A New Quintic Spline Method for Integro Interpolation and Its Error Analysis

Abstract

Abstract:In this paper, to overcome the innate drawbacks of some old methods, we present a new quintic spline method for integro interpolation. The method is free of any exact end conditions, and it can reconstruct a function and its first order to fifth order derivatives with high accuracy by only using the given integral values of the original function. The approximation properties of the obtained integro quintic spline are well studied and examined. The theoretical analysis and the numerical tests show that the new method is very effective for integro interpolation.

On a New Kind of Quartic Integro-Spline Over a Uniform Partition

“…To construct an integro-spline, we generally need n integral values I j ( j = 0, 1, • • •, n − 1) and several boundary conditions. In recent years, there are many studies about integro-spline interpolation ( [1,2,3,4,5,6,7,8,9,10,11,12]).…”

Some new super convergence of a quartic integro-spline at the mid-knots of a uniform partition

“…In [8], it was proved that the super convergence (7) at the knots still hold even if the exact boundary function values y(x 0 ), y(x 1 ), y(x n−1 ), y(x n ) in (3), (4), ( 5) and ( 6) are replaced respectively by the approximate boundary function values y(x 0 ), y(x 1 ), y(x n−1 ), y(x n ) given in ( 8), ( 9), (10) and (11).…”

“…Approximating y = y(x) and its derivatives by using the integral values (1) is called integroapproximation. Splines have been widely used for this problem, see the works of Behforooz [1,2], Zhanlav [3][4][5], Mijiddorj [6,7], Lang [8][9][10], Xu [11,12], Haghighi [13,14], and Wu [15][16][17]. Generally, the obtained integro-splines have good approximation abilities.…”

“…Later, the super convergence of some other integro-splines at the knots of a uniform partition has also been studied. The super convergence of sextic integro-spline in approximating y (k) (x j ) (k = 0, 2, 4) was presented in [15] and the super convergence of quintic integro-spline in approximating y (k) (x j ) (k = 1, 3) was given in [3,9,10]. Do some integro-splines have super convergence properties at some other points?…”