Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations
Abstract
:1. Introduction
2. Phase Lag and Stability Analysis
3. Derivation of the New Method
4. Results
- Problem 1 (Prothero–Robinson problem)
- Problem 2 (Duffing equation)
- Problem 3 (The well-known two-body problem)
- Problem 4 (Kramarz’s system)
5. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | |||
0 | 0 | |||||
0 | ||||||
Step-Size | MEHM | EHM5IIPA |
---|---|---|
0.4 | 8.12463 × 10−6 | 3.03912 × 10−5 |
0.2 | 4.72859 × 10−7 | 1.19831 × 10−6 |
0.1 | 2.80407 × 10−8 | 4.23368 × 10−8 |
0.05 | 1.69979 × 10−9 | 1.41116 × 10−9 |
0.025 | 1.04445 × 10−10 | 4.55621 × 10−11 |
Step-Size | MEHM | EHM5IIPA |
---|---|---|
0.4 | 2.48225 × 10−14 | 1.02736 |
0.2 | 5.51845 × 10−13 | 2.71483 × 10−1 |
0.1 | 2.95522 × 10−13 | 8.20955 × 10−2 |
0.05 | 3.76672 × 10−12 | 2.97346 × 10−3 |
0.025 | 4.66915 × 10−12 | 9.77418 × 10−5 |
Step-Size | MEHM | EHM5IIPA |
---|---|---|
0.4 | 1.42361 × 10−2 | 2.29762 × 10−1 |
0.2 | 9.29187 × 10−4 | 1.98607 × 10−3 |
0.1 | 6.00156 × 10−5 | 1.82083 × 10−4 |
0.05 | 3.81442 × 10−6 | 6.89947 × 10−6 |
0.025 | 2.40430 × 10−7 | 2.27004 × 10−7 |
Step-Size | MEHM | EHM5IIPA |
---|---|---|
0.05 | 1.16031 × 10−16 | 1.74602 × 10−16 |
0.025 | 1.72165 × 10−16 | 1.67037 × 10−15 |
0.0125 | 5.41637 × 10−15 | 5.97021 × 10−16 |
0.00625 | 7.41002 × 10−15 | 1.49427 × 10−14 |
0.003125 | 2.45548 × 10−14 | 5.03433 × 10−14 |
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Samat, F.; Ismail, E.S. Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations. Mathematics 2021, 9, 1028. https://doi.org/10.3390/math9091028
Samat F, Ismail ES. Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations. Mathematics. 2021; 9(9):1028. https://doi.org/10.3390/math9091028
Chicago/Turabian StyleSamat, Faieza, and Eddie Shahril Ismail. 2021. "Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations" Mathematics 9, no. 9: 1028. https://doi.org/10.3390/math9091028