Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 8509860, 7 pages
https://doi.org/10.1155/2018/8509860

Research Article

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

Yong-Hong Fan and Lin-Lin Wang

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

Correspondence should be addressed to Lin-Lin Wang; wangll_1994@sina.com

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 hand

thus

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 into

Solving the above equations, we can obtain

Let ; by recurrence formula (26), we can obtain all values of ; we calculate it in Table 1 for and .

Table 1: The comparison of exact value and approximate value.

(a)

Time domain

Exact value

Approximate value

Error ()

Absolute error ()

(b)

(c)

From Table 1, we can see that the maximum error is and the average error is .

Figure 1 shows the comparison of exact value and approximate value of Example 11 when and .

Figure 1: For .

3.2. The Second-Order

Consider the following TVP on with :

Here represents delta derivative. Also we assume that (27) with (28) has only one solution.

First, we give a necessary condition for the existence of solution for (27) with (28).

Lemma 12. If is the solution of (27) and (28), then

Especially, if , then

Let ; then by (8), we know that On the other hand, notice that ; then

Let be a nature number; then using the Taylor formula on time scales, we have

thus

set , , and , ; let ; then

Note that the delta derivative of at can be calculated as

where

In the following, we give a method to solve the numerical solution of (27) with (28).

Algorithm 13. Step 1. Make the transformation and let ; by (8) and (12), (27) with (28) is transformed into

Step 2. Let ; set and ; then (38) will be changed into

notice that

then

Step 3. From (41), we can solve step by step.

Now we consider an example to illustrate the efficiency of the numerical method.

Example 14. Consider the following second-order terminal value problem on

It is easy to see that is the solution of (42) with the terminal value condition (43).
Let ; then (42) can be transformed into

Set ; by (41), we have

Therefore

where

Notice that ; then by recurrence formula (47), we can obtain all values of ; we calculate it in Table 2.

Table 2: The comparison of exact value and approximate value for and

(a)

Time domain

Exact value

Approximate value

(b)

From Table 2, we can see that the maximum error is and the average error is

Figure 2 shows the comparison of exact value and approximate value of Example 14 when and

Figure 2: For .

4. Discussion and Conclusion

In this paper we have presented a new algorithm for solving the terminal value problems on a -difference equations. It makes a considering improvement: this establishing method is easy to program implementation, since the numerical calculation formula we established is a relatively simple iteration formula; on the other hand, this establishing method could be generalized to the third-order case, the fourth order case, and so on.

For problem (20) or (37)-(38), suppose that is the exact solution and is the numerical solution; from Algorithms 10 and 13, we can see that ; that is, is the exact value. However, ; notice that ; the error of converges to zero exponentially. In this sense, we may say that the convergence speed of this numerical method is very fast. In fact, the calculation precision can be further improved. By (32), the computational accuracy depends on the number of terms on the right-hand side. The more a higher-order derivative of at zero is obtained, the smaller the error would be.

As far as we know, there are very few mathematicians involved in the terminal values problems of -difference equations. In [15], the authors examine “terminal" value problems for dynamic equations on time scales; that is, a dynamic equation whose solutions are asymptotic at infinity. They present a number of new theorems that guarantee the existence and uniqueness of solutions, as well as some comparison-type results. And in [16], the authors propose a nonpolynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order, and they study the order of convergence of the proposed algorithm. But for the numerical algorithm on the terminal values problems of -difference equations, we have not found the relevant conclusion.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Additional Points

2000 AMS Subject Classification is as follows: 39A12, 39A13, 37N30.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by NSF of Shandong Province (ZR2015AM026), the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07), and NSF of China (11201213).

References

F. H. Jackson, “q-difference equations,” American Journal of Mathematics, vol. 32, no. 4, pp. 305–314, 1910.

R. D. Carmichael, “The general theory of Linear q-difference equations,” The American Journal of Mathematics, vol. 34, no. 2, pp. 147–168, 1912.

T. Mason, “Properties of the Solutions of Linear q-difference Equations with Entire Function Coefficient,” The American Journal of Mathematics, vol. 37, no. 4, pp. 439–444, 1915.

C. R. Adams, “On the linear ordinary q-difference equation,” in Annals of Mathematics: Second Series, vol. 30, pp. 195–205, 1929.

W. J. Trjitzinsky, “Analytic theory of linear q-difference equations,” Acta Mathematica, vol. 61, no. 1, pp. 1–38, 1933.

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

H. Jafari, A. Haghbin, S. J. Johnston, and D. Baleanu, “A new algorithm for solving dynamic equations on a time scale,” Journal of Computational and Applied Mathematics, vol. 312, pp. 167–173, 2017.

S. Hilger, Ein Mab kettenkalkül mit Anwendung auf Zentrumsmannig faltigkeiten [Ph.D. thesis], Universität of Würzburg, 1988.

R. P. Agarwal, M. Bohner, and D. O'Regan, “Time scale boundary value problems on infinite intervals,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 27–34, 2002.

C. Wang and R. P. Agarwal, “Uniformly rd-piecewise almost periodic functions with applications to the analysis of impulsive D-dynamic system on time scales,” Applied Mathematics and Computation, vol. 259, pp. 271–292, 2015.

C. Wang and R. P. Agarwal, “Almost periodic dynamics for impulsive delay neural networks of a general type on almost periodic time scales,” Communications in Nonlinear Science and Numerical Simulation, vol. 36, pp. 238–251, 2016.

V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, Boston, Mass, USA, 1996.

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

G. Bangerezako, “An Introduction to q-Difference Equations,” Journal of Difference Equations and Applications, vol. 17, no. 9, pp. 1229–1249, 2011.

R. Hilscher and C. C. Tisdell, “Terminal value problems for first and second order nonlinear equations on time scales,” Electronic Journal of Differential Equations, No. 68, 21 pages, 2008.

N. J. Ford, M. L. Morgado, and M. Rebelo, “A nonpolynomial collocation method for fractional terminal value problems,” Journal of Computational and Applied Mathematics, vol. 275, pp. 392–402, 2015.