American Institute of Mathematical Sciences

Advanced
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

G. C. Yang 1, and  K. Q. Lan 2,

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.
Keywords: reversed flow solutions, Schauder fixed point theorem, Falkner-Skan equation, boundary layer equations., System of singular integral equations, Helly selection theorem.
Mathematics Subject Classification: Primary: 34B16, 34B40; Secondary: 47H10, 76D1.
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.

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.

[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.

[1]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022045
[2]

Sergei Ivanov. On Helly's theorem in geodesic spaces. Electronic Research Announcements, 2014, 21: 109-112. doi: 10.3934/era.2014.21.109
[3]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[4]

Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855
[5]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775
[6]

Patricia J.Y. Wong. Existence of solutions to singular integral equations. Conference Publications, 2009, 2009 (Special) : 818-827. doi: 10.3934/proc.2009.2009.818
[7]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687
[8]

Long Fan, Cheng-Jie Liu, Lizhi Ruan. Local well-posedness of solutions to the boundary layer equations for compressible two-fluid flow. Electronic Research Archive, 2021, 29 (6) : 4009-4050. doi: 10.3934/era.2021070
[9]

Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155
[10]

Wenxiong Chen, Congming Li. Regularity of solutions for a system of integral equations. Communications on Pure and Applied Analysis, 2005, 4 (1) : 1-8. doi: 10.3934/cpaa.2005.4.1
[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320
[12]

Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152
[13]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252
[14]

Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010
[15]

Oleksandr Boichuk, Victor Feruk. Boundary-value problems for weakly singular integral equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1379-1395. doi: 10.3934/dcdsb.2021094
[16]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248
[17]

Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Juan L. G. Guirao, Y. S. Hamed. Link theorem and distributions of solutions to uncertain Liouville-Caputo difference equations. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 427-440. doi: 10.3934/dcdss.2021083
[18]

Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure and Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511
[19]

L. Ke. Boundary behaviors for solutions of singular elliptic equations. Conference Publications, 1998, 1998 (Special) : 388-396. doi: 10.3934/proc.1998.1998.388
[20]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy