On the Solutions of Falkner-Skan Equation

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Labbaoui Fatma, Aiboudi Mohammed, On the Solutions of Falkner-Skan Equation, Int. J. Anal. Appl., 18 (3) (2020), 409-420.

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Labbaoui Fatma, Aiboudi Mohammed

Abstract

We consider the differential equation f''' +ff'' +β(f'^2 - 1) = 0, with β > 0. In order to prove the existence of solutions satisfying the boundary conditions f (0) = a ≥ 0, f'(0) = b ≥ 0 and f'(+∞) = -1 or 1 for 0 < β ≤ 1/2 . We use shooting technique and consider the initial conditions f (0) = a, f'(0) = b and f”˜'(0) = c. We prove that there exists an infinitely many solutions such that f'(+∞) = 1.

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References

  1. M. Aiboudi and B. Brighi, On the solutions of a boundary value problem arising in free convection with prescribed heat ux, Arch. Math. 93(2), 2009, 165-174.
  2. M. Aiboudi, K. Boudjema Djeffal, and B. Brighi, On the Convex and Convex-Concave Solutions of Opposing Mixed Convection Boundary Layer Flow in a Porous Medium, Abstr. Appl. Anal. 2018 (2018), Article ID 4340204, 5 pages.
  3. M. Aiboudi, I. Bensari and B. Brighi, Similarity solutions of mixed convection boundary-layer flows in a porous medium, Differ. Equ. Appl. 9(1)(2017), 69-85.
  4. B. Brighi, The equation f''' +ff'' +g(f') = 0 and the associated boundary value problems, Results Math. 61(3-4)(2012), 355-391.
  5. B. Brighi, A. Fruchard and T. Sari, On the Blasius problem, Adv. Differ. Equ, 13(5-6), 2008, 509-600.
  6. B. Brighi and J.D. Hoernel, On similarity solutions for boundary layer ows with prescribed heat flux, Math. Methods Appl. Sci. 28(4)(2005), 479-503.
  7. B. Brighi and J.D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium, Appl. Math. Lett. 19(2006), 69-74.
  8. B. Brighi and J-D. Hoernel, On a general similarity boundary layer equation, Acta Math. Univ. Comenian. 77(1)(2008), 9-22.
  9. D. R. Hartree, On an equation occurring in Falkner and Skan's approximate treatment of the equations of the boundary layer, Math. Proc. Cambridge Phil. Soc. 33(2)(1937), 223-239.
  10. G.C. Yang, A note on f''' + ff'' + β(f'^2 - 1) = 0 with λ ∈ (-1/2 , 0) arising in boundary layer theory, Appl. Math. Lett. 17(2004), 1261-1265.
  11. G.C. Yang and J. Li, A new result on f''' + ff'' + β(f'2 - 1) = 0 arising in boundary layer theory, Nonlinear Funct. Anal. Appl. 10(2005), 117-122.
  12. G.C. Yang, L.L. Shi and K.Q. Lan, Properties of positive solutions of the Falkner-Skan equation arising in boundary layer theory, in: Integral Methods in Science and Engineering, Birkhauser, Boston, Boston, MA, 2008, 277-283.
  13. G.C. Yang, Existence of solutions of laminar boundary layer equations with decelerating external flows, Nonlinear Anal., Theory Methods Appl. 72(2010), 2063-2075.
  14. G.C.Yang, An extension result of the opposing mixed convection problem arising in boundary layer theory, Appl. Math. Lett. 38(1)(2014), 180-185.
  15. H. Weyl, On the Differential Equations of the Simplest Boundary-Layer Problems, Ann. Math. 43(2)(1942), 381-407.
  16. W. A. Coppel, On a Differential Equation of Boundary-Layer Theory, Phil. Trans. R. Soc. Lond. Ser. A, Math. Phys. Eng. Sci. 253(1023)(1960), 101-136.
  17. P. Hartman, Ordinary Differential Equations, JohnWiley & Sons, New York, NY, USA, 1964.