Study of a similarity boundary layer equation by using the shooting method
DOI:
https://doi.org/10.66147/lnaa.20253256Abstract
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 DOI: 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 DOI: 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. DOI: https://doi.org/10.66147/lnaa.20242124
[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. DOI: https://doi.org/10.7153/dea-09-06
[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. DOI: https://doi.org/10.1016/S0997-7546(03)00059-1
[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 DOI: 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 DOI: 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 DOI: 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 DOI: 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 DOI: 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. DOI: https://doi.org/10.46939/J.Sci.Arts-22.2-a09
[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. DOI: https://doi.org/10.1007/3-7643-7384-9_9
[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. DOI: https://doi.org/10.1016/j.aml.2005.02.038
[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. DOI: https://doi.org/10.1007/s00025-011-0122-0
[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. DOI: https://doi.org/10.1016/0017-9310(77)90059-X
[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. DOI: https://doi.org/10.1016/0017-9310(76)90078-8
[20] M. Guedda, Multiple solutions of mixed convection boundary-layer approximations in a porous medium. Appl. Math. Lett. 19(1), (2006) 63-68. DOI: https://doi.org/10.1016/j.aml.2005.02.037
[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 DOI: https://doi.org/10.66147/lnaa.20231321
22] E.M. Sparrow, R. Eichhorn, J.L. Gregg, Combined forced and free convection in boundary layer flow, Phys. Fluids 2 (1959) 319-328. DOI: https://doi.org/10.1063/1.1705928
[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. DOI: https://doi.org/10.1016/j.aml.2014.05.007
[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. DOI: https://doi.org/10.1016/j.euromechflu.2013.08.005
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