On computing differential transform of nonlinear non-autonomous functions and its applications

TL;DR

This paper introduces a new general formula for computing the differential transform of nonlinear non-autonomous functions, enhancing the method's applicability to complex differential and integro-differential equations.

Contribution

A novel general formula and recurrence relations for differential transforms of nonlinear non-autonomous functions, extending existing methods to broader classes of functions.

Findings

Effective in solving complex nonlinear differential equations

Applicable to integro-differential equations with nonlinearities

Simplifies the computation process

Abstract

Although being powerful, the differential transform method yet suffers from a drawback which is how to compute the differential transform of nonlinear non-autonomous functions that can limit its applicability. In order to overcome this defect, we introduce in this paper a new general formula and its related recurrence relations for computing the differential transform of any analytic nonlinear non-autonomous function with one or multi-variable. Regarding, the formula in the literature was found not applicable to deal with the present non-autonomous functions. Accordingly, a generalization is presented in this paper which reduces to the corresponding formula in the literature as a special case. Several test examples for different types of nonlinear differential and integro-differential equations are solved to demonstrate the validity and applicability of the present method. The obtained results declare that the suggested approach not only effective but also a straight forward even in solving differential and integro-differential equations with complex nonlinearities.