensions, this paper proposes an approach in which all integrals are reduced to one and a single algebraic grid is used for it. The second positive effect of this
approach is related to the minimization of the calculation of the values of the kernel of the Fredholm equation of the second kind due to the use of Horner’s scheme.
The paper considers two methods for choosing a purely real algebraic field. The first method is based on specifying an irreducible polynomial with integer coefficients, all of whose roots are real numbers. The second method is based on using a tower of quadratic fields.

With both methods of choosing a purely real algebraic field, we were able to use a large dimensional algebraic grid to integrate a function of a smaller number of variables. An important role in this was played by the weight function, which allows replacing the integral of a function
from the class 𝐸𝛼 𝑠 over the cube 𝐺𝑠 with the integral of a function from the class 𝐸𝛼,0 𝑠 [−1, 1] over the cube 𝐾𝑠. It is important to note that the new function goes to zero on the boundary of this cube.