Optimised Differential Transform Method: ‘Goldilocks Order’ of Polynomial Solution

The Differential Transform Method (DTM) is used to find semi-analytical solutions to linear, non-linear including partial and fractional differential equations. However, it has some limitations, as we obtain a truncated series solution that may not exhibit the real behavior of the problem but, in most cases, provides an excellent approximation of the true solution in a very small region [1]. In this study, we have suggested a method to improvise the solutions without compromising on computation time. We observe that the curve of the DTM approximate is inaccurate for orders of the polynomial (r) which are too low or too high. This leads us to conclude that there is a value of r for which the DTM solution is the best. We propose to optimize the order of the series in such a way that the absolute value of the differential equation at the DTM solution is either minimum or less than the error of tolerance. Such a solution is compared with the generic DTM of order 6, RK4 method, and analytical solution (if it exists) concerning the accuracy and computational efficiency. We use two examples, including the Airy equation to show that the proposed method is accurate and efficient.