Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'s

Panovsky and Richardson [A family of implicit Chebyshev methods for the numerical integration of second-order differential equations, J. Comput. Appl. Math. 23 (1988) 35–51] presented a method based on Chebyshev approximations for numerically solving the problem y″=f(x,y)y″=f(x,y), being the steplength constant. Coleman and Booth [Analysis of a Family of Chebyshev Methods for y″=f(x,y)y″=f(x,y), J. Comput. Appl. Math. 44 (1992) 95–114] made an analysis of the above method and suggested the convenience to design a variable steplength implementation. As far as we know this goal has not been achieved until now. Later on we extended the above method (this journal, 2003), and obtained a scheme for numerically solving the equation y″-2gy′+(g2+w2)=f(x,y)y″-2gy′+(g2+w2)=f(x,y). The question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper.