IJPAM: Volume 46, No. 2 (2008)

Invited Lecture Delivered at
Forth International Conference of Applied Mathematics
and Computing (Plovdiv, Bulgaria, August 12-18, 2007)


A CLASS OF INITIAL VALUE METHODS FOR THE DIRECT
SOLUTION OF SECOND ORDER INITIAL VALUE PROBLEMS

Samuel N. Jator
Department of Mathematics
Austin Peay State University
P.O. Box 4626, Clarksville, TN 37044, USA
e-mail: Jators@apsu.edu


Abstract.A class of Linear Multistep Methods (LMMs) are developed and applied as Initial Value Methods (IVMs) to solve second order Initial Value Problems (IVPs). The main method is derived by interpolating the assumed approximate solution at $x=x_{n+j}$, $j=1,2,...,r-1$ and collocating the differential system at $x=x_{n+j}$, $j=1,2,...,s-1$ respectively, where $r$ and $s$ are the number of interpolation and collocation points. The derivation of the main method leads to a continuous approximation from which IVMs are obtained and simultaneously applied as numerical integrators IVPs. In particular, the Multiple Finite Difference Methods (MFDMs) are implemented without the need for either predictors or other methods to supply the starting values. A numerical example is given to illustrate the efficiency of the methods.

Received: August 17, 2007

AMS Subject Classification: 65L05, 65L06, 65L12

Key Words and Phrases: collocation, interpolation, second order, initial value methods, linear multistep

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 46
Issue: 2