- Research Article
- Open Access
Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First Case
© L. H. Cao and J. M. Zhang. 2010
- Received: 13 July 2010
- Accepted: 27 October 2010
- Published: 31 October 2010
We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation where and are integers, and have asymptotic expansions of the form , , for large values of , , and .
- Asymptotic Expansion
- Difference Equation
- Asymptotic Solution
- Error Bound
- Integer Variable
for large values of , , , and .
Unlike the method used by Olver  to treat asymptotic solutions of second-order linear differential equations, the method used in Wong and Li's papers cannot give us way to obtain error bounds of these asymptotic solutions. Only order estimations were given in their papers. The estimations of error bounds for these asymptotic solutions to (1.1) were given in  by Zhang et al. But the problem of obtaining error bounds for these asymptotic solutions to (1.2) is still open. The purpose of this and the next paper (Error bounds for asymptotic solutions of second-order linear difference equations II: the second case) is to estimate error bounds for solutions to (1.2). The idea used in this paper is similar to that of Olver to obtain error bounds to the Liouville-Green (WKB) asymptotic expansion of solutions to second-order differential equations. It should be pointed out that similar method appeared in some early papers, such as Spigler and Vianello's papers [6–9].
In Wong and Li's second paper , two different cases were given according to different values of parameters. The first case is devoted to the situation when , and in the second case as where . The whole proof of the result is too long to understand, so we divide the estimations into two parts, part I (this paper) and part II (the next paper), which correspond to the different two cases of , respectively.
In the rest of this section, we introduce the main results of  in the case that is positive. In the next section, we give two lemmas on estimations of bounds for solutions to a special summation equation and a first order nonlinear difference equation which will be often used later. Section 3 is devoted to the case when . And in Section 4, we discuss the case when . The next paper (Error bounds for asymptotic solutions of second-order linear difference equations II: the second case) is dedicated to the case when .
1.1. The Result in  When
1.2. The Result in  When
In the following sections, we will discuss in detail the error bounds of the proceeding asymptotic solutions of (1.2). Before discussing the error bounds, we consider some lemmas.
2.1. The Bounds for Solutions to the Summation Equation
So we get the bound of any solution for the "summary equation" (2.1). Next we consider a nonlinear first-order difference equation.
2.2. The Bound Estimate of a Solution to a Nonlinear First-Order Difference Equation
has a solution such that is bounded by a constant , when is big enough.
3.1. The Error Bound for the Asymptotic Expansion of
where is the reminder and the coefficient of is zero. So is a constant. And satisfies being a constant; here we have made use of the definitions of in (1.5), (1.7), and .
Equation (3.8) is a second-order linear difference equation with two known linear independent solutions. Its coefficients are quite similar to those in (3.3). This reminds us to rewrite (3.3) in the form similar to (3.8).
are finite. Equation (3.10) is a inhomogeneous second-order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the associated homogeneous linear difference equation(3.8).
Now we estimate the bound of the function .
is an integer which is large enough such that , when .
Here we have made use of .
Here we have made use of (1.5) and (1.7).
Here we also have made use of (1.5) and (1.7).
that is, and .
3.2. The Error Bound for the Asymptotic Expansion of ( )
where is the reminder and is a constant.
when . Thus the sequence is increasing when .
Thus we complete the estimate of error bounds to asymptotic expansions of solutions of (1.2) as .
4.1. The Error Bound for the Asymptotic Expansion of
Now let us come to the case when . This time a difference equation which has two known linear independent solutions is also constructed for the purpose of comparison for (1.2).
is a constant. Here we have made use of the definitions of , , , in (1.9), (1.11) and .
where has the property that is a constant. Equation (4.14) is an inhomogeneous second-order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the associated homogeneous linear difference equation (4.13).
is a solution of (4.14).
4.2. The Error Bound for the Asymptotic Expansion of
Now we completed the estimate of the error bounds for asymptotic solutions to second order linear difference equations in the first case. For the second case, we leave it to the second part of this paper: Error Bound for Asymptotic Solutions of Second-order Linear Difference Equation II: the second case.
The authors would like to thank Dr. Z. Wang for his helpful discussions and suggestions. The second author thanks Liu Bie Ju Center for Mathematical Science and Department of Mathematics of City University of Hong Kong for their hospitality. This work is partially supported by the National Natural Science Foundation of China (Grant no. 10571121 and Grant no. 10471072) and Natural Science Foundation of Guangdong Province (Grant no. 5010509).
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