November  2013, 12(6): 2465-2495. doi: 10.3934/cpaa.2013.12.2465

Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations

1. 

Department of Computation Science, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China

2. 

Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada

Received  May 2012 Revised  December 2012 Published  May 2013

We investigate existence of reversed flow solutions of the Falkner-Skan equations by considering a system of two singular Hammerstein integral equations. We prove that the reversed flow solutions exist for each parameter in $(-1/6,0)$. This is an extension of results on nonexistence of reversed flow solutions obtained recently by the authors. As applications of our new results, we obtain existence of reversed flow similarity solutions of the boundary layer equations governing the flow of fluids over surfaces often arising from engineering problems.
Citation: G. C. Yang, K. Q. Lan. Systems of singular integral equations and applications to existence of reversed flow solutions of Falkner-Skan equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2465-2495. doi: 10.3934/cpaa.2013.12.2465
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show all references

References:
[1]

Dyn. Cont. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 9 (2002), 481-488.  Google Scholar

[2]

Oxford Univ. Press, New York, 1990.  Google Scholar

[3]

Appl. Math. Lett., 19 (2006), 69-74. doi: dx.doi.org/10.1016/j.aml.2005.02.038.  Google Scholar

[4]

Mathematika, 13 (1966), 1-6. doi: dx.doi.org/10.1112/S0025579300004125.  Google Scholar

[5]

Phil. Trans. Roy. Soc. London, Ser. A, 253 (1960), 101-136. doi: dx.doi.org/10.1112/S0025579300005052.  Google Scholar

[6]

Mathematika, 19 (1972), 129-133. doi: dx.doi.org/10.1112/S0025579300005052.  Google Scholar

[7]

3rd, Marcel Dekker, New York, 2003. Google Scholar

[8]

Springer-Verlag, Berlin, 1985. doi: dx.doi.org/10.1007/978-3-662-00547-7.  Google Scholar

[9]

Fluid Dynam. Res., 38 (2006), 211-223. doi: dx.doi.org/10.1016/j.fluiddyn.2005.11.001.  Google Scholar

[10]

Appl. Math. Lett., 19 (2006), 63-68. doi: dx.doi.org/10.1016/j.aml.2005.02.037.  Google Scholar

[11]

Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.  Google Scholar

[12]

SIAM J. Math. Anal., 3 (1972), 120-147. doi: dx.doi.org/10.1137/0503014.  Google Scholar

[13]

J. Differential Equations, 9 (1971), 580-590. doi: dx.doi.org/10.1016/0022-0396(71)90025-8.  Google Scholar

[14]

SIAM J. Appl. Math., 22 (1972), 329-334. doi: dx.doi.org/10.1137/0122031.  Google Scholar

[15]

Canad. Math. Bull., 51 (2008), 386-398. doi: dx.doi.org/10.4153/CMB-2008-039-7.  Google Scholar

[16]

AIAA J., 5 (1967), 1040-1042. doi: dx.doi.org/10.2514/3.4130.  Google Scholar

[17]

Academic Press, 1979.  Google Scholar

[18]

8ed, Springer-Verlag, Berlin, 2000.  Google Scholar

[19]

Bull. Amer. Math. Soc., 78 (1972), 415-419. doi: dx.doi.org/10.1090/S0002-9904-1972-12923-9.  Google Scholar

[20]

J. Differential Equations, 183 (2002), 1-55. doi: dx.doi.org/10.1006/jdeq.2001.4100.  Google Scholar

[21]

Proc. Camb. Phil. Soc., 50 (1954), 454-465. doi: dx.doi.org/10.1017/S030500410002956X.  Google Scholar

[22]

J. Differential Equations, 119 (1995), 336-394. doi: dx.doi.org/10.1006/jdeq.1995.1094.  Google Scholar

[23]

Canad. Math. Bull., 13 (1970), 125-127. doi: dx.doi.org/10.4153/CMB-1970-026-8.  Google Scholar

[24]

J. Math. Anal. Appl., 233 (1999), 246-256. doi: dx.doi.org/10.1006/jmaa.1999.6290.  Google Scholar

[25]

Ann. Math., 43 (1942), 381-407. doi: dx.doi.org/10.2307/1968875.  Google Scholar

[26]

Nonlinear Anal., 74 (2011), 5327-5339. doi: dx.doi.org/10.1016/j.na.2011.05.017.  Google Scholar

[27]

J. Math. Anal. Appl., 328 (2007), 1297-1308. doi: dx.doi.org/10.1016/j.jmaa.2006.06.042.  Google Scholar

[28]

Nonlinear Anal., 72 (2010), 2063-2075. doi: dx.doi.org/10.1016/j.na.2009.10.006.  Google Scholar

[29]

Appl. Math. Lett., 16 (2003), 827-832. doi: dx.doi.org/10.1016/S0893-9659(03)90003-6.  Google Scholar

[30]

Appl. Math. Lett., 17 (2004), 1261-1265. doi: dx.doi.org/10.1016/j.aml.2003.12.005.  Google Scholar

[31]

Integral Methods in Science and Engineering, 277-283, Birkhuser Boston, Boston, MA, 2008.  Google Scholar

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