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)$y^{″}-2{\mathit{gy}}^{\prime }+(g^2+w^2)=f(x,y)$. The question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper.