Journal of Applied Mathematics

Volume 2013 (2013), Article ID 743973, 4 pages

http://dx.doi.org/10.1155/2013/743973

Research Article

Developing a Series Solution Method of -Difference Equations

Hsuan-Ku Liu

Department of Mathematics and Information Education, National Taipei University of Education, Taiwan

Received 16 November 2012; Accepted 14 April 2013

Academic Editor: Filomena Cianciaruso

Copyright © 2013 Hsuan-Ku Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The series solution is widely applied to differential equations on but is not found in -differential equations. Applying the Taylor and multiplication rule of two generalized polynomials, we develop a series solution of linear homogeneous -difference equations. As an example, the series solution method is used to find a series solution of the second-order -difference equation of Hermite’s type.

1. Introduction

Several works have been done recently for series solutions on certain time scales. One of the difficulties for developing a theory of series solutions for linear homogeneous equations on time scales is that the formula for multiplication by two generalized polynomials is not easily found. If the time scale has constant graininess, Haile and Hall [1] provided an exact formula for the multiplicity of two generalized polynomials. Using the obtained results, the series solutions for linear dynamic equations are proposed on the time scales and (difference equations with step size ). On generalized time scales, Mozyrska and Pawłuszewicz [2] presented the formula for the multiplicity of the generalized polynomials of degree one and degree .

Let and use the notations

where 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 . Precisely, the second-order -difference equation is described as

where and are both -analytic functions at 0 in the interval . As an example, the series solution method is applied to consider the -Hermite's equation of the form

with initial condition and .

This paper is organized as follows: in Section 2, basic ideas on -calculus are introduced. The series solution method is developed in Section 3 and is applied to consider the -Hermite's equation in Section 4. Finally, a concise conclusion is provided in Section 5.

2. A Basic Introduction to Time Scales

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.

Then, we introduce the delta derivative by starting to define the forward and backward jump operators.

Definition 1. Let be a time scale. For , we define the forward jump operator by

while the backward jump operator by

Definition 2. The graininess function is defined by

According to the basic definitions, we can give some useful relationships concerning the delta derivative.

Let and be real numbers such that . The -shift factorial [4] is defined by

Assume is a function and . The -derivative [5] at is defined by

A -difference equation is an equation that contains -derivatives of a function defined on .

Definition 3. On the time scale , the -polynomials are defined recursively as follows:

Hence, for each fixed , the delta derivative of with respect to satisfies

By computing the recurrence relation, the -polynomials can be represented as

on [5].

Agarwal and Bohner [6] give a Taylor's formula for functions on a general time scale. On , the Taylor's formula can be rewritten as the following form.

Theorem 4. Let . Suppose is times differentiable on . Let . One has

Before developing the series solution method, we introduce the -analytic function on .

Definition 5. A real-valued function is said to be -analytic at if and only if there is a power series centered at that converges to near ; that is, there exist coefficients and points such that and

for all .

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

Theorem 6. Let and be two -polynomials at zero. One has

Proof. Since

we have

This implies that

Proposition 7. Let and be any two -polynomials. One has

Proof. Without loss of generality, we suppose and have

This implies that

Therefore, we have

by Theorem 6.

3. Developing Series Solutions Method

Using the Taylor series on time scales, we develop a series solution method for solving -difference equations in this section.

Consider a second-order -difference equation

where and are both -analytic functions at 0 in the interval . Hence, there exist two sequences of coefficients and such that

for all .

One can find a power series solution of the form

by carrying out the following steps.

Step 1. Since

we get

Substituting (24) and (26) into (22), we get the equation

Step 2. Set the coefficients of the power series equal to zero. That gives a recurrence relation that relates later coefficients in the power series (24) to the earlier ones. That is,

Step 3. Find all coefficients in terms of the first two coefficients and , thus writing the -series in the form

where and are two linearly independent -series solutions.

4. Applications

In this section, the series solution method is applied to consider the -Hermite's equations with initial conditions.

Consider the -Hermite's equation of the form

with and .

Let

then and . Applying (31) into (30), we have

where .

This implies that

Hence, we get

By computing the Wronskian of and at , we get

This implies that and are two linearly independent solutions.

Example 8. Consider the -Hermite's equation with of the form

with and . Substituting (31) into (36) yields

and which implies that

5. Conclusion

One area which is the lack of development is the theory of series solutions on -difference equations. In this paper, we present the formula for the multiplicity of two -polynomials at . The purpose is to provide the basic mechanics for finding the series solutions of linear homogeneous -difference equation. As an example we consider series solution of the -Hermite's equation of the form , with the initial conditions and . Using the presented method, the series solution of the Hermite's equation can be obtained iteratively. In future studies, we would apply the presented method to find the series solution of other -difference equations on .

References

B. D. Haile and L. M. Hall, “Polynomial and series solutions of dynamic equations on time scales,” Dynamic Systems and Applications, vol. 12, no. 1-2, pp. 149–157, 2003.

D. Mozyrska and E. Pawłuszewicz, “Hermite's equations on time scales,” Applied Mathematics Letters, vol. 22, no. 8, pp. 1217–1219, 2009.

H.-K. Liu, “Application of a differential transformation method to strongly nonlinear damped -difference equations,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2555–2561, 2011.

T. H. Koornwinder, “-special functions: a tutorial,” http://arxiv.org/abs/math/9403216

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001.

R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics, vol. 35, no. 1-2, pp. 3–22, 1999.