A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems
Abstract
:1. Introduction
- Introduction;
- Theory of Runge–Kutta Pairs of Orders 6(5);
- Training the coefficients;
- Numerical Tests;
- Conclusions.
2. Theory of Runge–Kutta Pairs of Orders 6(5)
- Solve , for .
- Put .
- Solve and for .
- Substitute from .
- Since find from
- is given from .
- Solve simultaneously for , , , and the equations:
- From evaluate and .
- From evaluate and .
- From evaluate and .
- From evaluate .
- Finally, using FSAL (First Stage As Last) property, substitute .
3. Training the Coefficients
4. Numerical Tests
- DLMP6(5) 9-stages FSAL pair given in [19].
- ST6(4) 7-stages FSAL pair given in [24].
- NEW6(5) 9-stages FSAL presented here.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Kovalnogov, V.N.; Fedorov, R.V.; Khakhalev, Y.A.; Simos, T.E.; Tsitouras, C. A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems. Mathematics 2021, 9, 1842. https://doi.org/10.3390/math9161842
Kovalnogov VN, Fedorov RV, Khakhalev YA, Simos TE, Tsitouras C. A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems. Mathematics. 2021; 9(16):1842. https://doi.org/10.3390/math9161842
Chicago/Turabian StyleKovalnogov, Vladislav N., Ruslan V. Fedorov, Yuri A. Khakhalev, Theodore E. Simos, and Charalampos Tsitouras. 2021. "A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems" Mathematics 9, no. 16: 1842. https://doi.org/10.3390/math9161842
APA StyleKovalnogov, V. N., Fedorov, R. V., Khakhalev, Y. A., Simos, T. E., & Tsitouras, C. (2021). A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems. Mathematics, 9(16), 1842. https://doi.org/10.3390/math9161842