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

Some remarks on the differential transform method

“…The further development of the complex integral method to the solution of fractional differential equations has probably, in principle, been achieved already [4,7,18] as the Caputo fractional derivative [3] is intimately related to the Cauchy integral theorem, on which all of the above methodologies are explicitly or implicitly constructed. Interestingly, the Taylor-Cauchy transform [5,19] the differential transform [8][9][10][11][12]20] and the complex integral method [2] were all developed as techniques for the solution of nonlinear differential equations. However, as has been shown in, for example [13][14][15][16][17], such methods have exceptional efficacy in the solution of linear differential equations also.…”

“…and 161 references to the original literature on the subject, which is quite enough to be going on with for now. In the case of the Herrera complex integral method, and its first cousin the Taylor-Cauchy transform method discussed below [5,19], the literature is more compact and we leave it at that.…”

“…The underlying link between the differential transform method and the complex integral method is based on the so-called Taylor-Cauchy transform [5,19]. The basis of the Taylor-Cauchy transform is the assumption that the kth-order…”

“…In (2.2), the closed contour C encircles the point 0 z and defines the domain of definition of ) (z f [5,19]. Specifically, the Taylor-Cauchy transform is defined by the pair of equations (2.1) and (2.2) [5,19]; the operational properties and the conditions on the transform pair (2.1) and (2.2) are discussed in detail in [5] and [19].…”