Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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AN IMPROVED IMPLICIT EULER METHOD FOR SOLVING INITIAL VALUE PROBLEMS

  • YUN, BEONG IN (DEPARTMENT OF MATHEMATICS, KUNSAN NATIONAL UNIVERSITY)
  • Received : 2022.07.06
  • Accepted : 2022.08.06
  • Published : 2022.09.25
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Abstract

To solve the initial value problem we present a new single-step implicit method based on the Euler method. We prove that the proposed method has convergence order 2. In practice, numerical results of the proposed method for some selected examples show an error tendency similar to the second-order Taylor method. It can also be found that this method is useful for stiff initial value problems, even when a small number of nodes are used. In addition, we extend the proposed method by using weighted averages with a parameter and show that its convergence order becomes 2 for the parameter near $\frac{1}{2}$. Moreover, it can be seen that the extended method with properly selected values of the parameter improves the approximation error more significantly.

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Acknowledgement

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) [No. 2021R1F1A1047343].

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