Numerical technique of Dynamical models with Applications to Chaotic Systems
Keywords:
Fractional differential system, AB operators, Stability, chaotic.
Abstract
In this article a new numerical scheme for dynamical models with applications to chaotic systems was developed, we present the analysis of errors in general. The novel numerical method was used to address both linear and nonlinear fractional dynamical systems. The technique was used two (linear and nonlinear) systems to be solved of the fractional ordinary differential equations as well as fours nonlinear chaotic models.
Downloads
Download data is not yet available.
References
1. W. Leung, Systems of Nonlinear Partial Differential Equa-tions: Applications to Biology and Engineering, vol.49, SpringerScience&BusinessMedia,2013.
2. T. Roubíček, Nonlinear Partial Differential Equations withApplications, vol. 153, Birkhäuser Boston, 2013.
3. L. Debnath, Nonlinear Partial Differential Equations for Scien-tists and Engineers, Springer Science & Business Media, 2012.
4. G. Jumarie, “Modified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresults,” Computers & Mathematics with Applications,vol. 51, no. 9-10, pp. 1367–1376, 2006.
5. S. Miller and B. Ross, An Introduction to the Fractional Cal-culus and Fractional Differential Equations, Wiley, 1993.
6. H. KOGA and M. NAKAGAWA, “An Evaluation Method ofFractal Dimensions in Terms of Fractional Integro Differential Equations,” IEICE transactions on fundamentalsof electronics, communications and computer sciences, vol. 86, no. 4, p. 974, 2003.
7. Z. Feng, “Exact solutions to the Lienard equation and its appli-cations,” Chaos, Solitons & Fractals, vol. 21, no. 2, pp. 343–348,2004.
8. Abdoon, Mohamed A., Faeza Lafta Hasan, and Nidal E. Taha. "Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation." Abstract and Applied Analysis. Vol. 2022. Hindawi, 2022.
9. Hasan, Faeza L., and Mohamed A. Abdoon. "The generalized (2+ 1) and (3+ 1)-dimensional with advanced analytical wave solutions via computational applications." International Journal of Nonlinear Analysis and Applications 12.2 (2021): 1213-1241.
10. Abdoon, M.A., Hasan, F.L. (2022). Advantages of the differential equations for solving problems in mathematical physics with symbolic computation. Mathematical Modelling of Engineering Problems, Vol. 9, No. 1, pp. 268-276. https://doi.org/10.18280/mmep.090133.
11. A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2) (2016) 763–769
12. Toufik, Mekkaoui, and Abdon Atangana. "New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models." The European Physical Journal Plus 132.10 (2017): 1-16.
13. Al-Zahrani, S. M., Elsmih, F. E. I., Al-Zahrani, K. S., & Saber, S. (2022). A Fractional Order SITR Model for Forecasting of Transmission of COVID-19: Sensitivity Statistical Analysis. Malaysian Journal of Mathematical Sciences, 16(3), 517-536.
14. Abdoon, M. A. (2015). First integral method: a general formula for nonlinear fractional Klein-Gordon equation using advanced computing language. American Journal of Computational Mathematics, 5(02), 127.
15. Abdoon, M. A. (2015). Programming first integral method general formula for the solving linear and nonlinear equations. Applied Mathematics, 6(03), 568.
16. Guma, F. E., Badawy, O. M., Musa, A. G., Mohammed, B. O., Abdoon, M. A., Berir, M., & Salih, S. Y. M. (2023). Risk factors for death among COVID-19 Patients admitted to isolation Units in Gedaref state, Eastern Sudan: a retrospective cohort study. Journal of Survey in Fisheries Sciences, 10(3S), 712-722.
17. Alzahrani, S. M., Elsmih, F. E., Al-Zahrani, K. S., & Saber, S. (2021). Analysis of Confirmed Cases of Covid-19 in Saudi Arabia Using Poison Autoregressive and Holt Winter Models. International Journal of Difference Equations (IJDE), 16(2), 179-187.
18. Saber, S. A. Y. E. D., & Alahmari, A. B. D. U. L. L. A. H. (2023). Compactness Estimates for the∂-Neumann problem on a q-pseudoconvex domains in a Stein manifold. Kragujevac Journal of Mathematics, 47(4), 627-636.
19. Qazza, A., Abdoon, M., Saadeh, R., & Berir, M. (2023). A New Scheme for Solving a Fractional Differential Equation and a Chaotic System. European Journal of Pure and Applied Mathematics, 16(2), 1128-1139.
20. Saadeh, R., A. Abdoon, M., Qazza, A., & Berir, M. (2023). A Numerical Solution of Generalized Caputo Fractional Initial Value Problems. Fractal and Fractional, 7(4), 332.
21. Saber, S., Alghamdi, A. M., Ahmed, G. A., & Alshehri, K. M. (2022). Mathematical Modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies. AIMS Mathematics, 7(7), 12011-12049.
22. I Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, in: Mathematics in Science and Engineering, vol. 198, 1999.
23. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2) (2015) 73–85.
24. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2) (2015) 87–92.
2. T. Roubíček, Nonlinear Partial Differential Equations withApplications, vol. 153, Birkhäuser Boston, 2013.
3. L. Debnath, Nonlinear Partial Differential Equations for Scien-tists and Engineers, Springer Science & Business Media, 2012.
4. G. Jumarie, “Modified Riemann-Liouville derivative and frac-tional Taylor series of nondifferentiable functions furtherresults,” Computers & Mathematics with Applications,vol. 51, no. 9-10, pp. 1367–1376, 2006.
5. S. Miller and B. Ross, An Introduction to the Fractional Cal-culus and Fractional Differential Equations, Wiley, 1993.
6. H. KOGA and M. NAKAGAWA, “An Evaluation Method ofFractal Dimensions in Terms of Fractional Integro Differential Equations,” IEICE transactions on fundamentalsof electronics, communications and computer sciences, vol. 86, no. 4, p. 974, 2003.
7. Z. Feng, “Exact solutions to the Lienard equation and its appli-cations,” Chaos, Solitons & Fractals, vol. 21, no. 2, pp. 343–348,2004.
8. Abdoon, Mohamed A., Faeza Lafta Hasan, and Nidal E. Taha. "Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation." Abstract and Applied Analysis. Vol. 2022. Hindawi, 2022.
9. Hasan, Faeza L., and Mohamed A. Abdoon. "The generalized (2+ 1) and (3+ 1)-dimensional with advanced analytical wave solutions via computational applications." International Journal of Nonlinear Analysis and Applications 12.2 (2021): 1213-1241.
10. Abdoon, M.A., Hasan, F.L. (2022). Advantages of the differential equations for solving problems in mathematical physics with symbolic computation. Mathematical Modelling of Engineering Problems, Vol. 9, No. 1, pp. 268-276. https://doi.org/10.18280/mmep.090133.
11. A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. 20 (2) (2016) 763–769
12. Toufik, Mekkaoui, and Abdon Atangana. "New numerical approximation of fractional derivative with non-local and non-singular kernel: application to chaotic models." The European Physical Journal Plus 132.10 (2017): 1-16.
13. Al-Zahrani, S. M., Elsmih, F. E. I., Al-Zahrani, K. S., & Saber, S. (2022). A Fractional Order SITR Model for Forecasting of Transmission of COVID-19: Sensitivity Statistical Analysis. Malaysian Journal of Mathematical Sciences, 16(3), 517-536.
14. Abdoon, M. A. (2015). First integral method: a general formula for nonlinear fractional Klein-Gordon equation using advanced computing language. American Journal of Computational Mathematics, 5(02), 127.
15. Abdoon, M. A. (2015). Programming first integral method general formula for the solving linear and nonlinear equations. Applied Mathematics, 6(03), 568.
16. Guma, F. E., Badawy, O. M., Musa, A. G., Mohammed, B. O., Abdoon, M. A., Berir, M., & Salih, S. Y. M. (2023). Risk factors for death among COVID-19 Patients admitted to isolation Units in Gedaref state, Eastern Sudan: a retrospective cohort study. Journal of Survey in Fisheries Sciences, 10(3S), 712-722.
17. Alzahrani, S. M., Elsmih, F. E., Al-Zahrani, K. S., & Saber, S. (2021). Analysis of Confirmed Cases of Covid-19 in Saudi Arabia Using Poison Autoregressive and Holt Winter Models. International Journal of Difference Equations (IJDE), 16(2), 179-187.
18. Saber, S. A. Y. E. D., & Alahmari, A. B. D. U. L. L. A. H. (2023). Compactness Estimates for the∂-Neumann problem on a q-pseudoconvex domains in a Stein manifold. Kragujevac Journal of Mathematics, 47(4), 627-636.
19. Qazza, A., Abdoon, M., Saadeh, R., & Berir, M. (2023). A New Scheme for Solving a Fractional Differential Equation and a Chaotic System. European Journal of Pure and Applied Mathematics, 16(2), 1128-1139.
20. Saadeh, R., A. Abdoon, M., Qazza, A., & Berir, M. (2023). A Numerical Solution of Generalized Caputo Fractional Initial Value Problems. Fractal and Fractional, 7(4), 332.
21. Saber, S., Alghamdi, A. M., Ahmed, G. A., & Alshehri, K. M. (2022). Mathematical Modelling and optimal control of pneumonia disease in sheep and goats in Al-Baha region with cost-effective strategies. AIMS Mathematics, 7(7), 12011-12049.
22. I Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, in: Mathematics in Science and Engineering, vol. 198, 1999.
23. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2) (2015) 73–85.
24. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2) (2015) 87–92.
Published
2023-05-07
How to Cite
Salih Yousuf Mohamed Salih, & Shahinaz.A. Elsamani. (2023). Numerical technique of Dynamical models with Applications to Chaotic Systems. Central Asian Journal of Theoretical and Applied Science, 4(5), 170-179. https://doi.org/10.17605/OSF.IO/6EHXM
Section
Articles
This work is licensed under a Creative Commons Attribution 4.0 International License.
