The Petrov-Galerkin method generalization for solving a second kind Fredholm integral equations system

The numerical solution problem of a second kind Fredholm m integral equations linear system is considered. The Petrov-Galerkin projection method generalization for solving this problem is proposed the first time. n is a coordinate functions number of two linearly independent systems. It can be equal, more or less then the m - the integral equations number. This algorithm advantage is that it is not sensitive to the parameters λ smallness in the integral equations system. The algorithm requires the correct choice of two linearly independent coordinate functions systems and their number. The solution algorithm is reduced to a matrix solution for an antidiagonal problem. Two examples are solved for the antidiagonal problem, and numerical solutions of the problem coincide with the exact solutions. Two theorems are proved for sufficient conditions for the proposed numerical algorithms correctness in two cases. In the first case, an antidiagonal problem is considered with two Fredholm equations of the second kind. The second theorem considers well-posedness conditions for a general diagonal problem. Undoubtedly, the proposed algorithm will be useful in mechanics and computational mathematics problems.