Abstract
One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.
Highlights
Falkner Methods (BFFM) for the direct integration of the general second order initial value problem whose solution is oscillatory or periodic, in the latter case with the frequency known, or that can be estimated in advance
The coefficients of the adapted Falkner method depend on the nature of the fitting function I ( x ) on how the set Ω is chosen, which can be any of the types listed in Nguyen et al [50], where for any of the choices we have to take a total of k + 4 elements to determine the adapted block Falkner method on the basis that the approximations are of the form in (2)
We have proposed a family of Adapted block Falkner methods using third derivative for solving second order initial value problem with oscillatory solution directly numerically
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Some researchers have considered the direct integration of the general second order IVPs containing the first derivative This can be found in the works by Guo and. Falkner Methods (BFFM) for the direct integration of the general second order initial value problem whose solution is oscillatory or periodic, in the latter case with the frequency known, or that can be estimated in advance. This class, which is an adapted formulation of the methods in Ramos and Rufai [33], uses a basis function different from what can be seen in the reviewed literature.
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