Discrete Dynamics in Nature and Society

Volume 2018, Article ID 8509860, 7 pages

https://doi.org/10.1155/2018/8509860

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

School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, China

Correspondence should be addressed to Lin-Lin Wang; moc.anis@4991_llgnaw

Received 21 January 2018; Revised 30 March 2018; Accepted 11 April 2018; Published 14 May 2018

Academic Editor: Douglas R. Anderson

Copyright © 2018 Yong-Hong Fan and Lin-Lin Wang. 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

We propose a new algorithm for solving the terminal value problems on a -difference equations. Through some transformations, the terminal value problems which contain the first- and second-order delta-derivatives have been changed into the corresponding initial value problems; then with the help of the methods developed by Liu and H. Jafari, the numerical solution has been obtained and the error estimate has also been considered for the terminal value problems. Some examples are given to illustrate the accuracy of the numerical methods we proposed. By comparing the exact solution with the numerical solution, we find that the convergence speed of this numerical method is very fast.

#### 1. Introduction

The -difference equations theory appeared already at the beginning of the 20th century. Intensive works have been made especially by Jackson [1], Carmichael [2], Mason [3], Adams [4], Trjitzinsky [5], and others. But from the 1930s to the beginning of the 1980s, only nonsignificant interest in the area appeared.

The numerical solution of the -difference equations plays an important role in many fields of science and engineering, mainly in physical science, as their solutions can provide more insight into the physical aspects of the problem.

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:with The parameters represent the linear damping parameters, is the nonlinearity parameter, is the frequency of underdamped motion, and . They established a numerical method and obtained the numerical results of (1). Later, Jafari et al. [7] proposed a numerical algorithm to obtain the numerical solutions for the first- and second-order dynamic equations on As an application, they investigated the numerical solution of (1) and, through the error analysis, they proved the effectiveness of the numerical algorithm they proposed.

From the works of [6] and [7], one can easily see that when , the methods they proposed are no longer effective, since for the formula when , so we cannot estimate by . In this paper, we want to give a new numerical method to gain the numerical solution of the general first- and second-order dynamic equations on with We assume that always hold in the following unless otherwise is stated.

#### 2. Preliminary

For -difference equation, the theory of time scales plays an important role, which was initiated by Hilger [8] in his Ph.D. thesis to unify both difference and differential calculus in a consistent way. Since then many authors have investigated the dynamic equations on time scales (see [9–11], etc.). This theory is a powerful tool for mathematical analysis in economics, population models, quantum physics, and so on.

Before giving our main result, first we list some basic properties about time scales which could be found in ([8, 12, 13]).

*Definition 1. *A time scale is an arbitrary nonempty closed subset of the real number .

*Definition 2. *For we define the forward jump operator and the backward jump operator by respectively.

*Definition 3. *Let be a time scale, for ; if , we say that is right-scattered, while if , we say that is left-scattered. Also, if and , then is called right-dense, and if and , then is called left-dense.

*Definition 4. *A function is called rd-continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions is denoted by

*Definition 5. *Assume that and let , where Then we define to be the number (provided it exists) with property that given any , there is a neighborhood of such that for all . We call the delta (or Hilger) derivative of and it turns out that is the usual derivative if and is the usual forward difference operator if .

Lemma 6. *The delta derivative of on right-scattered point can be calculated by *

In the following, we give two lemmas which will be useful to prove our main result.

Lemma 7. *Assume that is delta differentiable at Let and denote ; then is delta differentiable at and*

*Proof. *By the definition of delta derivative of and Lemma 6, we know that, for , notice that then

Lemma 8. *Assume that is twice delta differentiable at Let and denote ; then is twice delta differentiable at and*

*Proof. *Since and on the other handthus that is, (12) holds true.

#### 3. Numerical Method

In this section we present a numerical method to solve the terminal value problems involving the first- and second-order -difference equations.

##### 3.1. The First-Order

Consider the following TVP on with :Here means that represents delta derivative; in fact, we can easily see that when , where is the -difference operator. For the concept of -difference operator, one can see [14]. We assume that (16) with (17) has an unique solution.

First, we give a necessary condition for the existence of (16) with (17).

Lemma 9. *If is the solution of (16) and (17), then Especially, if , then *

Now we present a numerical method to solve (16) and (17).

*Algorithm 10. **Step 1*. Make the transformation and let ; by (8), (16) with (17) is transformed into*Step 2*. Motivated by the work of Liu [6], let and ; then (20) will be changed into notice that then*Step 3*. From the above equality, set and solve , where obviously only depends on , , and ; thus we can obtain all the values of at each point .

*Example 11. *Consider the following -difference equation ():It is easy to see that is the solution of (24); we now use Algorithm 10 to obtain the numerical solution of (24).

Let and ; by (23), (24) can be transformed intoSolving the above equations, we can obtainLet ; by recurrence formula (26), we can obtain all values of ; we calculate it in Table 1 for and .