The differential transformation method and Miller's recurrence

TL;DR

This paper enhances the differential transformation method (DTM) by incorporating Miller's recurrence to efficiently handle exponentiation and exponential functions, improving the method's capability for solving nonlinear differential equations.

Contribution

It introduces a novel implementation of Miller's recurrence within DTM for exponentiation and exponential functions, addressing a gap in existing literature.

Findings

Efficient implementation of exponentiation in DTM

Concise method for exponential function within DTM

Improved solutions for nonlinear differential equations

Abstract

The differential transformation method (DTM) enables the easy construction of a power-series solution to a nonlinear differential equation. The exponentiation operation has not been specifically addressed in the DTM literature, and constructing it iteratively is suboptimal. The recurrence for exponentiating a power series by J.C.P. Miller provides a concise implementation of exponentiation by a positive integer for DTM. An equally-concise implementation of the exponential function is also provided.