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Solution of a Boundary Value Problem Using the Differential Transformation Method

Consider the following two boundary value problems, the first linear and the second nonlinear: (1) with and , and (2) with and .
This Demonstration applies the differential transformation method (DTM) to solve these two problems. You can vary the parameter as well as the order of the Taylor series expansion, . Excellent agreement is obtained between the result of the DTM (blue dashed curve) and its shooting technique counterpart (red curve) for large values of .

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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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