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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi

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2191-0294
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Volume 19, Issue 5

Issues

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

Ramiz Tapdigoglu
  • LaSIE, Faculté des Sciences et de Technologies, University of La Rochelle. A. M. Crépeau, 17042 La Rochelle, France
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/ Berikbol T. Torebek
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  • Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 50010 Almaty, Kazakhstan; Al-Farabi Kazakh National University, 71 Al-Farabi ave., 50040 Almaty, Kazakhstan
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Published Online: 2018-05-31 | DOI: https://doi.org/10.1515/ijnsns-2017-0018

Abstract

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.

Keywords: Blasius equation; fractional derivative; Shauder’s principle; completely continuity operator

MSC 2010: 2000; 35A09; 34K06

References

  • [1]

    B. Brighi, The equation f'''+ff''+g(f')=0 and the associated boundary value problems, Results in Math. 61 (2012), 355–391.Web of ScienceCrossrefGoogle Scholar

  • [2]

    H. Blasius, Grenzschichten in Flüssigkeiten mit kleiner Reibung, Z. Angew. Math. Phys. 56 (1908), 1–37.

  • [3]

    V. M. Falkner, S. W. Skan, Solutions of the boundary layer equations, Phill. Mag. 7 (12) (1931), 865–896.

  • [4]

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

  • [5]

    I. Podlubny, Fractional differential equations. Mathematics in Science and Engineering, V.198. Academic Press, 1999, -356 p.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.

  • [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.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.

  • [9]

    H. Schlichting, Boundary Layer Theory, 7 th edition, McGraw Hill, 1951.

  • [10]

    L. D. Landau, E. M. Livshitz, Hydrodynamics, Moskow. Nauka. 1988.

  • [11]

    V. A. Trenogin, Functional analysis. Moskow, Nauka, 1980.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.Google Scholar

  • [13]

    R. Lin, F. Liu, Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Anal. 66 (2007), 856–869.

  • [14]

    K. Diethelm, N. J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265, (2002) 229–248.CrossrefWeb of ScienceGoogle Scholar

  • [15]

    V. Lakshmikanthama, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. 69 (2008), 2677–2682.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.Web of ScienceCrossrefGoogle 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.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.

  • [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.

  • [20]

    J. Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Math. 2 (1930), 171–180.Google Scholar

  • [21]

    A. Friedman, Foundations of modern analysis. Dover Publications, Inc. New York, 1970.Google Scholar

About the article

Accepted: 2018-05-20

Received: 2017-01-22

Published Online: 2018-05-31

Published in Print: 2018-07-26


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 5, Pages 493–498, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0018.

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