Temperature Distribution in Convective Fins with Variable Thermal Conductivity

Consider a convective straight fin with temperature-dependent thermal conductivity. The dimensionless problem reduces to (see [4] for details) with , , where the dimensionless quantities and parameters are defined as follows: , , , and .
The plots of the temperature distribution are obtained using the shooting technique (red curve) and the differential transformation method (dashed blue curve).
Excellent agreement is found between the two methods.

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DETAILS

The -order Taylor series expansion function of is of the form . At , this is the Maclaurin series, . The differential transformation of the function is given by . The differential inverse transform of is defined by . Table 1 in [2] gives a list of properties of the differential transformation. For instance, if then . In this Demonstration, and .
References
[1] M. J. Jang, C. L. Chen, and Y. C. Liu, "Analysis of the Response of a Strongly Nonlinear Damped System Using a Differential Transformation Technique," Applied Mathematics and Computation, 88(2–3), 1997 pp. 137–151. doi: 10.1016/S0096-3003(96)00308-6.
[2] C. L. Chen and Y. C. Liu, "Solution of Two-Boundary-Value Problems Using the Differential Transformation Method," Journal of Optimization Theory and Applications, 99(1), 1998 pp. 23–35. doi: 10.1023/A:1021791909142.
[3] M. J. Jang, C. L. Chen, and Y. C. Liu, "On Solving the Initial-Value Problems Using the Differential Transformation Method," Applied Mathematics and Computation, 115(2–3), 2000
pp. 145–160. doi: 10.1016/S0096-3003(99)00137-X.
[4] A. A. Joneidi, D. D. Ganji, and M. Babaelahi, "Differential Transformation Method to Determine Fin Efficiency of Convective Straight Fins with Temperature Dependent Thermal Conductivity," International Communications in Heat and Mass Transfer, 36(7), 2009 pp. 757–62. doi: 10.1016/j.icheatmasstransfer.2009.03.020.
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