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
    2015-10-07
    https://doi.org/10.14419/ijamr.v4i4.4691
  • 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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      [4] J.C. Butcher, Trees and numerical methods for ordinary differential equation, Numerical Algorithms 53 (2010) 153-170.

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      [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.

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  • How to Cite

    Hassani, H., & Dahaghin, M. S. (2015). Extending high order derivatives for special differential equations of the form \(y’ = f(y)\) by using monotonically labeled rooted trees. International Journal of Applied Mathematical Research, 4(4), 513-518. https://doi.org/10.14419/ijamr.v4i4.4691

    Received date: 2015-04-29

    Accepted date: 2015-09-14

    Published date: 2015-10-07