--ZusammenfassungOn Quartic Splines with Application to Quadratures. This paper presents a formulation and a study of an interpolatory quartic spline which interpolates the first and second derivatives of a given function. This formulation can be applied, in particular, to quadratures.
AMS Subject Classifications
A new procedure based on sixth degree (Hexic) C 2 -Spline for the numerical integration of the second order initial value problems (IVPs) y 00 ¼ f ðx; yÞ, including those possessing oscillatory solutions, is presented. The proposed method is essentially an implicit sixth order one-step method. Stability analysis shows that the method possesses ð0; 75:3Þ S ð130:2; 201:9Þ as interval of periodicity and=or absolute stability. In addition, the method has phase-lag (dispersion) of order six with actual phase-lag H 6 =774144. Convergence results yield error bounds k s ðrÞ À y ðrÞ k ¼ O h 6 À Á ; r ¼ 0; 1, in the uniform norm, provided y 2 C 8 ½0; b. Furthermore, it turns out that the method is a continuous extension of a sixth order four-stage Runge-Kutta (-Nyström) method. Numerical experiments will also be considered.
This paper is concerned with a construction of a "variable" one-parameter cubic ~' -s~l i n e collocation method for solving the initial value problem (IVP), viz where the collocation point xi+o = xi+ ph, p E (0, 11, 0 # 112. The presented method will be shown to be strongly unstable if 4 < 1,'2. It turns out that the proposed method is a continuous extension of the A-stabilized version of Simpson's rule introduced in [I], if 9 = 1. Moreover, the method is of order three, for ~( 1 / 2 , 1 ) and is of order four, if D = 1 and 1 : E c5[0, b].
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