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| Title: | 非線性微分方程式 t^2u";=u^p
On the nonlinear differential equation t^2u";=u^p |
| Authors: | 姚信宇 |
| Contributors: | 李明融
謝宗翰
姚信宇 |
| Keywords: | 正解的爆炸時間
正解的最大存在時間
Emden-Fowler方程式
blow-up time for positive solution
the life-span for positive solution
Emden-Fowler equation |
| Date: | 2011 |
| Issue Date: | 2012-10-30 11:07:20 (UTC+8) |
| Abstract: | 回顧一個重要的非線性二階方程式
d/dt(t^p(du/dt))+(-)t^(sigma)u^n=0,
這個方程式有許多有趣的物理應用，以Emden方程式的形式發生在天體物理學中；也以Fermi-Thomas方程式的形式出現在原子物理內。對於此類型的非線性方程式可以用來更頻繁且深入的探討數學物理，雖然目前仍存在著些許不確定性，不過如果在未來能有更全面的了解，這將有助於用來決定物理解的性質。
在這篇論文當中，我們討論微分方程式
t^2u"=u^p，p屬於N-{1},
其正解的性質。這個方程式是著名的 Emden-Fowler 方程式的一種特殊情形， 我們可以得到其解的一些有趣的現象及結果。
Recall the important nonlinear second-order equation
d/dt(t^p(du/dt))+(-)t^(sigma)u^n=0,
this equation has several interesting physical applications, occurring in astrophysics in the form of the Emden equation and in atomic physics in the form of the Fermi-Thomas equation. These seems a little doubt that nonlinear equations of this type would enter with greater frequency into mathematical physics, were it more widely known with what ease the properties of the physical solutions can be determined.
In this paper we discuss the property of positive solution of the ordinary differential equation
t^2u"=u^p, p belongs to N-{1},
this equation is a special case of the well-known Emden-Fowler equation, we obtain some interesting phenomena and resulits for solutions. |
| Reference: | 1. M. R. Li, Nichlineare Wellengleichungen 2. Ordnung auf beschränkten Gebieten. PhD-Dissertation Täbingen 1994.
2. M. R. Li, Estimates for the life-span of solutions for semilinear wave equations. Proceedings of the Workshop on Differential Equations V. National Tsing-Hua Uni. Hsinchu, Taiwan, Jan. 10-11, 1997.
3. M. R. Li, On the blow-up time and blow-up rate of positive solutions of semilinear wave equations □u-u^{p}=0. in 1-dimensional space. Commun Pure Appl Anal, to appear.
4. M. R. Li, Estimates for the life-span of solutions of semilinear wave equations. Commun Pure Appl Anal, 2008, 7(2): 417-432.
5. M. R. Li, On the semilinear wave equations. Taiwanese J Math, 1998, 2(3): 329-345.
6. M. R. Li, L. Y. Tsai, On a system of nonlinear wave equations. Taiwanese J Math, 2003, 7(4): 555-573 .
7. M. R. Li, L. Y. Tsai, Existence and nonexistence of global solutions of some systems of semilinear wave equations. Nonlinear Analysis, 2003, 54: 1397-1415.
8. M. R. Li, J. T. Pai, Quenching problem in some semilinear wave equations. Acta Math Sci, 2008, 28B(3): 523-529.
9. R. Duan, M. R. Li, T. Yang, Propagation of singularities in the solutions to the Boltzmann equation near equilibrium. Math Models Methods Appl Sci, 2008, 18(7): 1093-1114.
10. M. R. Li, On the generalized Emden-Fowler Equation u"(t)u(t)=c1+c2u`(t)² with c1>=0, c2>=0. Acta Math Sci, 2010 30B(4): 1227-1234.
11. T.H. Shieh, M. R. Li, Numerical treatment of contact discontinuously with multi-gases. J Comput Appl Math, 2009, 230(2): 656-673 .
12. M. R. Li, Y.J. Lin, T.H. Shieh, The flux model of the movement of tumor cells and health cells using a system of nonlinear heat equations. Journal of Computational Biology, vol. 18, No. 12, 2011, pp.1831-1839.
13. M. R. Li, T. H. Shieh, C. J. Yue, P. Lee, Y. T. Li, Parabola method in ordinary differential equation. Taiwanese J Math, vol. 15, No 4, 2011, pp.1841-1857.
14. R. Bellman, Stability Theory of Differential Equations. Yew York: McGraw-Hill, 1953.
15. E. Hille, National Academy of Sciences of the United States of America, Volume 62, issue1, 1968, pp.7-10. |
| Description: | 碩士
國立政治大學
應用數學研究所
95751013
100 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0095751013 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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| 101301.pdf | | 366Kb | Adobe PDF2 | 643 | View/Open | | 101302.pdf | | 732Kb | Adobe PDF2 | 618 | View/Open | | 101303.pdf | | 366Kb | Adobe PDF2 | 595 | View/Open |
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