zbMATH — the first resource for mathematics

Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity. (English) Zbl 1261.35140
Summary: This study investigates the exact solutions of a nonlinear fin problem with temperature-dependent thermal conductivity and the heat transfer coefficient. Both the conduction and heat transfer terms are given by the same power law in one case and the distinct power law in the other. Classical Lie symmetry techniques are employed to construct the exact solutions which satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.

35Q79 PDEs in connection with classical thermodynamics and heat transfer
35A30 Geometric theory, characteristics, transformations in context of PDEs
80A20 Heat and mass transfer, heat flow (MSC2010)
Full Text: DOI
[1] ()
[2] Aziz, A.; Na, T.Y., Periodic heat transfer in fins with variable thermal parameters, Int. J. heat mass transfer, 24, 1397-1404, (1981) · Zbl 0458.76076
[3] Bokhari, A.H.; Kara, A.H.; Zaman, F.D., A note on a symmetry analysis and exact solution for a nonlinear fin equation, Appl. math. lett., 19, 1356-1360, (2006) · Zbl 1143.35311
[4] Pakdemirli, M.; Sahin, A.Z., Group classification of fin equation with variable thermal properties, Int. J. eng. sci., 42, 1875-1889, (2004) · Zbl 1211.35141
[5] Pakdemirli, M.; Sahin, A.Z., Similarity analysis of a nonlinear fin equation, Appl. math. lett., 19, 378-384, (2006) · Zbl 1114.80003
[6] Vaneeva, O.O.; Johnpillai, A.G.; Popovych, R.O.; Sophocleous, C., Group analysis of nonlinear fin equation, Appl. math. lett., 21, 248-253, (2008) · Zbl 1160.35320
[7] Ovisianikov, L.V., Group analysis of differential equations, (1982), Academic press New York
[8] Popovych, R.O.; Sophocleous, C.; Vaneeva, O.O., Exact solutions of a remarkable fin equation, Appl. math. lett., 21, 209-214, (2008) · Zbl 1149.76047
[9] Khani, F.; Ahmadzadeh Raji, M.; Hamedi Nejad, H., Analytic solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, Commun. nonlinear sci. numer. simul., 14, 3327-3338, (2009) · Zbl 1221.74083
[10] Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman & Hall, CRC Press Boca Raton
[11] Allan, F.M., Derivation of the adomain decomposition method using the homotopy analysis method, Appl. math. comput., 190, 1, 6-14, (2007) · Zbl 1125.65063
[12] Sajid, M.; Hayat, T., The application of homotopy analysis method to thin film flows of a third order fluid, Chaos solitons fractals, 38, 506-515, (2008) · Zbl 1146.76588
[13] Sajid, M.; Hayat, T.; Asghar, S., On the analytic solution of the steady flow of a fourth grade fluid, Phys. lett. A, 355, 18-24, (2006)
[14] Sajid, M.; Hayat, T.; Asghar, S., Comparison of HAM and HPM solutions of film flows of non-Newtonian fluids on a moving belt, Nonlinear dyn., 50, 27-35, (2007) · Zbl 1181.76031
[15] Zhu, S.P., An exact and explicit solution for the valuationof American put option, Quant. finance, 6, 229-242, (2006)
[16] F. Khani, M. Ahmadzadeh Raji, H. Hamedi-Nezhad, A series solution of the fin problem with a temperature-dependent conductivity, 14 (7) (2009) 3007-3017
[17] Mahomed, F.M., Symmetry group classification of ordinary differential equations: survey of some results, Math. methods appl. sci., 30, 1995-2012, (2007) · Zbl 1135.34029
[18] Kim, S.; Huang, C.H., A series solution of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, J. phys. D: appl. phys., 40, 2979-2987, (2007) · Zbl 1144.82346
[19] Olver, P.J., Applications of Lie groups to differential equations, (1986), Springer New York · Zbl 0588.22001
[20] Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York, USA · Zbl 0698.35001
[21] Bluman, G.W.; Anco, S.C., Symmetry and integration methods for differential equations, (2002), Springer-Verlag New York · Zbl 1013.34004
[22] Bagderina, Yu.Yu., Equivalence of ordinary differential equations \(y'' = R(x, y) y {}^\prime^2 + 2 Q(x, y) y^\prime + P(x, y)\), Differ. equ., 43, 5, 595-604, (2007) · Zbl 1170.34320
[23] Ibragimov, N.H.; Torrisi, M.; Valenti, A., Preliminary group classification of equations \(v_t t = f(x, v_x) v_{x x} + g(x, v_x)\), J. math. phys., 32, 11, 2988-2995, (1991) · Zbl 0737.35099
[24] Ibragimov, N.H.; Säfström, N., The equivalence group and invariant solutions of a tumour growth model, Commun. nonlinear sci. numer. simul., 9, 61-68, (2004) · Zbl 1032.92017
[25] Joneidi, A.A.; Ganji, D.D.; Babaelahi, M., Differential transformation method to determine fin efficiency of convective straight fin with temperature dependent thermal conductivity, Int. comm. heat and mass trans., 36, 757-762, (2009)
[26] Hearn, A.C., Reduce user’s manual version 3.4, (1985), Rand Publication CP78, The Rand Cooporation Santa Monica, CA
[27] Lie, S., Klassifikation und integration von gewo¨nlichen differentialgleichugen zwischen \(x, y\) die eine gruppe von transformationen gestaten, Arch. math., VIII, IX, 187, (1883)
[28] Abramowitz; Stegun, Handbook of mathematical functions, (1972), Dover New York · Zbl 0543.33001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.