Abstract
The idea of symmetric super-implicit linear multi-step methods (SSILMMs) necessitates the use of not just past and present solution values of the ordinary differential equations (ODEs), but also, future values of the solution. Such methods have been proposed recently for the numerical solution of second-order ODEs. One technique to obtain more accurate integration process is to construct linear multi-step methods with hybrid points employing future solution values. In this regard, we construct families of Stӧrmer-Cowell type hybrid SSILMMs having higher order than that of the symmetric super-implicit method recently proposed for the same step number using the Taylors series approach. The newly derived hybrid SSILMMs are p-stable with accurate results when tested on some special second order IVPs.
Highlights
Consider the initial value problem (IVP), =, ; =, (1)in ordinary differential equations (ODEs) in which there is no explicit first derivative appearing
The linear multi-step methods for solving the second order IVP (1) is
The work in [2] further stressed on the work in [1], by considering the free parameters available in their proposed linear multi-step methods which can reduce the work to two functional evaluations, and reduces the work with respect to implementation for nonlinear problems of (1)
Summary
In ordinary differential equations (ODEs) in which there is no explicit first derivative appearing. As [1] further noted, [12] claimed to have derived high order p-stable linear multi-step methods but their concept of p-stability is considerably different from that given in [14]. The work in [2] further stressed on the work in [1], by considering the free parameters available in their proposed linear multi-step methods which can reduce the work to two functional evaluations, and reduces the work with respect to implementation for nonlinear problems of (1). Neta [16] considered a very special class of (2), the symmetric super-implicit linear multi-step method given by,. The Taylors series approach in the sense of the work in [1] will be used to derive the new hybrid extension of (11) in [16] while using MATHEMATICA v 8 [11]
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