Construction of local integro quintic splines

Abstract

In this paper, we show that the integro quintic splines can locally be constructed without solving any systems of equations. The new construction does not require any additional end conditions. By virtue of these advantages the proposed algorithm is easy to implement and effective. At the same time, the local integro quintic splines possess as good approximation properties as the integro quintic splines. In this paper, we have proved that our local integro quintic spline has superconvergence properties at the knots for the first and third derivatives. The orders of convergence at the knots are six (not five) for the first derivative and four (not three) for the third derivative.

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

“…In other words, A (27) is invertible, and the theorem is proven. determined by (17)- (19). It can be constructed as follows:…”

“…In the next section, (7)- (9), (11) and (15) will be used as the artificial end conditions for integro interpolation; see (18) and (19).…”

“…In this paper, we will mainly focus on the quintic spline methods; see [2,14,[17][18][19] for the existing ones.…”

“…The problem arises in many fields, such as numerical analysis, mathematical statistics, environmental science, mechanics, electricity, climatology, oceanography, and so on. We refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] for its applied backgrounds and some recent developments.…”

“…On the one hand, it needed n + 5 artificial values, which brought higher computational cost; on the other hand, the obtained quintic spline did not agree with the given integral values (2) over the subintervals. In [19], a local integro quintic spline method was given. It was also not dependent on exact end conditions and was able to produce good approximations.…”