Application of the Variational Iteration Method to Strongly Nonlinear q‐Difference Equations

Abstract

The theory of approximate solution lacks development in the area of nonlinear -difference equations. One of the difficulties in developing a theory of series solutions for the homogeneous equations on time scales is that formulas for multiplication of two -polynomials are not easily found. In this paper, the formula for the multiplication of two -polynomials is presented. By applying the obtained results, we extend the use of the variational iteration method to nonlinear -difference equations. The numerical results reveal that the proposed method is very effective and can be applied to other nonlinear -difference equations.

Developing a Series Solution Method of -Difference Equations

“…The production rule of two -polynomials at 0 which will be used to derive the series solution in following sections [3]. Proof.…”

“…where N denotes the set of positive integers. Liu [3] presented a formula for the multiplication of two -polynomials. The obtained results are used to develop a series solution method of the second-order difference equations on N .…”

“…A time scale means an arbitrary nonempty closed subset of the real numbers. The calculus of time scales was initiated by Liu [3] in order to create a theory that can unify discrete and continuous analysis.…”

A New Algorithm for Solving Terminal Value Problems of q-Difference Equations

“…Recently, many authors focus on the study of -difference equations, especially on the numerical solution problems, such as [6,7]. In [6], by using the differential transformation method, the authors studied the following strongly nonlinear -difference equation:…”

“…From the works of [6] and [7], one can easily see that when > 1, the methods they proposed are no longer effective, since for the formula…”