Abstract
In this paper, the relationships between the Hopf algebra and the B-series approach are illustraded in details. The Hopf Algebra was proposed to solve problems in wider fields: non-commutative geometry, combinatorics of renormalization in quantum field theory. These two approaches generate the recursive composition rules based on the dual structure. We proposed to reformulate the composition rule on the tree space. It is found to be much simpler and easier without using the dual structure. In this paper, the B-series approach is represented in a way that applications are based on the understanding of the meaning hiding behind the formulae. This also enables us to describe the Hopf algebra in B-series language without using the dual structure. In order to show that the B-series approach could be applied to solve problems which is not restricted to ordinary differential systems, we use an example which is solved using the Hopf Algebra approach. Through the demonstration of applying the B-series approach, we shows that the key of appying the B-series approach is to construct the elementary differential of the system. It turns out that the results we derived are generated in an interesting way.
Published Version
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