Main Article Content
Boundary value problems arise in many real applications such as nanofluids and other areas of applied sciences. The temperature/nanoparticles concentration are usually expressed as singular 2ndorder ODEs. So, it is a challenge to obtain the exact solution of these problems due to the difficulty of the singularity encountered in the governing equations. By means of a suitable transformation, a direct approach is introduced to solve a general class of 2nd-order ODEs. The efficiency of the obtained results is validated through selected problems in the literature. It is found that several existing solutions can be deduced as special cases of our generalized one. Moreover, the present results may be invested for similar future problems in fluid mechanics, especially nanofluids.
- M.A.A. Hamad, Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field, Int. Commun. Heat Mass Transfer 38 (2011), 487-492.
- P.K. Kameswaran, M. Narayana, P. Sibanda and P.V.S.N. Murthy, Hydromagnetic nanofluid flow due to a stretching or shrinking sheet with viscous dissipation and chemical reaction effects, Int. J. Heat Mass Transfer, 55 (2012), 7587-7595.
- E.H. Aly and A. Ebaid, Exact analytical solution for suction and injection flow with thermal enhancement of five nanofluids over an isothermal stretching sheet with effect of the slip model: a comparative study, Abstr. Appl. Anal. 2013 (2013), Article ID 721578.
- E.H. Aly and A. Ebaid, New exact solutions for boundary-layer flow of a nanofluid past a stretching sheet, J. Comput. Theor. Nanosci. 10 (11) (2013), 2591-2594.
- W.A. Khan, Z.H. Khan and M. Rahi, Fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary. Appl. Nanosci. 4 (2014), 633-641.
- A. Ebaid, F. Al Mutairi and S.M. Khaled, Effect of velocity slip boundary condition on the flow and heat transfer of Cu-water and TiO2-water nanofluids in the presence of a magnetic field, Adv. Math. Phys. 2014 (2014), Article ID 538950.
- A. Ebaid and M. Al Sharif, Application of Laplace transform for the exact effect of a magnetic field on heat transfer of carbon-nanotubes suspended nanofluids, Z. Naturforsch. A. 70 (6) (2015), 471-475.
- E.R. EL-Zahar, A.M. Rashad and A.M. Gelany, Studying High Suction Effect on Boundary-Layer Flow of a Nanofluid on Permeable Surface via Singular Perturbation Technique, J. Comput. Theor. Nanosci. 12 (11) (2015), 4828-4836.
- E.H. Aly and A. Ebaid, Exact analysis for the effect of heat transfer on MHD and radiation Marangoni boundary layer nanofluid flow past a surface embedded in a porous medium, J. Molecular Liquids, 215 (2016), 625-639.
- J.P. Boyd, Pade-approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Comput. Phys. 11 (3) (1997), 299-303.
- A.M. Wazwaz, The modified decomposition method and Pade approximants for solving the Thomas-Fermi equation, Appl. Math. Comput. 105 (1) (1999), 11-19.
- A.M. Wazwaz, The modified decomposition method and Pade approximants for a boundary layer equation in unbounded domain, Appl. Math. Comput. 177 (2) (2006), 737-744.
- E.R. EL-Zahar and Y.S. Hamed, An Algorithm for Solving Second Order Nonlinear Singular Perturbation Boundary Value Problems, J. Mod. Methods Numer. Math. 2 (1-2) (2011), 21-31.
- E.R. EL-Zahar, Approximate Analytical Solutions for Singularly Perturbed Boundary Value Problems by Multi-Step Differential Transform Method, J. Appl. Sci. 12 (19) (2012), 2026-2034.
- E.R. El-Zahar, Applications of Adaptive Multi step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering, Appl. Math. Inform. Sci. 9 (1) (2015), 223-232.
- E.R. EL-Zahar and S.M.M. EL-Kabeir, Approximate Analytical Solution of Nonlinear Third-Order Singularly Perturbed BVPs using Homotopy Analysis-Pade Method, J. Comput. Theor. Nanosci. 13 (2016), 8917-8927.
- A. Ebaid and E.H. Aly, Exact analytical solution of the peristaltic nanofluids flow in an asymmetric channel with flexible walls: Application to cancer treatment, Comput. Math. Methods Med. 2013 (2013), Article ID 825376.
- A. Ebaid, Remarks on the homotopy perturbation method for the peristaltic flow of Jeffrey fluid with nano-particles in an asymmetric channel, Comput. Math. Appl. 68 (3) (2014), 77-85.
- A. Ebaid and S.M. Khaled, An Exact Solution for a Boundary Value Problem with Application in Fluid Mechanics and Comparison with the Regular Perturbation Solution, Abstr. Appl. Anal. 2014 (2014), Article ID 172590.
- A. Ebaid, A.M. Wazwaz, E. Alali and B. Masaedeh, Hypergeometric Series Solution to a Class of Second-Order Boundary Value Problems via Laplace Transform with Applications to Nanouids, Commun. Theor. Phys. 67 (2017), 231.
- S.M. Khaled, The exact effects of radiation and joule heating on magnetohydrodynamic Marangoni convection over a flat surface, Therm. Sci. 22 (2018), 63-72.
- M. Qasim, Heat and mass transfer in a Jeffrey fluid over a stretching sheet with heat source/sink, Alexandria Eng. J. 52 (2013), 571-575.
- A. Ebaid, E. Alali, H.S. Ali, The exact solution of a class of boundary value problems with polynomial coefficients and its applications on nanofluids, J. Assoc. Arab Univ. Basic Appl. Sci. 24 (2017), 156-159.
- H.S. Ali, Elham Alali, A. Ebaid and F.M. Alharbi, Analytic Solution of a Class of Singular Second-Order Boundary Value Problems with Applications, Mathematics, 7 (2019), 172.
- Q.M. Al-Mdallal, N. Indumathi, B. Ganga and A.K. Abdul Hakeem, Marangoni radiative effects of hybrid-nanofluids flow past a permeable surface with inclined magnetic field, Case Stud. Therm. Eng. 17 (2020), 100571.