Extending high order derivatives for special differential equations of the form \(y' = f(y)\) by using monotonically labeled rooted trees.


  • Hossein Hassani Shahrekord University
  • Mohammad Shafie Dahaghin Shahrekord University






Labeled rooted trees, Monotonically labeled trees, Elementary differentials, Initial value problems, Derivatives.


This paper presents a review of the role played by labeled rooted trees to obtain derivatives for numerical solution of initial value problems in special case \(y' = f(y), y(x_0) = y_0\). We extend a process to find successive derivatives according to monotonically labeled rooted trees, and prove some relevant lemmas and theorems. In this regard, the  derivatives, of the monotonically labeled rooted trees with n vertices are presented by using the monotonically labeled rooted trees with k + n vertices. Eventually, this process is applied to trees without labeling.


[1] J.C. Butcher, The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, volume 2, John Wiley & sons, Chichester and New York, 1987.

[2] J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & sons, 2008.

[3] J.C. Butcher, T.M.H. Chan, The tree and forest spaces with applications to initial-value problem methods, BIT Numerical Mathematics 50 (2010) 713-728.

[4] J.C. Butcher, Trees and numerical methods for ordinary differential equation, Numerical Algorithms 53 (2010) 153-170.

[5] J.C. Butcher, Runge-kutta Method and Banach Algebras, Computational and Applied Mathematics 236 (2012) 3931-3936.

[6] J.M. Franco, An embedded pair of exponentially fitted explicit Runge-Kutta Methods, Journal of Computational and Applied Mathematics 149 (2002) 407-414.

[7] E. Hairer, G. Wanner, Solving Ordinary Differential Equation II. Stiff and Differential-Algebraic Problems, Springer,1996.

[8] S. Koikari, Rooted tree analysis of Runge-Kutta methods with exact treatment of linear terms, Journal of Computational and Applied Mathematics 177 (2005) 427-453.

View Full Article: