×

The spectral homotopy analysis method extended to systems of partial differential equations. (English) Zbl 1468.65215

Summary: The spectral homotopy analysis method is extended to solutions of systems of nonlinear partial differential equations. The SHAM has previously been successfully used to find solutions of nonlinear ordinary differential equations. We solve the nonlinear system of partial differential equations that model the unsteady nonlinear convective flow caused by an impulsively stretching sheet. The numerical results generated using the spectral homotopy analysis method were compared with those found using the spectral quasilinearisation method (SQLM) and the two results were in good agreement.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Motsa, S. S.; Sibanda, P.; Shateyi, S., A new spectral-homotopy analysis method for solving a nonlinear second order BVP, Communications in Nonlinear Science and Numerical Simulation, 15, 9, 2293-2302 (2010) · Zbl 1222.65090 · doi:10.1016/j.cnsns.2009.09.019
[2] Motsa, S. S.; Sibanda, P.; Awad, F. G.; Shateyi, S., A new spectral-homotopy analysis method for the MHD Jeffery-Hamel problem, Computers & Fluids, 39, 7, 1219-1225 (2010) · Zbl 1242.76363 · doi:10.1016/j.compfluid.2010.03.004
[3] Motsa, S. S., On the pratical use of the spectral homotopy analysis method and local linearisation method for unsteady boundary-layer flows caused by an impulsively stretching plate, Numerical Algorithms (2013) · Zbl 1299.76192 · doi:10.1007/s11075-013-9766-z
[4] Bellman, R. E.; Kalaba, R. E., Quasilinearization and Nonlinear Boundary-Value Problems. Quasilinearization and Nonlinear Boundary-Value Problems, Modern Analytic and Computional Methods in Science and Mathematics, Vol. 3 (1965), Elsevier · Zbl 0139.10702
[5] Kumari, M.; Slaouti, A.; Takhar, H. S.; Nakamura, S.; Nath, G., Unsteady free convection flow over a continuous moving vertical surface, Acta Mechanica, 116, 75-82 (1996) · Zbl 0866.76085
[6] Ishak, A.; Nazar, R.; Pop, I., Unsteady mixed convection boundary layer flow due to a stretching vertical surface, The Arabian Journal for Science and Engineering B, 31, 2, 165-182 (2006)
[7] Pop, I.; Na, T.-Y., Unsteady flow past a stretching sheet, Mechanics Research Communications, 23, 4, 413-422 (1996) · Zbl 0893.76017 · doi:10.1016/0093-6413(96)00040-7
[8] Wang, C. Y.; Du, Q.; Miklavčič, M.; Chang, C. C., Impulsive stretching of a surface in a viscous fluid, SIAM Journal on Applied Mathematics, 57, 1, 1-14 (1997) · Zbl 0869.76013 · doi:10.1137/S0036139995282050
[9] Mahdy, A., Unsteady mixed convection boundary layer flow and heat transfer of nanofluids due to stretching sheet, Nuclear Engineering and Design, 249, 248-255 (2012) · doi:10.1016/j.nucengdes.2012.03.025
[10] Bachok, N.; Ishak, A.; Pop, I., Unsteady boundary-layer flow and heat transfer of a nanofluid over a permeable stretching/shrinking sheet, International Journal of Heat and Mass Transfer, 55, 7-8, 2102-2109 (2012) · doi:10.1016/j.ijheatmasstransfer.2011.12.013
[11] Sharma, R.; Ishak, A.; Pop, I., Partial slip flow and heat transfer over a stretching sheet in a nanofluid, Mathematical Problems in Engineering, 2013 (2013) · doi:10.1155/2013/724547
[12] Bird, R. B.; Stewar, W. R.; Lightfoot, E. N., Transport Phenomena (1960), Tokyo, Japan: Toppan, Tokyo, Japan
[13] Streeter, V. L., Handbook of Fluid Dynamics, 6 (1961), New York, NY, USA: McGraw-Hill, New York, NY, USA
[14] Barrow, H.; Rao, T. L. S., Effect of variation in the volumetric expansion coefficient on free convection heat transfer, British Chemical Engineering, 16, 8, 704-709 (1971)
[15] Brown, A., Effect on laminar free convection heat transfer of the temperature dependence of the coefficient of volumetric expansion, Journal of Heat Transfer, 97, 1, 133-135 (1975)
[16] Partha, M. K., Nonlinear convection in a non-Darcy porous medium, Applied Mathematics and Mechanics. English Edition, 31, 5, 565-574 (2010) · Zbl 1378.76111 · doi:10.1007/s10483-010-0504-6
[17] Prasad, K. V.; Vajravelu, K.; Van Gorder, R. A., Non-Darcian flow and heat transfer along a permeable vertical surface with nonlinear density temperature variation, Acta Mechanica, 220, 1-4, 139-154 (2011) · Zbl 1385.76013 · doi:10.1007/s00707-011-0474-2
[18] Karcher, C.; Müller, U., Bénard convection in a binary mixture with a nonlinear density-temperature relation, Physical Review E, 49, 5, 4031-4043 (1994) · doi:10.1103/PhysRevE.49.4031
[19] Elbeleze, A. A.; Kılıçman, A.; Taib, B. M., Homotopy perturbation method for fractional Black-Scholes European option pricing equations using Sumudu transform, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.91179 · doi:10.1155/2013/524852
[20] Javidi, M.; Raji, M. A., Combination of Laplace transform and homotopy perturbation method to solve the parabolic partial differential equations, Communications in Fractional Calculus, 3, 10-19 (2012)
[21] Trefethen, L. N., Spectral methods in MATLAB. Spectral methods in MATLAB, Software, Environments, and Tools, 10 (2000), Society for Industrial and Applied Mathematics (SIAM) · Zbl 0953.68643 · doi:10.1137/1.9780898719598
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.