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 The second order convergence of the obtained generalized Newmark schemes is justified mathematically and demonstrated computationally through problems of practical interest.