Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System
Abstract
:1. Introduction
2. Preliminary
2.1. Bratu’s Problem
2.2. The Rsller-Type System
3. The Optimal Parametric Iteration Method
3.1. Preliminary
3.2. Semi-Analytical Solutions Using OPIM Technique
4. Numerical Results and Validation
OPIM Solutions versus Iterative Solutions
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
References
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t | ||||
---|---|---|---|---|
0 | 5.68434 × | 8.12291 | 4.54747 | 7.10543 |
1/2 | 1.56141 | 8.68649 | 2.21405 | 4.85037 |
1 | 1.48405 | 1.41248 | 2.6783 | 5.0592 |
3/2 | 6.04191 | 3.44 | 1.44322 | 6.4853 |
2 | 3.3371 | 9.82754 | 3.82636 | 5.22813 |
5/2 | 6.32447 | 3.4972 | 3.48186 | 2.96143 |
3 | 3.62685 | 1.43278 | 7.30771 | 7.67939 |
7/2 | 4.35175 | 4.90013 | 7.79294 | 7.20673 |
4 | 4.14164 | 1.68089 | 1.48269 | 4.27381 |
9/2 | 3.19588 | 1.68538 | 2.11227 | 1.03521 |
5 | 1.32596 | 1.90962 | 2.91648 | 9.71333 |
11/2 | 1.90914 | 3.46814 | 1.51662 | 4.52946 |
6 | 1.3966 | 2.01006 | 3.01251 | 8.10717 |
13/2 | 3.33686 | 8.48484 | 4.2035 | 9.86731 |
7 | 1.15607 | 1.38123 | 8.79764 | 7.25709 |
15/2 | 3.18879 | 9.57228 | 2.69659 | 1.10401 |
8 | 5.01597 | 1.00957 | 5.19338 | 1.25896 |
17/2 | 6.856 | 5.66887 | 2.75871 | 9.05699 |
9 | 3.01073 | 5.25215 | 5.99213 | 1.1485 |
19/2 | 6.08924 | 1.05709 | 6.60261 | 1.42172 |
10 | 4.58559 | 2.81366 | 1.80462 | 1.40391 |
t | |||
---|---|---|---|
0 | −8.12291 | −7.10542 | 7.10542 |
1/2 | −0.1009833721 | −0.1009828870 | 4.85037 |
1 | −0.5875537666 | −0.5875532607 | 5.05920 |
3/2 | −1.2588228636 | −1.2588222150 | 6.48530 |
2 | −1.8816621245 | −1.8816616017 | 5.22813 |
5/2 | −2.2683521270 | −2.2683518309 | 2.96142 |
3 | −2.3108289294 | −2.3108281614 | 7.67938 |
7/2 | −1.9975070462 | −1.9975063255 | 7.20673 |
4 | −1.4151229407 | −1.4151225133 | 4.27380 |
9/2 | −0.7351867600 | −0.7351857248 | 1.03520 |
5 | −0.1849885987 | −0.1849876273 | 9.71333 |
t | |||
---|---|---|---|
0 | 0.25 | 0.2499999249 | 7.50267 |
1/2 | −0.6367781227 | −0.6367801438 | 2.02113 |
1 | −1.2376076804 | −1.2376093968 | 1.71634 |
3/2 | −1.3673785582 | −1.3673786474 | 8.91577 |
2 | −1.0603932046 | −1.0603910929 | 2.11171 |
5/2 | −0.4497705143 | −0.4497711768 | 6.62540 |
3 | 0.2837908478 | 0.2837896153 | 1.23259 |
7/2 | 0.9401129558 | 0.9401146702 | 1.71439 |
4 | 1.3313637430 | 1.3313651699 | 1.42692 |
9/2 | 1.3107276683 | 1.3107251137 | 2.55464 |
5 | 0.8126645233 | 0.8126672218 | 2.69845 |
t | |||
---|---|---|---|
0 | 0.25 | 0.2499999249 | 0.25 |
0.35 | −0.3847286943 | −0.3847266999 | −0.3823724677 |
0.7 | −0.9282966040 | −0.9282953550 | −0.9683700083 |
1.05 | −1.2724487484 | −1.2724506529 | −1.4370077000 |
1.4 | −1.3794335466 | −1.3794320704 | −1.7324569333 |
1.75 | −1.2616050449 | −1.2616062739 | −1.8234993489 |
2.1 | −0.9572932068 | −0.9572918609 | −1.7129807750 |
2.45 | −0.5195989669 | −0.5196005038 | −1.4472651645 |
2.8 | −0.0116471526 | −0.0116453617 | −1.1256885333 |
3.15 | 0.4978649029 | 0.4978635857 | −0.9100128968 |
3.5 | 0.9401129558 | 0.9401146702 | −1.0338802083 |
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Ene, R.-D.; Pop, N. Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System. Mathematics 2024, 12, 1308. https://doi.org/10.3390/math12091308
Ene R-D, Pop N. Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System. Mathematics. 2024; 12(9):1308. https://doi.org/10.3390/math12091308
Chicago/Turabian StyleEne, Remus-Daniel, and Nicolina Pop. 2024. "Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System" Mathematics 12, no. 9: 1308. https://doi.org/10.3390/math12091308
APA StyleEne, R.-D., & Pop, N. (2024). Semi-Analytical Closed-Form Solutions for Dynamical Rössler-Type System. Mathematics, 12(9), 1308. https://doi.org/10.3390/math12091308