Solution of a PDE Using the Differential Transformation Method

Consider the partial differential equation (PDE) with initial condition and boundary conditions and , , and , where is the thermal diffusivity. This problem represents the transient heat conduction in a slab. This Demonstration obtains the temperature profile for user-set values of the dimensionless time and the thermal diffusivity . The red curve and the dashed blue curve are obtained using Mathematica's built-in function NDSolve and the differential transformation method (DTM), respectively. Here, the DTM gives reasonably good results despite its simplicity.

The -order Taylor series expansion of is of the form . If one takes , the Maclaurin series of is . The differential transformation of the function is given by . The differential inverse transform of is defined by . Table 1 in [1] gives a list of properties of the differential transformation. In this Demonstration, and .

Reference

[1] 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.