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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 80
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping and C.A. Mota Soares
Paper 58
A Collocation Type Implicit Taylor Series Algorithm for ODE Initial Value Problems G. Molnárka and E. Miletics
Department of Mathematics, Széchenyi István University, Gyor, Hungary Full Bibliographic Reference for this paper
, "A Collocation Type Implicit Taylor Series Algorithm for ODE Initial Value Problems", in B.H.V. Topping, C.A. Mota Soares, (Editors), "Proceedings of the Fourth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 58, 2004. doi:10.4203/ccp.80.58
Keywords: implicit Taylor series methods, numerical methods, initial values problems, collocation methods, parallel algorithms, piecewise polynomials.
Summary
In this paper we give implicit generalization of several explicit Taylor series methods.
This generalization means, that the classical truncated Taylor
series we extend with extra terms
and the unknown coefficients in these extra terms we determine by several collocation
conditions. These collocation conditions give the implicit extension of the methods.
The main idea of the rehabilitation of the Taylor series algorithms is based on the new possibility to calculate formally the truncated Taylor series as approximate solutions, but approximate calculation of higher derivatives using well-known technique for the partial differential equations gives more chance to use this old technique [1]. We regard the following ordinary differential equation initial value problem (ODE IVP).
We suppose, that the solution of the problem
where
where
In the paper we show that the following theorem is valid.
If the solution of the problem (18) has solution
The algorithm given by (19) and (20) is a collocation type implicit algorith for the solution of ODE initial value problems. We describe a symple iteration algorithm for the solution for one step of the implicit method. Let us introduce the following notation:
Using these notations the system of equations (20) defining the suggested numerical algorithm can be rewrited into the following form:
where
We prove if the function References
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