Taylor-Cauchy Transforms for Analysis of a Class of Nonlinear Systems

This paper presents a new transform calculus for analyzing a certain class of nonlinear systems. The method, which can be justified rigorously by the partition theory, essentially transforms a nonlinear differential equation having certain conditions imposed on its linear, nonlinear, and driving terms into an algebraic equation. The latter is easily solved recursively owing to its symmetry and convolution properties. The transform pair is based on a combination of the Cauchy integral theorem and Taylor's series in complex form. To illustrate the method a number of examples are solved and a table of transforms is included. Because the results of this transform technique are the same as those given by the partition method under certain circumstances, the two are compared. It is then seen that the solution can be obtained uniquely and exactly. The Taylor-Cauchy transform method can be compared with the Laurent-Cauchy transform method, given in a companion paper, for the solution of linear systems described by differential-difference and sum equations.