1Nhatrang Educational College, 01 Nguyen Chanh Street, Nhatrang City, Vietnam 2Department 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
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.
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 [1–3]).
Equation (1.1)1 is related to the Kirchhoff equation
presented by Kirchhoff in 1876 (see ). This equation is an extension of the classical D’Alembert 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, 5–14] and references therein.
In , 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 , 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 .
Lemma 2.1. There exist two constants and such that, for all , we have(i),
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),
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 .
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
With , we put , and .
3. The Hight Order Iterative Schemes
Fix , we make the following assumptions: ();
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
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 ). Consider as in Lemma 2.5 the basis for and put
where the coefficients satisfy the system of the following nonlinear differential equations:
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)–(3.11) has a unique solution on an interval . Proof of Lemma 3.2. The system of (3.8)–(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:
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 ,
On the other hand, by
By Lemma 2.1, (iii), and the assumption (), we deduce from (3.16) that
It follows that
Hence, we obtain choosing and, such that
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
In order to consider , we also note that
and satisfy the following inequality:
It implies that
It remains to estimate . By
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)–(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
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)–(3.54) that
where are defined by (3.49)1. Proof (iv). We have
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
(iv.1.2) Estimating . It follows from
(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