A unified approach for the development of -step block Falkner-type methods for solving general second-order initial-value problems in ODEs

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• 作者：Higinio Ramos^a ; ^b ; ^{higra@usal.es} ; Shubham Mehta^c ; ^{shanuuma@iitr.ac.in} ; J. Vigo-Aguiar^a ; ^d ; ^{jvigo@usal.es}
• 关键词：65L05 ; 65L20
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：July 2017
• 年：2017
• 卷：318
• 期：Complete
• 页码：550-564
• 全文大小：525 K
• 卷排序：318

文摘

A family of kk-step block multistep methods where the main formulas are of Falkner type is proposed for the direct integration of the general second order initial-value problem where the differential equation is of the general form y″=f(x,y,y′)y″=f(x,y,y′). The two main Falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k+1k+1 points. The main characteristics of the methods are discussed through their formulation in vector form. Each method is formulated as a group of 2k2k simultaneous formulas over kk non-overlapping intervals. In this way, the method produces the approximation of the solution simultaneously at kk points on these intervals. As in other block methods, there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge–Kutta methods). The resulting family is efficient and competitive compared with other existing methods in the literature, as may be seen from the numerical results.