Geophysical Journal | 2002 volume 24 4

A theory of spline-iteration methods of computer physics. Part 1. Methods of solving linear integral equations

© V.N. Starkov

The work is devoted to the development of the theory of spline-iteration methods of computational physics, more exactly: new spline-iteration methods of the solution of linear and nonlinear integral equations. A short comparative characteristic of spline-functions and algebraic polynomials is given along with spline-quadrature and cubature formulas, which have been derived using approximate parabolic and cubic splines with the estimation of errors of these formulas for a series of smooth functions classes. The first part of the work includes the grounds for spline-iteration methods of the solution of second kind linear equations. An analysis of the existent methods of employing of spline-functions for an approximate solution of Fredholm equations of the second kind. A conclusion about the efficiency of the use of approximate parabolic and cubic splines in algorithms of the solving of second kind linear integral equation is drawn. It is proposed and founded a modification of the classical method of simple iterations for the solution of Fredholm equations of the second kind based on the use of approximate parabolic and cubic splines. A spline-quadrature algorithm of the method is studied in detail and a priori and a posteriori estimates of errors of the solution are analyzed. A new method of determining the total error of solving the Fredholm equation of the second kind by the spline-iteration method is proposed. A new version of approximate solution of the Volterra equation of the second kind is suggested which is based on two significant points. Firstly, each of the sections of the original grid knots is covered with an additional grid needed for the use of spline-quadrature formulas. Secondly, a new notion of the boundary conditions is introduced for approximate splines. Estimations of errors of the approximate solution of the Volterra spline-iteration method are obtained.

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