# Solution of a Boundary Value Problem Using the Differential Transformation Method

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Consider the following two boundary value problems, the first linear and the second nonlinear: (1) with and , and (2) with and .

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Contributed by: Housam Binous, Ahmed Bellagi, and Brian G. Higgins (October 2013)

Open content licensed under CC BY-NC-SA

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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 .

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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