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

 
 
 
  • Abstract
  • Keywords
  • References
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  • 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.


  • Keywords


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

  • References


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      [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.

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      [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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Article ID: 4691
 
DOI: 10.14419/ijamr.v4i4.4691




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