Taylor theory associated with Hahn difference operator

Abstract

In this paper, we establish Taylor theory based on Hahn's difference operator D q,ω which is defined by D q,ω f (t) = f (qt+ω)-f (t) t(q-1)+ω , t = ω 1-q , where q ∈ (0, 1) and ω is a positive number.

On q,ω -differential transform method

“…Similar to the q, ω-Taylor expansion of a single variable function (see [30]), we can give the same for a multivariate function. If all q, ω-partial derivatives of u (x, t) exists in some neighborhood of (ω 0 , ω 0 ) , then…”

“…Let ϕ be a function defined on J and is q, ω-differentiable infinitely many times at ω 0 . Under conditions similar to the convergence conditions given in theorem 2.10 in [30], ϕ has the q, ω-Taylor expansion…”

“…1.4. q, ω-Taylor expansion [30] In [30], the authors established the q, ω-Taylor theory using Hahn's difference operator. Let ϕ be a function defined on J and is q, ω-differentiable infinitely many times at ω 0 .…”

Asymptotic iteration method for solving Hahn difference equations

“…For example, if , then this lemma implies A slightly different form of the following theorem was given in [ 34 , Lemma 2.1]; however, we give here a direct proof based on the method of mathematical induction.…”