WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 23, 2024
About One Multistep Multiderivative Method of Predictor-Corrector Type Constructed for Solving İnitial-Value Problem for ODE of Second Order
Authors: ,
Abstract: Considering the wide application of the initial-value problem for Ordinary Differential Equations second-order with a special structure, here for solving this problem constructed the special Multistep Multiderivative Methods. Many scientists studied this problem , but the most distinguishing is the Ştörmer. To solve this problem here is proposed to use the Multistep Second derivative Method with a special structure. This method has been generalized by many authors, which is called as the linear Multistep Multiderivative Methods with the constant coefficients. Many authors shave shown that the Multistep Second derivative Method can be applied to solve the initial-value problem for ODEs of the first order. Euler himself using his famous method discovered that, in his method when moving from one point to another local truncation errors add up, the results of which reach a very large value. To solve this problem, he suggested using more accurate methods. For this aim, Euler proposed calculating the next term in the Taylor series of the solutions of the investigated problem. Developing this idea and papulation of the Multistep Multiderivative Methods here to solve the named problem it is suggested to use MultistepThriedderivative Methods, taking into account that methods of this type are more accurate. For the demonstration above, receiving results here have constructed some concrete methods. Also by using some of Dahlquist’s and Ibrahimov’s results for Multistep Methods with the maximum order of accuracy were compared. Proven that the MultistepThriedderivative Methods are more accurate than the others. By using model problems have illustrated some results received here.
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Keywords: Initial-value problem for ODEs, Stability and Degree, Multistep Multiderivative Methods (MMM), Local Truncation Error, Störmer Method, Multistep Secondderivative Methods (MSM)
Pages: 599-607
DOI: 10.37394/23206.2024.23.63