Numerical methods for ordinary differential systems: the initial value problem
Numerical methods for ordinary differential systems: the initial value problem
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Analysis of a family of Chebyshev methods for y′′=fx,y
Journal of Computational and Applied Mathematics
Selected papers of the sixth conference on Numerical Treatment of Differential Equations
Variable-order, variable-step methods for second-order initial-value problems
Journal of Computational and Applied Mathematics
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Although it is possible to integrate a special second-order differential equation of the form y''(x) = f(x,y(x)), y(x0) = y0, y'(x0) = y'0 (1) by reducing it to a first order system and applying one of the methods available for those systems, it seems more natural to provide numerical methods to integrate (1) directly without using first derivatives.The advantage of these approaches has to be with the fact that they are able to exploit special information about ODES, and this results in an increase in efficiency (that is, high accuracy at low cost). For instance, it is well-known that Runge-Kutta-Nyström methods for (1) involve a real improvement compared to standard Runge-Kutta methods, for a given number of stages [ 4, p. 285]. On the contrary, a linear k-step method for first-order ODEs becomes a 2k-step method for (1), [4, p. 461], increasing the computational work.