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

Authors

  • Hossein Hassani Shahrekord University
  • Mohammad Shafie Dahaghin Shahrekord University

DOI:

https://doi.org/10.14419/ijamr.v4i4.4691

Published:

2015-10-07

Keywords:

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

Abstract

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.

References

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