Araştırma Makalesi
Yıl 2025,
Cilt: 15 Sayı: 1, 80 - 98, 30.06.2025
Öz
Kaynakça
- J. D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem. Chichester, UK: Wiley, 1991.
- P. Onumanyi, D. O. Awoyemi, S. N. Jator, and U. W. Sirisena, “New linear multistep methods with continuous coefficients for first order initial value problems,” J. Nig. Math. Soc., vol. 13, no. 1, pp. 37–51, 1994.
- O. A. Akinfenwa, S. N. Jator, and N. M. Yao, “A self‑starting block Adams methods for solving stiff ODEs,” in Proc. 14th IEEE Int. Conf. Comput. Sci. Eng., 2011, p. 156.
- D. G. Yakubu, G. M. Kumlen, and S. Markus, “Second derivative Runge‑Kutta collocation methods based on Lobatto nodes for stiff systems,” J. Mod. Methods Numer. Math., vol. 8, no. 2, pp. 118–138, 2017.
- K. Mehdizadeh, N. Nasehi, and G. Hojjati, “A class of second derivative multistep methods for stiff systems,” Acta Univ. Apulensis, vol. 30, no. 1, pp. 171–188, 2012.
- J. Garba and U. Mohammed, “Derivation of a new one‑step numerical integrator with three intra‑step points for solving first order ordinary differential equations,” Niger. J. Math. Appl., vol. 30, no. 1, pp. 155–172, 2020.
- U. Mohammed, J. Garba, and M. E. Semenov, “One‑step second derivative block intra‑step method for stiff system of ordinary differential equations,” J. Niger. Math. Soc., vol. 40, no. 1, pp. 47–57, 2021.
- K. M. Ibrahim, R. K. Jamal, and F. H. Ali, “Chaotic behaviour of the Rossler model and its analysis by using bifurcations of limit cycles and chaotic attractors,” J. Phys.: Conf. Ser., 2018, doi:10.1088/1742-6596/1003/1/012099.
- G. Chen and J. L. Moiola, “An overview of bifurcation, chaos and nonlinear dynamics in control systems,” J. Franklin Inst., vol. 331, no. 6, pp. 819–858, 1994.
- A. Farshidianfar and A. Saghafi, “Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems,” Nonlinear Dyn., vol. 75, no. 4, pp. 783–806, 2014.
- L. V. Stepanova and S. A. Igonin, “Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium,” Appl. Math. Model., vol. 38, no. 14, pp. 3436–3455, 2014.
- M. Jiya, “Application of homotopy perturbation method (HPM) for the solution of non‑linear differential equations,” Pac. J. Sci. Technol., vol. 11, no. 2, pp. 268–272, 2010.
- J. Giné, J. Llibre, K. Wu, and X. Zhang, “Averaging methods of arbitrary order periodic solutions and integrability,” J. Differ. Equations, vol. 260, no. 5, pp. 4130–4156, 2016.
- H. de la Cruz, R. J. Biscay, J. C. Jimenez, and F. Carbonell, “Local linearization–Runge–Kutta methods: a class of A‑stable explicit integrators for dynamical systems,” Math. Comput. Model., vol. 57, nos. 3–4, pp. 720–740, 2013.
- H. H. Goldstine, A History of Numerical Analysis from the 16th through the 19th Century. New York, NY, USA: Springer, 2012.
- K. Atkinson, W. Han, and D. E. Stewart, Numerical Solution of Ordinary Differential Equations. Pure Appl. Math., vol. 108, 2011.
- G. G. Dahlquist, “A special stability problem for linear multistep methods,” BIT, vol. 3, no. 1, pp. 27–43, 1963.
- P. Henrici, Discrete Variable Methods in Ordinary Differential Equations. Hoboken, NJ, USA: Wiley, 1962.
- R. T. Keller and D. U. Qiang, “Discovery of dynamics using linear multistep methods,” SIAM, vol. 59, no. 1, pp. 429–455, 2021.
- M. Musa and M. A. Unwala, “Extended 3‑point super class of block backward differentiation formula for solving stiff initial value problems,” Abacus (Math. Sci. Ser.), vol. 44, no. 1, pp. 584–591, 2019.
- B. S. H. Kashkari and M. I. Syam, “Optimization of one step block method with three hybrid points for solving first‑order ordinary differential equations,” Results Phys., vol. 12, pp. 592–596, 2019.
- S. B. G. Karakoç, A. Saha, and D. Y. Sucu, “A collocation algorithm based on septic B‑splines and bifurcation of traveling waves for Sawada Kotera equation,” Math. Comput. Simul., vol. 203, pp. 12–27, 2023.
- R. Jiwari and S. A. Pandey, “Collocation algorithm based on septic B‑splines and bifurcation of traveling waves for Sawada–Kotera equation,” Math. Comput. Simul., vol. 203, no. 1, pp. 12–27, 2023.
- A. Kiliçman, A. E. Ismail, and Z. Ismail, “A novel scheme based on collocation finite element method to generalised Oskolkov equation,” J. Sci. Arts, vol. 25, no. 4, pp. 895–908, 2021.
- B. M. Saka and F. Erdogan, “A septic B‑spline collocation method for solving the generalized equal width wave equation,” Kuwait J. Sci., vol. 43, no. 1, pp. 20–31, 2016.
A Two-step with First and Second Derivative Scheme for Numerical Solution of First-Order Problems in Dynamical Systems
Öz
One of the numerical techniques used to solve differential equations is the linear multistep method (LMM). A two-step second-derivative intra-point block numerical method of uniform order ten is proposed for solving dynamical systems in ordinary differential equations. The derived two-step method with multi-derivatives effectively addresses the challenges in solving nonlinear dynamical systems – exhibiting phenomena such as multiple steady states, oscillations, and chaos. The inclusion of second derivative in the block method makes sure more information about the ODE is used in generating the solution thereby improving the accuracy of the method. The method is A-stable, making it suitable for solving nonlinear dynamic systems in ordinary differential equations (ODEs). In addition, the method possesses a higher order of accuracy, and the associated error constants are very small. This block method generates numerical solutions that provide solution profiles and phase portraits for the problems considered under various situations of dynamical systems. The results generated from this method underscore its potential as a robust and versatile tool for solving a wide range of practical problems arising in real-life.
Anahtar Kelimeler
Dynamical system Intra-step point Non-linear Numerical technique Profile solution Phase portrait
Kaynakça
- J. D. Lambert, Numerical Methods for Ordinary Differential Systems, The Initial Value Problem. Chichester, UK: Wiley, 1991.
- P. Onumanyi, D. O. Awoyemi, S. N. Jator, and U. W. Sirisena, “New linear multistep methods with continuous coefficients for first order initial value problems,” J. Nig. Math. Soc., vol. 13, no. 1, pp. 37–51, 1994.
- O. A. Akinfenwa, S. N. Jator, and N. M. Yao, “A self‑starting block Adams methods for solving stiff ODEs,” in Proc. 14th IEEE Int. Conf. Comput. Sci. Eng., 2011, p. 156.
- D. G. Yakubu, G. M. Kumlen, and S. Markus, “Second derivative Runge‑Kutta collocation methods based on Lobatto nodes for stiff systems,” J. Mod. Methods Numer. Math., vol. 8, no. 2, pp. 118–138, 2017.
- K. Mehdizadeh, N. Nasehi, and G. Hojjati, “A class of second derivative multistep methods for stiff systems,” Acta Univ. Apulensis, vol. 30, no. 1, pp. 171–188, 2012.
- J. Garba and U. Mohammed, “Derivation of a new one‑step numerical integrator with three intra‑step points for solving first order ordinary differential equations,” Niger. J. Math. Appl., vol. 30, no. 1, pp. 155–172, 2020.
- U. Mohammed, J. Garba, and M. E. Semenov, “One‑step second derivative block intra‑step method for stiff system of ordinary differential equations,” J. Niger. Math. Soc., vol. 40, no. 1, pp. 47–57, 2021.
- K. M. Ibrahim, R. K. Jamal, and F. H. Ali, “Chaotic behaviour of the Rossler model and its analysis by using bifurcations of limit cycles and chaotic attractors,” J. Phys.: Conf. Ser., 2018, doi:10.1088/1742-6596/1003/1/012099.
- G. Chen and J. L. Moiola, “An overview of bifurcation, chaos and nonlinear dynamics in control systems,” J. Franklin Inst., vol. 331, no. 6, pp. 819–858, 1994.
- A. Farshidianfar and A. Saghafi, “Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems,” Nonlinear Dyn., vol. 75, no. 4, pp. 783–806, 2014.
- L. V. Stepanova and S. A. Igonin, “Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium,” Appl. Math. Model., vol. 38, no. 14, pp. 3436–3455, 2014.
- M. Jiya, “Application of homotopy perturbation method (HPM) for the solution of non‑linear differential equations,” Pac. J. Sci. Technol., vol. 11, no. 2, pp. 268–272, 2010.
- J. Giné, J. Llibre, K. Wu, and X. Zhang, “Averaging methods of arbitrary order periodic solutions and integrability,” J. Differ. Equations, vol. 260, no. 5, pp. 4130–4156, 2016.
- H. de la Cruz, R. J. Biscay, J. C. Jimenez, and F. Carbonell, “Local linearization–Runge–Kutta methods: a class of A‑stable explicit integrators for dynamical systems,” Math. Comput. Model., vol. 57, nos. 3–4, pp. 720–740, 2013.
- H. H. Goldstine, A History of Numerical Analysis from the 16th through the 19th Century. New York, NY, USA: Springer, 2012.
- K. Atkinson, W. Han, and D. E. Stewart, Numerical Solution of Ordinary Differential Equations. Pure Appl. Math., vol. 108, 2011.
- G. G. Dahlquist, “A special stability problem for linear multistep methods,” BIT, vol. 3, no. 1, pp. 27–43, 1963.
- P. Henrici, Discrete Variable Methods in Ordinary Differential Equations. Hoboken, NJ, USA: Wiley, 1962.
- R. T. Keller and D. U. Qiang, “Discovery of dynamics using linear multistep methods,” SIAM, vol. 59, no. 1, pp. 429–455, 2021.
- M. Musa and M. A. Unwala, “Extended 3‑point super class of block backward differentiation formula for solving stiff initial value problems,” Abacus (Math. Sci. Ser.), vol. 44, no. 1, pp. 584–591, 2019.
- B. S. H. Kashkari and M. I. Syam, “Optimization of one step block method with three hybrid points for solving first‑order ordinary differential equations,” Results Phys., vol. 12, pp. 592–596, 2019.
- S. B. G. Karakoç, A. Saha, and D. Y. Sucu, “A collocation algorithm based on septic B‑splines and bifurcation of traveling waves for Sawada Kotera equation,” Math. Comput. Simul., vol. 203, pp. 12–27, 2023.
- R. Jiwari and S. A. Pandey, “Collocation algorithm based on septic B‑splines and bifurcation of traveling waves for Sawada–Kotera equation,” Math. Comput. Simul., vol. 203, no. 1, pp. 12–27, 2023.
- A. Kiliçman, A. E. Ismail, and Z. Ismail, “A novel scheme based on collocation finite element method to generalised Oskolkov equation,” J. Sci. Arts, vol. 25, no. 4, pp. 895–908, 2021.
- B. M. Saka and F. Erdogan, “A septic B‑spline collocation method for solving the generalized equal width wave equation,” Kuwait J. Sci., vol. 43, no. 1, pp. 20–31, 2016.
Toplam 25 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Diferansiyel ve İntegral Denklemlerin Sayısal Çözümü, Sayısal Analiz |
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 30 Haziran 2025 |
| Gönderilme Tarihi | 12 Şubat 2025 |
| Kabul Tarihi | 24 Haziran 2025 |
| Yayımlandığı Sayı | Yıl 2025 Cilt: 15 Sayı: 1 |
