Solution of the Fredholm Integral Equations Method of Replacing the Integral by a Quadrature with the Twelveth Order of Error in Matrix Form

Abstract

An algorithm for the numerical solution of the Fredholm equation of the second kind with a continuous kernel by the method of integral replacement and the matrix solution of SLAE with a quadrature formula of the twelfth order of error with a number of integration intervals divisible by ten is proposed. The new formula, compared to Simpson's formula, gives 15 significant digits for the nodal values of the solution function, even with a small number of intervals of 10.20 on a segment in a finite number of elementary operations. The resulting algorithm has double precision and minimal computation time. While Simpson's formula,together with the matrix method for solving SLAE, gives only 6 significant digits with twenty integration intervals. Moreover, double precision is not available for Simpson's formula (15 zeros in the infinite norm of the solution residual), since the FORTRAN language allows maximum matrix arrays of 200×200. Estimates are obtained for the upper bound of the admissible parameter |λ| for the matrix of the Fredholm equation with strict diagonal dominance or with a small norm of the integral kernel.