# American Institute of Mathematical Sciences

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 and 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. [2] D. J. Acheson, "Elementary Fluid Dynamics," Oxford Univ. Press, New York, 1990. [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. [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. [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. [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. [7] I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, Marcel Dekker, New York, 2003. [8] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. doi: dx.doi.org/10.1007/978-3-662-00547-7. [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. [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. [11] P. Hartman, "Ordinary Differential Equations," Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. [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. [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. [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. [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. [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. [17] T. Y. Na, "Computational Methods in Engineering Boundary Value Problems," Academic Press, 1979. [18] H. Schlichting and K. Gersten, "Boundary-Layer Theory," 8ed, Springer-Verlag, Berlin, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [2] D. J. Acheson, "Elementary Fluid Dynamics," Oxford Univ. Press, New York, 1990. [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. [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. [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. [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. [7] I. G. Currie, "Fundamental Mechanics of Fluids," 3rd, Marcel Dekker, New York, 2003. [8] K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, Berlin, 1985. doi: dx.doi.org/10.1007/978-3-662-00547-7. [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. [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. [11] P. Hartman, "Ordinary Differential Equations," Classics in Applied Mathematics, 38, the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. [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. [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. [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. [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. [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. [17] T. Y. Na, "Computational Methods in Engineering Boundary Value Problems," Academic Press, 1979. [18] H. Schlichting and K. Gersten, "Boundary-Layer Theory," 8ed, Springer-Verlag, Berlin, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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