Numerical verification of the orders of accuracy of truncated asymptotic expansion solutions to the van der Pol equation. (English) Zbl 1470.34147
Summary: We propose a numerical method to verify the orders of accuracy of truncated asymptotic expansion solutions to the van der Pol equation. Our proposed method is simpler than an existing method [E. Deeba and S. Xie, J. Comput. Anal. Appl. 3, No. 2, 165–171 (2001; Zbl 1024.65051)]. Our proposed method does not need any technique of a least-squares fit of error data, whereas the existing one does. Numerical simulations confirm that our proposed method works effectively.
MSC:
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
Keywords:
asymptotic expansion; formal expansion; order of accuracy; perturbation method; van der Pol equationCitations:
Zbl 1024.65051References:
| [1] | Olabode, DL; Miwadinou, CH; Monwanou, VA; Chabi Orou, JB, Effects of passive hydrodynamics force on harmonic and chaotic oscillations in nonlinear chemical dynamics, Phys. D Nonlinear Phenomena., 386-387, 49-59 (2019) · Zbl 1415.92219 · doi:10.1016/j.physd.2018.09.001 |
| [2] | Olabode, DL; Miwadinou, CH; Monwanou, AV; Chabi Orou, JB, Horseshoes, chaos and its passive control in dissipative nonlinear chemical dynamics, Phys. Scr., 93, 8, 85203 (2018) · doi:10.1088/1402-4896/aacef0 |
| [3] | Verhulst, F., Nonlinear Differential Equations and Dynamical Systems (1996), Berlin: Springer, Berlin · Zbl 0854.34002 · doi:10.1007/978-3-642-61453-8 |
| [4] | Miwadinou, CH; Monwanou, AV; Yovogan, J.; Hinvi, LA; Nwagoum Tuwa, PR; Chabi Orou, JB, Modeling nonlinear dissipative chemical dynamics by a forced modified van der Pol-Duffing oscillator with asymmetric potential: Chaotic behaviors predictions, Chin. J. Phys., 56, 3, 1089-1104 (2018) · Zbl 07819495 · doi:10.1016/j.cjph.2018.03.033 |
| [5] | Shchepakina, E.; Korotkova, O., Canard explosion in chemical and optical systems, Discrete Contin. Dyn. Syst. B, 18, 2, 495-512 (2013) · Zbl 1270.34161 · doi:10.3934/dcdsb.2013.18.495 |
| [6] | Spasic, AM; Lazarevic, MP; Mitrovic, MV; Krstic, DN, Electron and momentum transfer phenomena at developed deformable and rigid liquid-liquid interfaces, Chem. Ind. Chem. Eng. Q., 12, 2, 123-132 (2006) · doi:10.2298/CICEQ0602123S |
| [7] | Koper, MTM, Some simple bifurcation sets of an extended van der Pol model and their relation to chemical oscillators, J. Chem. Phys., 102, 13, 5278-5287 (1995) · doi:10.1063/1.469253 |
| [8] | Samardzija, N.; Greller, LD; Wasserman, E., Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems, J. Chem. Phys., 90, 4, 2296-2304 (1989) · doi:10.1063/1.455970 |
| [9] | Anan, Y.; Go, N., Solitary wave and spatially locked solitary pattern in a chemical reaction system, J. Theor. Biol., 80, 2, 171-183 (1979) · doi:10.1016/0022-5193(79)90203-0 |
| [10] | Deeba, E.; Xie, S., The asymptotic expansion and numerical verification of van der Pol’s equation, J. Comput. Anal. Appl., 3, 2, 165-171 (2001) · Zbl 1024.65051 · doi:10.1023/A:1010189225921 |
| [11] | Abdel-Halim Hassan, IH, The asymptotic expansion and numerical verification method for linear and nonlinear initial value problem, Appl. Math. Comput., 180, 1, 29-37 (2006) · Zbl 1103.65078 · doi:10.1016/j.amc.2005.11.146 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
