The Differential Transform

One of the methods of mathematical analysis in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator d/dt and solved a number of applied problems. However, he did not give operational calculus a mathematical basis; this was done with the aid of the Laplace transform; J. Mikusiński (1953) put operational calculus into algebraic form, using the concept of a function ring [1]. Thereupon I’m suggesting here the use of the integration operator dt to make an extension for the systematic use of operational calculus. Throughout designing and analyzing a control system, we need to treat the functionality of the system with respect to time. The reaction of the system instructs us how to stable it by amplifiers and feedbacks [2], thereafter the Differential Transform is a good tool for doing this, and it’s a technique to frustrate difficulties we may bump into, also it has the methods to find the immediate reaction of the system with respect to infinitesimal (tiny) time which spares us from the hard work in finding the time dependent function, this could be done by producing finite series.