Study of a similarity boundary layer equation by using the shooting method

Authors

Mohamed Boulekbache
Abdelkrim Salim Hassiba Benbouali University of Chlef Corresponding Author

Abstract

In this paper we are interested in the study of the existence or nonexistence and uniqueness or nonuniqueness of the solutions of the boundary value problem involving a third order ordinary nonlinear autonomous differential equation satisfying a boundary conditions. Its solutions are the similarity solutions of a problem of boundary-layer theory.

References

[1] S. Abbas, B. Ahmad, M. Benchohra and A. Salim, Fractional Difference, Differential Equations and Inclusions: Analysis and Stability, Morgan Kaufmann, Cambridge, 2024. https://doi.org/10.1016/C2023-0-00030-9

[2] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On the solution of a boundary value problem associated with a fractional differential equation, Math. Methods Appl. Sci. 47 (2024), 10928-10939. https://doi.org/10.1002/mma.6652

[3] H. Afshari and M. N. Sahlan, The existence of solutions for some new boundary value problems involving the q-derivative operator in quasi-b-metric and b-metric-like spaces, Lett. Nonlinear Anal. Appl. 2 (1) (2024), 16-22.

[4] M. Aiboudi, I. Bensari-Khelil, B. Brighi, Similarity solutions of mixed convection boundary-layer flows in a porous medium. Differ. Equa. and Appl., 9(1)(2017), 69-85.

[5] M. Aiboudi, K. Boudjema Djeffal, B. Brighi, On the convex and convex-concave solutions of opposing mixed convection boundary layer flow in a porous medium. Abst. and Appl. Anal., (2008) ID 4340204.

[6] E.H. Aly, L. Elliott, D.B. Ingham, Mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. Eur. Jour. Mech. B/Fluids 22,(2003) 529-543.

[7] M. Benchohra, S. Bouriah, A. Salim and Y. Zhou, Fractional Differential Equations: A Coincidence Degree Approach, Berlin, Boston: De Gruyter, 2024. https://doi.org/10.1515/9783111334387

[8] M. Benchohra, E. Karapınar, J. E. Lazreg and A. Salim, Advanced Topics in Fractional Differential Equations: A Fixed Point Approach, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-26928-8

[9] M. Benchohra, E. Karapınar, J. E. Lazreg and A. Salim, Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives, Springer, Cham, 2023. https://doi.org/10.1007/978-3-031-34877-8

[10] M. Benchohra, G. M N’Guérékata and A. Salim, Advanced Topics on Semilinear Evolution Equations, Hackensack, NJ, World Scientific, 2025. https://doi.org/10.1142/14092

[11] M. Benchohra, A. Salim and Y. Zhou, Integro-Differential Equations: Analysis, Stability and Controllability, Berlin, Boston: De Gruyter, 2024. https://doi.org/10.1515/9783111437910

[12] M. Boulekbache, K. Boudjema Djeffal, M. Aiboudi, On the solutions of boundary value problem arising in mixed convection. Appl. Math. E-notes, 22(2022), 585-591.

[13] K. Boudjema Djeffal, K. Bouazzaoui, Aiboudi M., An extension result of the mixed convection problem boundary layer flow over a vertical permeable surface embedded in a porous medium. Int. Jour. Math. Compt. meth. vol. 5 (2020).

[14] B. Brighi, J.-D. Hoernel, Recent advances on similarity solutions arising during free convection, Prog. in Nonlin. Diff. Equ. and Their Appl. Vol. 63, (2005), 83-92.

[15] B.Brighi, J.-D. Hoernel, On general similarity boundary layer equation. Acta Math. Univ. Comenian. 77(2008),9-22.

[16] B. Brighi, J.-D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium. Appl. Math. Lett. 19(1),(2006) 69-74.

[17] B. Brighi, The equation f ′′′ + f f ′′ + g(f ′) = 0 and the associated boundary value problems. Results Math. 61 (3-4),(2012) 355-391.

[18] P. Cheng, Similarity solutions for mixed convection from horizontal impermeable surfaces in saturated porous media. Int. J. Heat Mass Transfer. Vol 20 (1976) 893-898.

[19] P. Cheng, I-Dee Chang, Buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces. Int. J. Heat Mass Transfer. Vol 19 (1976) 1267-1272.

[20] M. Guedda, Multiple solutions of mixed convection boundary-layer approximations in a porous medium. Appl. Math. Lett. 19(1), (2006) 63-68.

[21] A. Salim and M. Benchohra, A study on tempered (k, ψ)-Hilfer fractional operator, Lett. Nonlinear Anal. Appl. 1 (3) (2023), 101-121. https://doi.org/10.5281/zenodo.8361961

22] E.M. Sparrow, R. Eichhorn, J.L. Gregg, Combined forced and free convection in boundary layer flow, Phys. Fluids 2 (1959) 319-328.

[23] 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.

[24] G.C. Yang, L. Zhang, L.F. Dang, Existence and nonexistence of solutions on opposing mixed convection problems in boundary layer theory. Eur. Jour. of Mech. B/Fluids 43, (2014)148-153.