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

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 ) , being the steplength constant. Coleman and Booth [Analysis of a Family of Chebyshev Methods for 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 ″ - 2 gy ′ + ( 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.