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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 290287, 2 pages
Periodic Solution of the Hematopoiesis Equation
National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 Ren-Ai Road, Suzhou 215123, China
Received 4 December 2012; Accepted 30 December 2012
Copyright © 2013 Ji-Huan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Wu and Liu (2012) presented some results for the existence and uniqueness of the periodic solutions for the hematopoiesis model. This paper gives a simple approach to find an approximate period of the model.
Equation (1) admits periodic solutions as revealed in . Hereby we suggest a simple approach to the search for an approximate period of (1) using a simple amplitude-frequency formulation [2–5]. To this end, we rewrite (1) in the form
Assume that the periodic solution can be expressed in the form
Setting , , and , , respectively, we have
This formulation has been widely used to solve periodic solutions of various nonlinear oscillators [6–13], and it is often called as He’s frequency formulation, He’s amplitude-frequency formulation, or He’s frequency-amplitude formulation. In case , no period solution is admitted. A similar criterion is given for a nonlinear equation arising in electrospinning process .
The work is supported by PAPD (a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions), National Natural Science Foundation of China under Grant no. 10972053, and Project for Six Kinds of Top Talents in Jiangsu Province, China (Grant no. ZBZZ-035).
- J. Wu and Y. Liu, “Fixed point theorems and uniqueness of the periodic solution for the hematopoiesis models,” Abstract and Applied Analysis, vol. 2012, Article ID 387193, 7 pages, 2012.
- J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.
- J. H. He, “Asymptotic methods for Solitary Solutions and Compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.
- J. H. He, “Comment on ‘He's frequency formulation for nonlinear oscillators’,” European Journal of Physics, vol. 29, no. 4, pp. L19–L22, 2008.
- J. H. He, “An improved amplitude-frequency formulation for nonlinear oscillators,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 211–212, 2008.
- Z. F. Ren, “He’s frequency-amplitude formulation for nonlinear oscillators,” International Journal of Modern Physics B, vol. 25, no. 17, pp. 2379–2382, 2011.
- D. Younesian, H. Askari, Z. Saadatnia, and M. KalamiYazdi, “Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3222–3228, 2010.
- A. E. Ebaid, “Analytical periodic solution to a generalized nonlinear oscillator: application of He's frequency-amplitude formulation,” Mechanics Research Communications, vol. 37, no. 1, pp. 111–112, 2010.
- H.-L. Zhang, “Application of He's amplitude-frequency formulation to a nonlinear oscillator with discontinuity,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2197–2198, 2009.
- X.-C. Cai and W.-Y. Wu, “He's frequency formulation for the relativistic harmonic oscillator,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2358–2359, 2009.
- Y.-N. Zhang, F. Xu, and L.-l. Deng, “Exact solution for nonlinear Schrödinger equation by He's frequency formulation,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2449–2451, 2009.
- A. E. Ebaid, “Oscillations in an potential via he's frequency-amplitude formulation,” Zeitschrift fur Naturforschung A, vol. 64, no. 12, pp. 877–878, 2009.
- Z. F. Ren, G. Q. Liu, Y. X. Kang et al., “Application of He's amplitude-frequency formulation to nonlinear oscillators with discontinuities,” Physica Scripta, vol. 80, no. 4, Article ID 045003, 2009.
- J. H. He, H. Y. Kong, R. R. Yang, et al., “Review on fiber morphology obtained by the bubble electrospinning and Blown bubble spinning,” Thermal Science, vol. 16, no. 5, pp. 1263–1279, 2012.