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
References:
[1]

R. P. Agarwal and D. O'Regan, Singular integral equations arising in Homann flow, Dyn. Cont. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 9 (2002), 481-488.  Google Scholar

[2]

D. J. Acheson, "Elementary Fluid Dynamics," Oxford Univ. Press, New York, 1990.  Google Scholar

[3]

B. Brighi and J. D. Hoernel, On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett., 19 (2006), 69-74. doi: dx.doi.org/10.1016/j.aml.2005.02.038.  Google Scholar

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S. N. Brown and K. Stewartson, On the reversed flow solutions of the Falkner-Skan equations, Mathematika, 13 (1966), 1-6. doi: dx.doi.org/10.1112/S0025579300004125.  Google Scholar

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A. H. Craven and L. A. Peletier, On the uniqueness of solutions of the Falkner-Skan equation, Mathematika, 19 (1972), 129-133. doi: dx.doi.org/10.1112/S0025579300005052.  Google Scholar

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I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, Marcel Dekker, New York, 2003. Google Scholar

[8]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. doi: dx.doi.org/10.1007/978-3-662-00547-7.  Google Scholar

[9]

M. Guedda and Z. Hammouch, On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers, Fluid Dynam. Res., 38 (2006), 211-223. doi: dx.doi.org/10.1016/j.fluiddyn.2005.11.001.  Google Scholar

[10]

M. Guedda, Multiple solutions of mixed convenction boundary-layer approximations in a porous medium, Appl. Math. Lett., 19 (2006), 63-68. doi: dx.doi.org/10.1016/j.aml.2005.02.037.  Google Scholar

[11]

P. Hartman, "Ordinary Differential Equations," Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.  Google Scholar

[12]

P. Hartman, On the existence of similar solutions of some boundary layer problems, SIAM J. Math. Anal., 3 (1972), 120-147. doi: dx.doi.org/10.1137/0503014.  Google Scholar

[13]

S. P. Hastings, An existence theorem for a class of nonlinear boundary value problems including that of Falkner and Skan, J. Differential Equations, 9 (1971), 580-590. doi: dx.doi.org/10.1016/0022-0396(71)90025-8.  Google Scholar

[14]

S. P. Hastings, Reversed flow solutions of the Falkner-Skan equations, SIAM J. Appl. Math., 22 (1972), 329-334. doi: dx.doi.org/10.1137/0122031.  Google Scholar

[15]

K. Q. Lan and G. C. Yang, Positive solutions of the Falker-Skan equation arising in boundary layer theory, Canad. Math. Bull., 51 (2008), 386-398. doi: dx.doi.org/10.4153/CMB-2008-039-7.  Google Scholar

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P. A. Libby and T. M. Liu, Further solutions of the Falkner-Skan equation, AIAA J., 5 (1967), 1040-1042. doi: dx.doi.org/10.2514/3.4130.  Google Scholar

[17]

T. Y. Na, "Computational Methods in Engineering Boundary Value Problems," Academic Press, 1979.  Google Scholar

[18]

H. Schlichting and K. Gersten, "Boundary-Layer Theory," 8ed, Springer-Verlag, Berlin, 2000.  Google Scholar

[19]

K. Schrader, A generalization of the Helly selection theorem, Bull. Amer. Math. Soc., 78 (1972), 415-419. doi: dx.doi.org/10.1090/S0002-9904-1972-12923-9.  Google Scholar

[20]

C. Sparrow and H. P. F. Swinnerton-Dyer, The Falkner-Skan equation. II. Dynamics and the bifurcations of $P$- and $Q$-orbits, J. Differential Equations, 183 (2002), 1-55. doi: dx.doi.org/10.1006/jdeq.2001.4100.  Google Scholar

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K. Stewartson, Further solutions of the Falkner-Skan equation, Proc. Camb. Phil. Soc., 50 (1954), 454-465. doi: dx.doi.org/10.1017/S030500410002956X.  Google Scholar

[22]

H. P. F. Swinnerton-Dyer and C. T. Sparrow, The Falkner-Skan equation. I. The creation of strange invariant sets, J. Differential Equations, 119 (1995), 336-394. doi: dx.doi.org/10.1006/jdeq.1995.1094.  Google Scholar

[23]

K. K. Tam, A note on the existence of a solution of the Falkner-Skan equation, Canad. Math. Bull., 13 (1970), 125-127. doi: dx.doi.org/10.4153/CMB-1970-026-8.  Google Scholar

[24]

J. Wang, W. Gao and Z. Zhang, Singular nonlinear boundary value problems arising in boundary layer theory, J. Math. Anal. Appl., 233 (1999), 246-256. doi: dx.doi.org/10.1006/jmaa.1999.6290.  Google Scholar

[25]

H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. Math., 43 (1942), 381-407. doi: dx.doi.org/10.2307/1968875.  Google Scholar

[26]

G. C. Yang and K. Q. Lan, Nonexistence of the reversed flow solutions of the Falkner-Skan equations, Nonlinear Anal., 74 (2011), 5327-5339. doi: dx.doi.org/10.1016/j.na.2011.05.017.  Google Scholar

[27]

G. C. Yang and K. Q. Lan, The velocity and shear stress functions of the Falkner-Skan equation arising in boundary layer theory, J. Math. Anal. Appl., 328 (2007), 1297-1308. doi: dx.doi.org/10.1016/j.jmaa.2006.06.042.  Google Scholar

[28]

G. C. Yang, Existence of solutions of laminar boundary layer equations with decelerating external flows, Nonlinear Anal., 72 (2010), 2063-2075. doi: dx.doi.org/10.1016/j.na.2009.10.006.  Google Scholar

[29]

G. C. Yang, Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory, Appl. Math. Lett., 16 (2003), 827-832. doi: dx.doi.org/10.1016/S0893-9659(03)90003-6.  Google Scholar

[30]

G. C. Yang, A note on $f'''+ff''+\lambda (1-f^{'2})=0$ with $\lambda\in (-\frac{1}{2},0)$ arising in boundary layer theory, Appl. Math. Lett., 17 (2004), 1261-1265. doi: dx.doi.org/10.1016/j.aml.2003.12.005.  Google Scholar

[31]

G. C. Yang, L. L. Shi and K. Q. Lan, Properties of positive solutions of the Falkner-Skan equation arising in boundary layer theory, Integral Methods in Science and Engineering, 277-283, Birkhuser Boston, Boston, MA, 2008.  Google Scholar

show all references

References:
[1]

R. P. Agarwal and D. O'Regan, Singular integral equations arising in Homann flow, Dyn. Cont. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 9 (2002), 481-488.  Google Scholar

[2]

D. J. Acheson, "Elementary Fluid Dynamics," Oxford Univ. Press, New York, 1990.  Google Scholar

[3]

B. Brighi and J. D. Hoernel, On the concave and convex solutions of a mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett., 19 (2006), 69-74. doi: dx.doi.org/10.1016/j.aml.2005.02.038.  Google Scholar

[4]

S. N. Brown and K. Stewartson, On the reversed flow solutions of the Falkner-Skan equations, Mathematika, 13 (1966), 1-6. doi: dx.doi.org/10.1112/S0025579300004125.  Google Scholar

[5]

W. A. Coppel, On a differential equation of boundary-layer theory, Phil. Trans. Roy. Soc. London, Ser. A, 253 (1960), 101-136. doi: dx.doi.org/10.1112/S0025579300005052.  Google Scholar

[6]

A. H. Craven and L. A. Peletier, On the uniqueness of solutions of the Falkner-Skan equation, Mathematika, 19 (1972), 129-133. doi: dx.doi.org/10.1112/S0025579300005052.  Google Scholar

[7]

I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, Marcel Dekker, New York, 2003. Google Scholar

[8]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. doi: dx.doi.org/10.1007/978-3-662-00547-7.  Google Scholar

[9]

M. Guedda and Z. Hammouch, On similarity and pseudo-similarity solutions of Falkner-Skan boundary layers, Fluid Dynam. Res., 38 (2006), 211-223. doi: dx.doi.org/10.1016/j.fluiddyn.2005.11.001.  Google Scholar

[10]

M. Guedda, Multiple solutions of mixed convenction boundary-layer approximations in a porous medium, Appl. Math. Lett., 19 (2006), 63-68. doi: dx.doi.org/10.1016/j.aml.2005.02.037.  Google Scholar

[11]

P. Hartman, "Ordinary Differential Equations," Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.  Google Scholar

[12]

P. Hartman, On the existence of similar solutions of some boundary layer problems, SIAM J. Math. Anal., 3 (1972), 120-147. doi: dx.doi.org/10.1137/0503014.  Google Scholar

[13]

S. P. Hastings, An existence theorem for a class of nonlinear boundary value problems including that of Falkner and Skan, J. Differential Equations, 9 (1971), 580-590. doi: dx.doi.org/10.1016/0022-0396(71)90025-8.  Google Scholar

[14]

S. P. Hastings, Reversed flow solutions of the Falkner-Skan equations, SIAM J. Appl. Math., 22 (1972), 329-334. doi: dx.doi.org/10.1137/0122031.  Google Scholar

[15]

K. Q. Lan and G. C. Yang, Positive solutions of the Falker-Skan equation arising in boundary layer theory, Canad. Math. Bull., 51 (2008), 386-398. doi: dx.doi.org/10.4153/CMB-2008-039-7.  Google Scholar

[16]

P. A. Libby and T. M. Liu, Further solutions of the Falkner-Skan equation, AIAA J., 5 (1967), 1040-1042. doi: dx.doi.org/10.2514/3.4130.  Google Scholar

[17]

T. Y. Na, "Computational Methods in Engineering Boundary Value Problems," Academic Press, 1979.  Google Scholar

[18]

H. Schlichting and K. Gersten, "Boundary-Layer Theory," 8ed, Springer-Verlag, Berlin, 2000.  Google Scholar

[19]

K. Schrader, A generalization of the Helly selection theorem, Bull. Amer. Math. Soc., 78 (1972), 415-419. doi: dx.doi.org/10.1090/S0002-9904-1972-12923-9.  Google Scholar

[20]

C. Sparrow and H. P. F. Swinnerton-Dyer, The Falkner-Skan equation. II. Dynamics and the bifurcations of $P$- and $Q$-orbits, J. Differential Equations, 183 (2002), 1-55. doi: dx.doi.org/10.1006/jdeq.2001.4100.  Google Scholar

[21]

K. Stewartson, Further solutions of the Falkner-Skan equation, Proc. Camb. Phil. Soc., 50 (1954), 454-465. doi: dx.doi.org/10.1017/S030500410002956X.  Google Scholar

[22]

H. P. F. Swinnerton-Dyer and C. T. Sparrow, The Falkner-Skan equation. I. The creation of strange invariant sets, J. Differential Equations, 119 (1995), 336-394. doi: dx.doi.org/10.1006/jdeq.1995.1094.  Google Scholar

[23]

K. K. Tam, A note on the existence of a solution of the Falkner-Skan equation, Canad. Math. Bull., 13 (1970), 125-127. doi: dx.doi.org/10.4153/CMB-1970-026-8.  Google Scholar

[24]

J. Wang, W. Gao and Z. Zhang, Singular nonlinear boundary value problems arising in boundary layer theory, J. Math. Anal. Appl., 233 (1999), 246-256. doi: dx.doi.org/10.1006/jmaa.1999.6290.  Google Scholar

[25]

H. Weyl, On the differential equations of the simplest boundary-layer problems, Ann. Math., 43 (1942), 381-407. doi: dx.doi.org/10.2307/1968875.  Google Scholar

[26]

G. C. Yang and K. Q. Lan, Nonexistence of the reversed flow solutions of the Falkner-Skan equations, Nonlinear Anal., 74 (2011), 5327-5339. doi: dx.doi.org/10.1016/j.na.2011.05.017.  Google Scholar

[27]

G. C. Yang and K. Q. Lan, The velocity and shear stress functions of the Falkner-Skan equation arising in boundary layer theory, J. Math. Anal. Appl., 328 (2007), 1297-1308. doi: dx.doi.org/10.1016/j.jmaa.2006.06.042.  Google Scholar

[28]

G. C. Yang, Existence of solutions of laminar boundary layer equations with decelerating external flows, Nonlinear Anal., 72 (2010), 2063-2075. doi: dx.doi.org/10.1016/j.na.2009.10.006.  Google Scholar

[29]

G. C. Yang, Existence of solutions to the third-order nonlinear differential equations arising in boundary layer theory, Appl. Math. Lett., 16 (2003), 827-832. doi: dx.doi.org/10.1016/S0893-9659(03)90003-6.  Google Scholar

[30]

G. C. Yang, A note on $f'''+ff''+\lambda (1-f^{'2})=0$ with $\lambda\in (-\frac{1}{2},0)$ arising in boundary layer theory, Appl. Math. Lett., 17 (2004), 1261-1265. doi: dx.doi.org/10.1016/j.aml.2003.12.005.  Google Scholar

[31]

G. C. Yang, L. L. Shi and K. Q. Lan, Properties of positive solutions of the Falkner-Skan equation arising in boundary layer theory, Integral Methods in Science and Engineering, 277-283, Birkhuser Boston, Boston, MA, 2008.  Google Scholar

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