THE CONVERGENCE ESTIMATES FOR GALERKIN-WAVELET SOLUTION OF PERIODIC PSEUDODIFFERENTIAL INITIAL VALUE PROBLEMS

Using the discrete Fourier transform and Galerkin-Petrov scheme, we get some results on the solutions and the convergence estimates for periodic pseudodiﬀer-ential initial value problems


Introduction.
In recent years, wavelets have been developing intensively and have become a powerful tool to study mathematics and technology, for example, the theory of the singular integral, singular integro-differential equations, the areas such as sound analysis, image compression, and so on (see [9,10] and references therein).In this paper, we use a scaling function and a multilevel approach to estimate the error of the problem ∂u(x, t) ∂t = a • Au(x, t), x ∈ n , t > 0, a ∈ R, u(x, 0) = u 0 (x), x ∈ n , (1.1) where A is a pseudodifferential operator (see [1,2,3,4,6,8,9,12]) with a symbol σ ∈ C ∞ (R n ), σ is positively homogeneous of degree r > 0 such that n = R n /Z n , and [u 0 ](x) = k∈Z n u 0 (x + k) is a periodic operator.We discuss only problem (1.1) with the following condition: aσ (ξ) ≤ 0, ∀ξ ∈ Z n . (1.3)

Preliminaries and notations. The continuous Fourier transform of the function
with the inverse Fourier formula (see [4,8,11]).
The discrete Fourier transform of the function and the inverse Fourier transform is (see [6]).Some simple properties of the discrete Fourier transform are where where where then H s ( n ) is the Sobolev space endowed with the norm and the inner product Here, we also define the discrete Sobolev space H s d (R n ), s ∈ R, of the functions f ∈ H s (R n ) such that the following norm is finite: ( Using the variable separate method and the discrete Fourier transform, the solution of problem (1.1) can be represented as where E(t) is a differentiable function and We recall that a multiresolution approximation (MRA) of L 2 (R n ) is, as a definition, an increasing sequence V j , j ∈ Z, of closed linear subspaces of L 2 (R n ) with the following properties: There exists a function, called the scaling function (SF) φ(x) ∈ V 0 , such that the sequence is a Riesz basic of V 0 (see [5,9]).An SF φ is called µ-regular (µ ∈ N) if, for each m ∈ N, there exists c m such that the following condition holds: It follows from (2.14), (2.15), (2.16), and (2.17) that V j = span{φ jk (x), k ∈ Z n }, j ∈ Z.
(2) For each µ ∈ N, there exists an SF φ(x) with compact support, and φ(x) is µ-regular; so in what follows, we always assume that φ has compact support and is µ-regular (see [9]).
Using the periodic operator and an MRA of L 2 (R n ), we can build an MRA of L 2 ( n ) with the SF [φ] as follows. Denote ) where [6]).
For each j ≥ 0, let P j : L 2 ( n ) → [V j ] be the orthogonal projection from L 2 ( n ) on [V j ], which has the following property.
for all u ∈ H q ( n ), where c is independent of j and u. (2.23)

The Galerkin-wavelet solution.
Fix a distribution with compact support η ∈ H −s (Γ ), where s ≥ 0 satisfying AV h ⊂ H s ( n ) and where Γ ⊂ R n is some fixed compact domain such as a hypercube.For f ∈ H s ( n ), define The space is contained in (AV h ) , which is the dual of AV h .The corresponding Galerkin-Petrov-wavelet scheme is then given by where Then the scheme (3.3) and (3.4) provides an algebra equation system and the solution can be solved by Fourier series.
Lemma 3.1.The following formulas hold true: ( Proof.(a) It follows from (2.3) and (2.19) that The proof of the lemma is complete.(3.12)(b) Similarly, we can get the second assertion.
The following lemma is extracted from [6].

Write
We have thus If the triplet (σ ,φ,η) is admissible, then it follows from (3.22) and Lemma 3.4 that δ(ξ) Theorem 4.2.Suppose that r +s ≤ s ≤ p, 0 ≤ m ≤ s, and it is assumed that the triplet (σ ,φ,η) is admissible.Then, for where c is independent of u, h, and u 0 .
Proof.It follows from (4.8) that The equality Hence, from (4.7) and (4.12), we obtain By (1.3) and the admissibility of the triplet (σ ,φ,η), inequality (4.13) is also valid for all ξ ∈ Z n .Hence, for each 0 ≤ m ≤ s, r + s ≤ s ≤ p, and 0 ≤ t ≤ T , we get  The theorem is thus proved.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Theorem 4 . 4 .
If all the hypotheses of Theorem 4.2 and assumption (4.17) are satisfied, thenu − u h m ≤ ch s−r u 0 m+s,d + ch s u 0 m+s ,(4.18) which has compact support, or any function, for which k+[0,1] n |f (x)| 2 dx decays exponentially as |k| tends to infinity, belongs to ᏸ 2 .The periodic operator [u] is totally defined if u ∈ ᏸ 2 .Here, we assume that u 0 ∈ ᏸ 2 .