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Classical Lie point symmetry analysis of a steady nonlinear one-dimensional fin problem. (English) Zbl 1234.35277

Summary: We consider the one-dimensional steady fin problem with the Dirichlet boundary condition at one end and the Neumann boundary condition at the other. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended. Some invariant solutions are constructed. The effects of thermogeometric fin parameter and the exponent on temperature are studied. Also, the fin efficiency is analyzed.

MSC:

34C14 Symmetries, invariants of ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
80A20 Heat and mass transfer, heat flow (MSC2010)

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[1] A. D. Kraus, A. Aziz, and J. Welte, Extended Surface Heat Transfer, John Wiley & Sons, New York, NY, USA, 2001.
[2] F. Khani, M. A. Raji, and H. H. Nejad, “Analytical solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 8, pp. 3327-3338, 2009. · Zbl 1221.74083 · doi:10.1016/j.cnsns.2009.01.012
[3] F. Khani, M. A. Raji, and S. Hamedi-Nezhad, “A series solution of the fin problem with a temperature-dependent thermal conductivity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 7, pp. 3007-3017, 2009. · doi:10.1016/j.cnsns.2008.11.004
[4] S. Kim and C. H. Huang, “A series solution of the non-linear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient,” Journal of Physics D, vol. 40, no. 9, pp. 2979-2987, 2007. · doi:10.1088/0022-3727/40/9/046
[5] R. J. Moitsheki, T. Hayat, and M. Y. Malik, “Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity,” Nonlinear Analysis, vol. 11, no. 5, pp. 3287-3294, 2010. · Zbl 1261.35140 · doi:10.1016/j.nonrwa.2009.11.021
[6] A. H. Bokhari, A. H. Kara, and F. D. Zaman, “A note on a symmetry analysis and exact solutions of a nonlinear fin equation,” Applied Mathematics Letters, vol. 19, no. 12, pp. 1356-1360, 2006. · Zbl 1143.35311 · doi:10.1016/j.aml.2006.02.003
[7] M. Pakdemirli and A. Z. Sahin, “Group classification of fin equation with variable thermal properties,” International Journal of Engineering Science, vol. 42, no. 17-18, pp. 1875-1889, 2004. · Zbl 1211.35141 · doi:10.1016/j.ijengsci.2004.04.005
[8] M. Pakdemirli and A. Z. Sahin, “Similarity analysis of a nonlinear fin equation,” Applied Mathematics Letters, vol. 19, no. 4, pp. 378-384, 2006. · Zbl 1114.80003 · doi:10.1016/j.aml.2005.04.017
[9] O. O. Vaneeva, A. G. Johnpillai, R. O. Popovych, and C. Sophocleous, “Group analysis of nonlinear fin equations,” Applied Mathematics Letters, vol. 21, no. 3, pp. 248-253, 2008. · Zbl 1160.35320 · doi:10.1016/j.aml.2007.02.023
[10] R. O. Popovych, C. Sophocleous, and O. O. Vaneeva, “Exact solutions of a remarkable fin equation,” Applied Mathematics Letters, vol. 21, no. 3, pp. 209-214, 2008. · Zbl 1149.76047 · doi:10.1016/j.aml.2007.03.009
[11] R. J. Moitsheki, “Steady heat transfer through a radial fin with rectangular and hyperbolic profiles,” Nonlinear Analysis, vol. 12, no. 2, pp. 867-874, 2011. · Zbl 1205.80035 · doi:10.1016/j.nonrwa.2010.08.011
[12] S. Kim, J.-H. Moon, and C.-H. Huang, “An approximate solution of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient,” Journal of Physics D, vol. 40, no. 14, pp. 4382-4389, 2007. · doi:10.1088/0022-3727/40/14/038
[13] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, NY, USA, 1986. · Zbl 0588.22001
[14] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer, New York, NY, USA, 1989. · Zbl 0698.35001
[15] H. Stephani, Differential Equations: Their Solution Using Symmetries, Cambridge University Press, Cambridge, UK, 1989. · Zbl 0704.34001
[16] N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, New York, NY, USA, 1999. · Zbl 1047.34001
[17] G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, NY, USA, 2002. · Zbl 1013.34004
[18] G. W. Bluman, A. F. Cheviakov, and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York, NY, USA, 2010. · Zbl 1223.35001
[19] A. C. Hearn, “Reduce user’s manual version 3.4,” Rand Publication CP78, The Rand Cooporation, Santa Monica, Calif, USA, 1985.
[20] N. H. Ibragimov, M. Torrisi, and A. Valenti, “Preliminary group classification of equations vtt=f(x,vx)vxx+g(x,vx),” Journal of Mathematical Physics, vol. 32, no. 11, pp. 2988-2995, 1991. · Zbl 0737.35099 · doi:10.1063/1.529042
[21] F. M. Mahomed, “Symmetry group classification of ordinary differential equations: survey of some results,” Mathematical Methods in the Applied Sciences, vol. 30, no. 16, pp. 1995-2012, 2007. · Zbl 1135.34029 · doi:10.1002/mma.934
[22] R. H. Yeh and S. P. Liaw, “An exact solution for thermal characteristics of fins with power-law heat transfer coefficient,” International Communications in Heat and Mass Transfer, vol. 17, no. 3, pp. 317-330, 1990. · doi:10.1016/0735-1933(90)90096-3
[23] R. J. Moitsheki and C. Harley, “Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient,” Pramana, vol. 77, no. 3, pp. 519-532, 2011. · doi:10.1007/s12043-011-0172-6
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