Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 31, 2018

On a Non-linear Boundary-Layer Problem for the Fractional Blasius-Type Equation

  • Ramiz Tapdigoglu and Berikbol T. Torebek EMAIL logo


In this paper, we consider a non-linear sequential differential equation with Caputo fractional derivative of Blasius type and we reduce the problem to the equivalent non-linear integral equation. We prove the complete continuity of the non-linear integral operator. The theorem on the existence of a solution of the problem for the Blasius equation of fractional order is also proved.

MSC 2010: 2000; 35A09; 34K06


The authors are grateful to Professor Mokhtar Kirane for valuable advices during discussions of the results of the present work. The final version of this paper was completed when Berikbol T. Torebek was visiting the University of La Rochelle. The authors would like to thank the editor and referees for their valuable comments and remarks, which led to a great improvement of the article. The second named author is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant No.AP05131756).


[1] B. Brighi, The equation f'''+ff''+g(f')=0 and the associated boundary value problems, Results in Math. 61 (2012), 355–391.10.1007/s00025-011-0122-0Search in Google Scholar

[2] H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Angew. Math. Phys. 56 (1908), 1–37.Search in Google Scholar

[3] V. M. Falkner, S. W. Skan, Solutions of the boundary layer equations, Phill. Mag. 7 (12) (1931), 865–896.10.1080/14786443109461870Search in Google Scholar

[4] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations. Elsevier, North-Holland. Mathematics studies, 2006, -539p.Search in Google Scholar

[5] I. Podlubny, Fractional differential equations. Mathematics in Science and Engineering, V.198. Academic Press, 1999, -356 p.Search in Google Scholar

[6] K. M. Furati, Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative. Fract. Calc. Appl. Anal. 16 (1) (2013), 171–188.10.2478/s13540-013-0012-0Search in Google Scholar

[7] B. Kh. Turmetov, B. T. Torebek, On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation. New York J. Math. 20 (2014), 1237–1251.Search in Google Scholar

[8] M. B. Glauert, M. J. Lighthill, The axisymmetric boundary layer on a long thin cylinder, Proc. R. Soc. London 320 (1955), 188–203.Search in Google Scholar

[9] H. Schlichting, Boundary Layer Theory, 7 th edition, McGraw Hill, 1951.Search in Google Scholar

[10] L. D. Landau, E. M. Livshitz, Hydrodynamics, Moskow. Nauka. 1988.Search in Google Scholar

[11] V. A. Trenogin, Functional analysis. Moskow, Nauka, 1980.Search in Google Scholar

[12] A. I. Dreglea, N. A. Sidorov, Continuous solutions of some boundary layer problem, Proc. Appl. Math. Mech. 7 (1) (2007), P.2150037–P.2150038.10.1002/pamm.200700194Search in Google Scholar

[13] R. Lin, F. Liu, Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Anal. 66 (2007), 856–869.10.1016/ in Google Scholar

[14] K. Diethelm, N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, (2002) 229–248.10.1006/jmaa.2000.7194Search in Google Scholar

[15] V. Lakshmikanthama, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), 2677–2682.10.1016/ in Google Scholar

[16] B. Ahmad, On nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order, Results Math. 63 (1) (2013), 183–194.10.1007/s00025-011-0187-9Search in Google Scholar

[17] A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (1) (2012), 403–411.10.1016/j.jmaa.2011.11.065Search in Google Scholar

[18] B.-P. Moghaddam, J.-A. Tenreiro Machado, A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels. Fractional Calculus Appl. Anal. 20 (4) (2017), 1023–1042.10.1515/fca-2017-0053Search in Google Scholar

[19] A. Dabiri, E. A. Butcher, Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods. Appl. Math. Modell. 56 (2018), 424–448.Search in Google Scholar

[20] J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180.10.4064/sm-2-1-171-180Search in Google Scholar

[21] A. Friedman, Foundations of modern analysis. Dover Publications, Inc. New York, 1970.Search in Google Scholar

Received: 2017-01-22
Accepted: 2018-05-20
Published Online: 2018-05-31
Published in Print: 2018-07-26

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 7.6.2023 from
Scroll to top button