The method of finding solutions of partial dynamic equations on time scales

Hsuan-Ku Liu

doi:10.1186/1687-1847-2013-141

Abstract

On time scales, one area lacking of development is the method of finding solutions on partial dynamic equations. This paper proposes a method for finding the exact solution of linear partial dynamic equations on arbitrage time scales. We modify the variational iteration method on ℝ to find an approximation of the nonlinear partial dynamic equation on q N ¯ . As an example, the modified variational iteration method is applied to q-Berger equations and to q-Fisher equations. Their numerical results reveal that the proposed method is very effective.

Highlights

A time scale is a nonempty closed subset of real numbers
On nonlinear partial dynamic equations, approximate solutions obtained by the variational iteration method are not found yet
When the initial condition can be represented as a finite series of generalized polynomials, we have proposed a useful method of finding the exact solution of partial dynamic equations on time scales

Summary

Introduction

A time scale is a nonempty closed subset of real numbers. On time scale calculus, notations and theorems have been well established for the univariate case [ ]. Methods of finding solutions are not mentioned for partial dynamic equations on time scales. On nonlinear partial dynamic equations, approximate solutions obtained by the variational iteration method are not found yet. We derive a product rule of two generalized polynomials on qZ, which provides an idea for developing a series solutions on q-calculus. We extend the variational iteration method from the set of real numbers R to the time scales qZ. In Section , a product rule of two generalized polynomials at is derived on qZ and the variational iteration method is applied to find an approximate solution of the Burger equation and the Fisher equation. According to the forward jump operator and the gain function, the delta derivative on the time scale T is given as follows.

The linear operator L is selected as

Conclusion and future direction