A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations
| Autor: | A.V. Kunynets, M.V. Kutniv, Andrzej Włoch, Bohdan Datsko |
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| Rok vydání: | 2020 |
| Předmět: |
Numerical Analysis
Differential equation Applied Mathematics Numerical analysis 010103 numerical & computational mathematics Interval (mathematics) Grid 01 natural sciences 010101 applied mathematics Computational Mathematics symbols.namesake Singularity Taylor series symbols Initial value problem Applied mathematics Node (circuits) 0101 mathematics Mathematics |
| Zdroj: | Applied Numerical Mathematics. 148:140-151 |
| ISSN: | 0168-9274 |
| DOI: | 10.1016/j.apnum.2019.09.006 |
| Popis: | A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. |
| Databáze: | OpenAIRE |
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