The differential transformation method and Miller's recurrence

AbstractThe differential transformation method (DTM) enables the easy construction of a power-seriessolution to a nonlinear differential equation. The exponentiation operation has not been specif-ically addressed in the DTM literature, and constructing it iteratively is suboptimal. The recur-rence for exponentiating a power series by J.C.P. Miller provides a concise implementation ofexponentiation by a positive integer for DTM. An equally-concise implementation of the expo-nential function is also provided.Keywords: differential transformation method, DTM, power series, exponentiation, exponentialfunction, differential equations1. IntroductionConstructing power-series solutions to differential equations, especially those which do notadmit a closed-form solution, has long been an important, and widely-used, solution technique.Traditionally,computingpower-seriessolutionsrequireda fair amountof“boiler-plate”symbolicmanipulation, especially in the setup of the power-matching phase. The differential transforma-tion method (DTM) enables the easy construction of a power-series solution by specifying aconversion between the differential equation and a recurrence relation for the power-series coef-ficients [1].The table in the current literature which specifies the trans lation between the terms of thedifferential equation and the recurrence relation has a striking omission: it contains no expo-nentiation operation. Exponentiation by a positive integer can be constructed iteratively usingthe table entry for multiplication (i.e. multiplying the function with itself n times), but such aconstruction is suboptimal because it leads to n − 1 nested sums. Using a recurrence for expo-nentiating a power series by J.C.P. Miller, a table entry for positive-integer exponentiation canbe provided which introduces only a single sum. A single-sumrecurrence for the exponentialfunction of a power series can be similarly constructed.