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International Journal of Differential Equations

VolumeΒ 2011Β (2011), Article IDΒ 679528, 31 pages

http://dx.doi.org/10.1155/2011/679528

## A High Order Iterative Scheme for a Nonlinear Kirchhoff Wave Equation in the Unit Membrane

^{1}Nhatrang Educational College, 01 Nguyen Chanh Street, Nhatrang City, Vietnam^{2}Department of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Street, District 5, Ho Chi Minh City, Vietnam

Received 5 May 2011; Accepted 16 October 2011

Academic Editor: BashirΒ Ahmad

Copyright Β© 2011 Le Thi Phuong Ngoc and Nguyen Thanh Long. 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

A high-order iterative scheme is established in order to get a convergent sequence at a rate of order () to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.

#### 1. Introduction

In this paper we consider the initial and boundary value problem where are given functions satisfying conditions specified later, , and is a given constant.

Equation (1.1)_{1} herein is the bidimensional nonlinear wave equation describing nonlinear vibrations of the unit membrane . In the vibration process, the area of the unit membrane and the tension at various points change in time. The condition on the boundary describes elastic constraints, where the constant has a mechanical signification. The boundary condition is satisfied automatically if is a classical solution of the problem (1.1), for example, with . This condition is also used in connection with Sobolev spaces with weight (see [1β3]).

Equation (1.1)_{1} is related to the Kirchhoff equation
presented by Kirchhoff in 1876 (see [4]). This equation is an extension of the classical DβAlembert wave equation which considers the effects of the changes in the length of the string during the vibrations. The parameters in (1.2) have the following meanings: is the lateral deflection, is the length of the string, is the area of the cross-section, is the Young modulus of the material, is the mass density, and is the initial tension.

The Kirchhoff wave equation of the form (1.1)_{1} received much attention. Many interesting results about the existence, stability, regularity in time variable, asymptotic behavior, and asymptotic expansion of solutions can be found, for example, in [2, 3, 5β14] and references therein.

In [2], in a special case, sufficient conditions were established for a quadratic convergence to the solution of (1.1) with and . Based on the ideas about recurrence relations for a third-order method for solving the nonlinear operator equation in [15], we extend the above result by the construction of a high-order iterative scheme for (1.1)_{1}, where and are more generalized.

In this paper, we associate with (1.1)_{1} a recurrent sequence defined by, with satisfying (1.1)_{2-3}. The first term is chosen as . If and , we prove that the sequence converges at a rate of order to a unique weak solution of the problem (1.1). This result is a relative generalization of [2, 3, 8, 9, 14, 16].

#### 2. Preliminary Results, Notations, Function Spaces

Put . We omit the definitions of the usual function spaces , and . For any function we define as and define the space as completion of the space with respect to the norm . Similarly, for any function we define as and define the space as completion of the space with respect to the norm . Note that the norms and can be defined, respectively, from the inner products Identifying with its dual we obtain the dense and continuous embedding . The inner product notation will be reutilized to denote the duality pairing between and .

We then have the following lemmas, the proofs of which can be found in [1].

Lemma 2.1. *There exist two constants and such that, for all , we have*(i)*,
*(ii)*,
*(iii)*. *

Lemma 2.2. *The embedding is compact.*

*Remark 2.3. *In Lemma 2.1, the two constants and can be given explicitly as and . We also note that for all (see [17, page 128/Lemma 5.40]). On the other hand, by and for all , it follows that . From both relations we deduce that for all .

Now, let the bilinear form be defined by
where is a positive constant. Then, there exists a unique bounded linear operator such that for all . We then have the following lemma.

Lemma 2.4. *The symmetric bilinear form defined by (2.2) is continuous on and coercive on , that is,*(i)*,
*(ii)*,
**for all , where and .*

The proof of Lemma 2.4 is straightforward and we omit it.

Lemma 2.5. *There exists an orthonormal Hilbert basis of the space consisting of eigenfunctions corresponding to eigenvalues such that*(i)*,
*(ii)*. **Note that it follows from (ii) that is automatically an orthonormal set in with respect to as inner product. The eigensolutions are indeed eigensolutions for the boundary value problem
*

The proof of Lemma 2.5 can be found in ([18, page 87, Theorem 7.7]) with and as defined by (2.2).

For any function we define as and define the space as completion of with respect to the norm . Note that is also a Hilbert space with respect to the scalar product and that can be defined also as .

We then have the following two lemmas the proof of which can be found in [1].

Lemma 2.6. *The embedding is compact.*

Lemma 2.7. *For all we have
*

For a Banach space , we denote by its norm, by its dual space, and by the Banach space of all real measurable functions such that

Let denote respectively.

With , we put , and .

#### 3. The Hight Order Iterative Schemes

Fix , we make the following assumptions: (); ()(i), (ii); (),

where . We put

With and satisfying assumptions and , respectively, for each given, we introduce the following constants:

For each and we get

We will choose as first initial term , suppose that and associate with the problem (1.1) the following variational problem.

Find so that where

Then, we have the following theorem.

Theorem 3.1. *Let assumptions hold. Then there exist a constant depending on and a constant depending on such that, for , there exists a recurrent sequence defined by (3.6), (3.7).*

*Proof. *The proof consists of several steps.*Step 1. *The Faedo-Galerkin approximation (introduced by Lions [19]). Consider as in Lemma 2.5 the basis for and put
where the coefficients satisfy the system of the following nonlinear differential equations:
where

Let us suppose that satisfies (3.5). Then we have the following lemma.

Lemma 3.2. *Let assumptions hold. For fixed and , then, the system (3.8)β(3.11) has a unique solution on an interval .**Proof of Lemma 3.2. *The system of (3.8)β(3.11) is rewritten in the form
and it is equivalent to the system of integral equations
for . Omitting the index , it is written as follows:
where ,

For every and that will be chosen later, we put , where , for each . Clearly is a closed nonempty subset in , and we have the operator . In what follows, we will choose and such that(i),
(ii). *Proof (i). *First we note that, for all ,
so
On the other hand, by
we have
By Lemma 2.1, (iii), and the assumption (), we deduce from (3.16) that

It follows that
Thus
where and

Hence, we obtain

choosing and, such that
where
Then
which means that maps into itself.*Proof (ii). *We now prove that, for all , for all ,
where is defined as (3.26).

Proof of (3.28) is as follows.

For all , for all , we have
where
so
in which
In order to consider , we also note that
where
and satisfy the following inequality:

It implies that
and then

It remains to estimate . By
we obtain
On the other hand,

Hence, we deduce from (3.39), (3.40) that

We deduce that
We note that

It follows from (3.28) that

By (3.25), it follows that is contractive. We deduce that has a unique fixed point in ; that is, the system (3.8)β(3.11) has a unique solution on an interval . The proof of Lemma 3.2 is complete.

The following estimates allow one to take constant for all and .*Step 2. *A priori estimates. Put
where
with is defined by (2.2). Then it follows that

We will now require the following lemma.

Lemma 3.3. *We have
**
where are defined as follows:
**Proof of Lemma 3.3. **Proof (i), (ii). *Note that
We deduce that
*Proof (iii). *We have
By (3.18)_{3}, we have
On the other hand, it follows from (3.49) and that

It follows from (3.52)β(3.54) that
where are defined by (3.49)_{1}.*Proof (iv). *We have

Hence
We shall estimate step by step the terms on the right-hand side of (3.57) as follows.*(iv.1) Estimating **.* We have
We will estimate step by step the terms as follows.*(iv.1.1) Estimating **.* We have
Hence
*(iv.1.2) Estimating **.* It follows from
that
*(iv.1.3) Estimating **.* Similarly, with
we obtain

It follows from (3.58), (3.60), (3.62), (3.64) that
*(iv.2) Estimating *. By the assumption , we deduce that