Numerical Solution of Fredholm Equations With Double Precision by the Integral Kernel Degeneracy Method

Abstract

The work for the first time proposed a modified method of degeneration of the integral nucleus to solve the integrated equations of Fredgolm of the second kind. The idea is to put the integral core in a series of Taylor along one variable x, and not in two variables x, s as in the classical method. The decomposition of the nucleus in the row is carried out at the middle point of the integration segment, which reduces the modules of the elements of the matrix C, as well as the area of disaster for the I – λC matrix. The system of degree basic functions is used at the integration segment. Three theorems are offered for sufficient conditions for the correctness of the proposed algorithm by the degeneration of the integral nucleus. The definition of the factorial norm of Chebyshev Vector-functions has been introduced. The factual norm for the system of private derivatives of the integral nucleus on the variable x and the parameter λ are included in the inequality of the third theorem – a sufficient condition for the correctness of the algorithm. The numerical algorithm proposed in the work was tested on three integral equations of Fredgolm with nuclei with exponential growth or with a periodic change in the sign of the nucleus. Numerical solutions coincide with accurate solutions in 15 significant signs in a uniform metric.