IJPAM: Volume 1, No. 3 (2002)

GENERALIZED NEWMARK SCHEMES
FOR SINGULAR SECOND ORDER
INITIAL-VALUE PROBLEMS

M.M. Chawla
Dept. of Mathematics & Computer Science
Kuwait University
P.O. Box 5969, Safat 13060, KUWAIT


Abstract.An important class of singular second order initial-value problems is $y^{\prime \prime }+\left( \alpha /x\right) y^{\prime }+f\left(
x,y\right) =0,$ $0 \lessdot x\lessdot x_{f},$ $y\left( 0\right) =a,$ $
y^{\prime }\left( 0\right) =0,$ with $\alpha =1$ (cylindrical symmetry) or $
\alpha =2$ (spherical symmetry). This class includes the well-known singular equations of Emden and Liouville which have found applications in electrohydrodynamics, thermal explosions and stellar stability. For regular second order initial-value problems (with $\alpha =0$), a well-known singly-implicit one-step integration scheme is due to Newmark. In the present paper, we describe Newmark-like singly-implicit one-step integration schemes for problems with $\alpha =1$ and $2.$ The second order convergence of the obtained generalized Newmark schemes is justified mathematically and demonstrated computationally through problems of practical interest.

Received: February 2, 2002

AMS Subject Classification: 65L05

Key Words and Phrases: singular second order initial-value problems, Emden and Liouville equations, Newmark scheme, generalized Newmark schemes

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 1
Issue: 3