Open Access

A Linear Difference Scheme for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

Boundary Value Problems20102010:781750

DOI: 10.1155/2010/781750

Received: 24 August 2010

Accepted: 14 November 2010

Published: 30 November 2010

Abstract

We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term by finite difference method. A linear three-level implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify that the method is accurate and efficient.

1. Introduction

A symmetric version of regularized long wave equation (SRLWE),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ1_HTML.gif
(1.1)
has been proposed to model the propagation of weakly nonlinear ion acoustic and space charge waves [1]. The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq1_HTML.gif solitary wave solutions are
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ2_HTML.gif
(1.2)
The four invariants and some numerical results have been obtained in [1], where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq2_HTML.gif is the velocity, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq3_HTML.gif . Obviously, eliminating https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq4_HTML.gif from (1.1), we get a class of SRLWE:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ3_HTML.gif
(1.3)

Equation (1.3) is explicitly symmetric in the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq6_HTML.gif derivatives and is very similar to the regularized long wave equation that describes shallow water waves and plasma drift waves [2, 3]. The SRLW equation also arises in many other areas of mathematical physics [46]. Numerical investigation indicates that interactions of solitary waves are inelastic [7]; thus, the solitary wave of the SRLWE is not a solution. Research on the wellposedness for its solution and numerical methods has aroused more and more interest. In [8], Guo studied the existence, uniqueness, and regularity of the numerical solutions for the periodic initial value problem of generalized SRLW by the spectral method. In [9], Zheng et al. presented a Fourier pseudospectral method with a restraint operator for the SRLWEs and proved its stability and obtained the optimum error estimates. There are other methods such as pseudospectral method, finite difference method for the initial-boundary value problem of SRLWEs (see [915]).

In applications, the viscous damping effect is inevitable, and it plays the same important role as the dispersive effect. Therefore, it is more significant to study the dissipative symmetric regularized long wave equations with the damping term
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq7_HTML.gif are positive constants, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq8_HTML.gif is the dissipative coefficient, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq9_HTML.gif is the damping coefficient. Equations (1.4)-(1.5) are a reasonable model to render essential phenomena of nonlinear ion acoustic wave motion when dissipation is considered. Existence, uniqueness, and wellposedness of global solutions to (1.4)-(1.5) are presented (see [1620]). But it is difficult to find the analytical solution to (1.4)-(1.5), which makes numerical solution important.

To authors' knowledge, the finite difference method to dissipative SRLWEs with damping term (1.4)-(1.5) has not been studied till now. In this paper, we propose linear three level implicit finite difference scheme for (1.4)-(1.5) with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ6_HTML.gif
(1.6)
and the boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ7_HTML.gif
(1.7)

We show that this difference scheme is uniquely solvable, convergent, and stable in both theoretical and numerical senses.

Lemma 1.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq11_HTML.gif , the solution of (1.4)–(1.7) satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq12_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq14_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq15_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq16_HTML.gif is a generic positive constant that varies in the context.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ8_HTML.gif
(1.8)
Multiplying (1.4) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq17_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq18_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ9_HTML.gif
(1.9)
According to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ10_HTML.gif
(1.10)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ11_HTML.gif
(1.11)
Then, multiplying (1.5) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq19_HTML.gif and integrating over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq20_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ12_HTML.gif
(1.12)
By
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ13_HTML.gif
(1.13)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ14_HTML.gif
(1.14)
Adding (1.14) to (1.11), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ15_HTML.gif
(1.15)

So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq21_HTML.gif is decreasing with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq22_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq24_HTML.gif . Then, it indicates that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq25_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq26_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq27_HTML.gif . It is followed from Sobolev inequality that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq28_HTML.gif .

2. Finite Difference Scheme and Its Error Estimation

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq30_HTML.gif be the uniform step size in the spatial and temporal direction, respectively. Denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq32_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq35_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq36_HTML.gif . We define the difference operators as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ16_HTML.gif
(2.1)
Then, the average three-implicit finite difference scheme for the solution of (1.4)–(1.7) is as follow:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ17_HTML.gif
(2.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ18_HTML.gif
(2.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ19_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ20_HTML.gif
(2.5)

Lemma 2.1.

Summation by parts follows [12, 21] that for any two discrete functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq37_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ21_HTML.gif
(2.6)

Lemma 2.2 (discrete Sobolev's inequality [12, 21]).

There exist two constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq39_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ22_HTML.gif
(2.7)

Lemma 2.3 (discrete Gronwall inequality [12, 21]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq41_HTML.gif are nonnegative functions and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq42_HTML.gif is nondecreasing. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq43_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ23_HTML.gif
(2.8)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq44_HTML.gif .

Theorem 2.4.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq46_HTML.gif , then the solution of (2.2)–(2.5) satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ24_HTML.gif
(2.9)

Proof.

Taking an inner product of (2.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq47_HTML.gif   (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq48_HTML.gif ) and considering the boundary condition (2.5) and Lemma 2.1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ25_HTML.gif
(2.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq49_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ26_HTML.gif
(2.11)
we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ27_HTML.gif
(2.12)
Taking an inner product of (2.3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq50_HTML.gif  (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq51_HTML.gif ), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ28_HTML.gif
(2.13)
Adding (2.12) to (2.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ29_HTML.gif
(2.14)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ30_HTML.gif
(2.15)
Equation (2.14) can be changed to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ31_HTML.gif
(2.16)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq52_HTML.gif , and (2.16) is changed to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ32_HTML.gif
(2.17)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq53_HTML.gif is sufficiently small which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq54_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ33_HTML.gif
(2.18)
Summing up (2.18) from 1 to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq55_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ34_HTML.gif
(2.19)

From Lemma 2.3, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq56_HTML.gif , which implies that, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq58_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq59_HTML.gif . By Lemma 2.2, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq60_HTML.gif .

Theorem 2.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq62_HTML.gif , the solution of difference scheme (2.2)–(2.5) satisfies:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ35_HTML.gif
(2.20)

Proof.

Differentiating backward (2.2)–(2.5) with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq63_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ36_HTML.gif
(2.21)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ37_HTML.gif
(2.22)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ38_HTML.gif
(2.23)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ39_HTML.gif
(2.24)
Computing the inner product of (2.21) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq64_HTML.gif   (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq65_HTML.gif ) and considering (2.24) and Lemma 2.1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ40_HTML.gif
(2.25)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq66_HTML.gif . It follows from Theorem 2.4 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ41_HTML.gif
(2.26)
By the Schwarz inequality and Lemma 2.1, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ42_HTML.gif
(2.27)
Noting that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ43_HTML.gif
(2.28)
it follows from (2.25) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ44_HTML.gif
(2.29)
Computing the inner product of (2.22) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq67_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq68_HTML.gif ) and considering (2.24) and Lemma 2.1, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ45_HTML.gif
(2.30)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ46_HTML.gif
(2.31)
then (2.30) is changed to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ47_HTML.gif
(2.32)
Adding (2.29) to (2.32), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ48_HTML.gif
(2.33)
Leting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq69_HTML.gif , we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq70_HTML.gif . Choosing suitable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq71_HTML.gif which is small enough to satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq72_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ49_HTML.gif
(2.34)
Summing up (2.34) from 1 to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq73_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ50_HTML.gif
(2.35)

By Lemma 2.3, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq74_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq76_HTML.gif . It follows from Theorem 2.4 and Lemma 2.2 that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq77_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq78_HTML.gif .

3. Solvability

Theorem 3.1.

The solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq79_HTML.gif of (2.2)–(2.5) is unique.

Proof.

Using the mathematical induction, clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq80_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq81_HTML.gif are uniquely determined by initial conditions (2.4). then select appropriate second-order methods (such as the C-N Schemes) and calculate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq83_HTML.gif (i.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq85_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq86_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq87_HTML.gif are uniquely determined). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq89_HTML.gif are the only solution, now consider https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq90_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq91_HTML.gif in (2.2) and (2.3):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ51_HTML.gif
(3.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ52_HTML.gif
(3.2)
Taking an inner product of (3.1) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq92_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ53_HTML.gif
(3.3)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ54_HTML.gif
(3.4)
then it holds
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ55_HTML.gif
(3.5)
Taking an inner product of (3.2) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq93_HTML.gif and adding to (3.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ56_HTML.gif
(3.6)

which implies that (3.1)-(3.2) have only zero solution. So the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq94_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq95_HTML.gif of (2.2)–(2.5) is unique.

4. Convergence and Stability

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq97_HTML.gif be the solution of problem (1.4)–(1.7); that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq99_HTML.gif , then the truncation of the difference scheme (2.2)–(2.5) is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ57_HTML.gif
(4.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ58_HTML.gif
(4.2)

Making use of Taylor expansion, it holds https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq100_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq101_HTML.gif .

Theorem 4.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq103_HTML.gif , then the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq105_HTML.gif in the senses of norms https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq106_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq107_HTML.gif , respectively, to the difference scheme (2.2)–(2.5) converges to the solution of problem (1.4)–(1.7) and the order of convergence is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq108_HTML.gif .

Proof.

Subtracting (2.2) from (4.1) subtracting (2.3) from (4.2), and letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq109_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq110_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ59_HTML.gif
(4.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ60_HTML.gif
(4.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ61_HTML.gif
(4.5)
Computing the inner product of (4.3) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq111_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ62_HTML.gif
(4.6)
According to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ63_HTML.gif
(4.7)
it follow from Lemma 1.1, Theorems 2.4, and 2.5 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ64_HTML.gif
(4.8)
By the Schwarz inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ65_HTML.gif
(4.9)
Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ66_HTML.gif
(4.10)
it follows from (4.9)–(4.10) and (4.6) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ67_HTML.gif
(4.11)
Computing the inner product of (4.4) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq112_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ68_HTML.gif
(4.12)
Adding (4.12) to (4.11), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ69_HTML.gif
(4.13)
Leting
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ70_HTML.gif
(4.14)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ71_HTML.gif
(4.15)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq113_HTML.gif is sufficiently small which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq114_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ72_HTML.gif
(4.16)
Summing up (4.16) from 1 to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq115_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ73_HTML.gif
(4.17)
Select appropriate second-order methods (such as the C-N Schemes), and calculate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq116_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq117_HTML.gif , which satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ74_HTML.gif
(4.18)
Noticing that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ75_HTML.gif
(4.19)
we then have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ76_HTML.gif
(4.20)
By Lemma 2.3, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ77_HTML.gif
(4.21)
This yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ78_HTML.gif
(4.22)
By Lemma 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ79_HTML.gif
(4.23)

Similarly to Theorem 4.1, we can prove the result as follows.

Theorem 4.2.

Under the conditions of Theorem 4.1, the solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq118_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq119_HTML.gif of (2.2)–(2.5) is stable in the senses of norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq120_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq121_HTML.gif , respectively.

5. Numerical Simulations

Since the three-implicit finite difference scheme can not start by itself, we need to select other two-level schemes (such as the C-N Scheme) to get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq123_HTML.gif . Then, reusing initial value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq125_HTML.gif , we can work out https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq126_HTML.gif . Iterative numerical calculation is not required, for this scheme is linear, so it saves computing time.

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq127_HTML.gif , the damping does not have an effect and the dissipative will not appear. So the initial conditions of (1.4)–(1.7) are same as those of (1.1):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Equ80_HTML.gif
(5.1)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq128_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq130_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq131_HTML.gif . Since we do not know the exact solution of (1.4)-(1.5), an error estimates method in [21] is used: a comparison between the numerical solutions on a coarse mesh and those on a refine mesh is made. We consider the solution on mesh https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq132_HTML.gif as the reference solution. In Table 1, we give the ratios in the sense of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq133_HTML.gif at various time steps.
Table 1

The error ratios in the sense of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq134_HTML.gif at various time steps.

  

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq135_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq136_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq137_HTML.gif

μ

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq138_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq139_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq140_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq141_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq142_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq143_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq144_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq145_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq146_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq147_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq148_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq149_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq150_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq151_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq152_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq153_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq154_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq155_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq156_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq157_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq158_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq159_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq160_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq161_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq162_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq163_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq164_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq165_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq166_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq167_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq168_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq169_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq170_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq171_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq172_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq173_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq174_HTML.gif

 

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq175_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq176_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq177_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq178_HTML.gif

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq179_HTML.gif , a wave figure comparison of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq180_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq181_HTML.gif at various time steps is as in Figures 1 and 2.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Fig1_HTML.jpg
Figure 1

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq182_HTML.gif , the wave graph of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq183_HTML.gif at various times.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_Fig2_HTML.jpg
Figure 2

When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq184_HTML.gif , the wave graph of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq185_HTML.gif at various times.

From Table 1, it is easy to find that the difference scheme in this paper is second-order convergent. Figures 1 and 2 show that the height of wave crest is more and more low with time elapsing due to the effect of damping and dissipativeness. It simulates that the continue energy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F781750/MediaObjects/13661_2010_Article_954_IEq186_HTML.gif of problem (1.4)–(1.7) in Lemma 1.1 is digressive. Numerical experiments show that the finite difference scheme is efficient.

Declarations

Acknowledgments

The work of Jinsong Hu was supported by the research fund of key disciplinary of application mathematics of Xihua University (Grant no. XZD0910-09-1). The work of Youcai Xu was supported by the Youth Research Foundation of Sichuan University (no. 2009SCU11113).

Authors’ Affiliations

(1)
School of Mathematics and Computer Engineering, Xihua University
(2)
School of Mathematics, Sichuan University

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© The Author(s) Jinsong Hu et al. 2010

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