# AN EFFICIENT SPECTRAL SOLUTION FOR UNSTEADY BOUNDARY LAYER FLOW AND HEAT TRANSFER DUE TO A STRETCHING SHEET

## Main Article Content

## Abstract

In this paper, an efficient Spectral Collocation method based on the shifted Legen- dre polynomials is applied to study the unsteady boundary-layer flow and heat transfer due to a stretching sheet. A similarity transformation is used to reduce the govern- ing unsteady boundary layer equations to a system of nonlinear ordinary differential equations. Then, the shifted Legendre polynomials and their operational matrix of derivative are used for producing an analytical aproximate solution of this system of nonlinear ordinary differential equations. The main advantage of the proposed method is that the need for guessing and correcting the initial values during the solution proce- dure is eliminated and a stable solution with good accuracy can be obtained by using the given boundary conditions in the problem. A very good agreement is observed between the obtained results by the proposed Spectral Collocation method and those of previously published ones.

## Article Details

**Thermal Science**, [S.l.], mar. 2017. ISSN 2334-7163. Available at: <http://thermal-science.tech/journal/index.php/thsci/article/view/2130>. Date accessed: 18 oct. 2017. doi: https://doi.org/10.2298/TSCI150329097M.

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Accepted 2017-03-13

Published 2017-03-13

## References

[2] E. Babolian, M. M. Hosseini, A Modified Spectral Method for Numerical Solution of Ordinary Differential Equations with Non-analytic Solution, Appl. Math. Comput., 132 (2002), 341351.

[3] F. Mohammadi, M. M. Hosseini, Syed Tauseef Mohyud-Din. Legendre wavelet galerkin method for solving ordinary differential equations with non-analytic solution. Int. J. Syst. Sci. 42 (4) (2011) 579-585.

[4] H. A. Khater, R. S. Temsah, M. M. Hassan. A Chebyshev spectral collocation method for solving Burgers-type equations. J. Comput. Appl. Math. 222 (2) (2008) 333-350.

[5] M. Kamrani, and S. M. Hosseini. Spectral collocation method for stochastic Burgers equation driven by additive noise, Math. Comput. Simul. 82 (9) (2012) 1630-1644.

[6] M. R. Malik, T. A. Zang, M.Y Hussaini. A spectral collocation method for the Navier- Stokes equations. J. Comput. Phys. 61 (1) (1985) 64-88.

[7] A. Karageorghis, T. N. Phillips, A. R. Davies. Spectral collocation methods for the primary two-point oundary value problem in modelling viscoelastic flows. Int. J. Numer. Methods. Eng. 26 (4) (1988) 805-813.

[8] H. Chen, Y. Su, B. D. Shizgal. A direct spectral collocation Poisson solver in polar and cylindrical coordinates. J. Comput. Phys. 160 (2) (2000) 453-469.

[9] Y. Chen, T. Tang, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput. 79 (269) (2010) 147-167.

[10] S. Nemati, P. M. Lima, Y. Ordokhani. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials. J. Comput. Appl. Math. 242 (2013) 53-69.

[11] C. D. Pruett, C. L. Streett, A spectral collocation method for compressible, non- similar boundary layers. Int. J. Numer. Methods. Fluids. 13 (6) (1991) 713-737.

[12] M. R. Malik, Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86 (2) (1990) 376-413.

[13] B. Bialecki, A. Karageorghis, Spectral Chebyshev-Fourier collocation for the Helmholtz and variable coefficient equations in a disk. J. Comput. Phys. 227 (19) (2008) 8588-8603.

[14] M. T. Darvishi, S. Kheybari, F. Khani, Spectral collocation method and Darvishi’s preconditionings to solve the generalized Burgers-Huxley equation. Commun. Non- linear. Sci. Numer. Simul. 13 (10) (2008) 2091-2103.

[15] A. Shidfar, R. Pourgholi, Numerical approximation of solution of an inverse heat conduction problem based on Legendre polynomials, Appl. Math. Comput 175 (2) (2006): 1366-1374.

[16] H. Khalil, R. Ali Khan, A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation, Comput. Math. Appl 67 (10) (2014) 1938-1953.

[17] Rong-Yeu Chang, Maw-Ling Wang, Shifted Legendre function approximation of differential equations; application to crystallization processes. Comput. Chem. Eng. 8 (2) (1984) 117-125.

[18] A. Saadatmandi, M. Dehghan. A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59 (3) (2010) 1326-1336.

[19] L. J. Crane, Flow past a stretching plate, J. Appl. Math. Phys., 21 (1970) 645-647.

[20] P. Carragher, L. J. Crane, Heat transfer on a continuous stretching sheet, J. Appl. Math. Mech., 62 (1982) 564-565.

[21] B. K. Dutta, P. Roy, A. S. Gupta, Temperature field in flow over a stretching surface with uniform heat flux, Int. Comm. Heat Mass Transfer, 12 (1985) 89-94.

[22] L. J. Grubka, K. M. Bobba, Heat transfer characteristic of a continuous stretching surface with variable temperature, J. Heat Transf., 107 (1985) 248-250.

[23] E. M. A. Elbashbeshy, Heat transfer over a stretching surface with variable surface heat flux, J. Phys. D Appl. Phys., 31 (1998) 1951-1954.

[24] E. M. A. Elbashbeshy, M. A. A. Bazid, Heat transfer over an unsteady stretching surface, Heat Mass Tran., 41 (2004) 1-4.

[25] S. Sharidan, T. Mahmood, I. Pop, Similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet, Int. J. Appl. Mech. Eng., 11 (2006) 647-654.

[26] M. M. Rashidi, M. Keimanesh, Using Differential Transform Method and Pade Approximant for Solving MHD Flow in a Laminar Liquid Film from a Horizontal Stretching Surface, Mathematical Problems in Engineering, Volume 2010 (2010).

[27] M. M. Rashidi, E. Erfani, The Modified Differential Transform Method for Investigating Nano Boundary-Layers over Stretching Surfaces, Int. J. Numer. Methods Heat Fluid Flow 21 (7) (2011) 864-883.

[28] M. M. Rashidi, N. Freidoonimehr, A. Hosseini, O. Anwar Beg, T. K. Hung, Homotopy Simulation of Nanofluid Dynamics from a Non-Linearly Stretching Isothermal Permeable Sheet with Transpiration, Meccanica 49 (2) (2014) 469-482.

[29] M. M. Rashidi, S. A. Mohimanian Pour, Analytic approximate solutions for unsteady boundary-layer flow and heat transfer due to a stretching sheet by homotopy analysis method, Nonlinear Anal. Model. Control. 15 (1) (2010) 83-95.

[30] W. Ibrahim, B. Shanker, Unsteady Boundary Layer Flow and Heat Transfer Due to a Stretching Sheet by Quasilinearization Technique, World Journal of Mechanics, 1 (6) (2011) 288-293.