A Comparative Study of Wavelets Methods for Solving Non-Linear Two- Dimensional Boussinesq System of Type BBM-BBM

In this paper, numerical techniques based on the wavelets methods are proposed for the numerical solution of non-linear two-dimensional BBM-BBM system and we compared between them. Two methods used in numerical solutions, are the Haar wavelets and Legendre wavelets methods.In addition, we derived formulas of integrals for Legendre wavelets analytically. Its efficiency is tested by solving an example for which the exact solution is known. The accuracy of the numerical solutions is quite high even if the number of calculation points is small, by increasing the number of collocation points, the error of the solution rapidly decreases. We have found that the Legendre wavelets method is better and closer to the exact solution than the Haar wavelets method.

Legendre wavelets is presented. Section 6 numerical results are presented. Concluding remarks are given in section 7.

Haarwavelets
As a powerful mathematical tool, Wavelet analysis has been widely used in image digital processing, quantum field theory, numerical analysis and many other fields in recent years.
The Haar functions are an orthogonal family of switched rectangular waveforms where amplitudes can differ from one function to another. They are defined in the interval [0,1) by [9]: The operational matrix of integration P, which is a 2M square matrix, is defined by the equation: [11] Lipek, U. found the general form of v-times of integrals [11]:  Integrate equation (3) from (0) to (1), we get the following notation: Usually, the series expansion of (7) contains infinite terms for a general smooth function u(x). However, If u(x) is approximated as piecewise constant during each subinterval, then u(x) will be terminated at finite m terms, that is: Where, T is referring to the transpose. can be expanded into Haar series by [16]: Where the coefficient matrix  (9) can be written into the discrete form by: by using equations (2) and (3), we get [16]: performing the double integration, we obtain: , k can assume any positive integer, m is the order for legendre polynomials and t is the normalized time. They are defined on the interval ) 1 , 0 [ by: [1,12] ( ) where m=0,1,….,M-1. in equation (1), the coefficient

Legendre Wavelets
Here Lm(t) are the well-known legendre polynomials of the order m, which are orthogonal to the weight function w(t)=1 and satisfy the following recursive formula: (15) such that the set of legendre wavelets are an orthonormal set. , in which (.,.) denotes the inner product.
If the infinite series in (16) are truncated, then (16) can be written as: may be expanded as:   Now, we derive the operational matrix of integration P, which is a ( 1 2 1  − M k ) square matrix is defined by the equation: These integrals can be evaluated by using equation (14), we get:  We also introduce the following notation:

Mathematical Model
We consider the non-linear two-dimensional coupled Benjamin-Bona-Mahony (BBM-BBM) system which has the form [8,15]: is a bounded open set in 2 IR , and b, d > 0 and the initial data: and zero Dirichlet homogenous boundary conditions [8,15]: is proportional to the deviation of the free surface from its rest, while )) , = is proportional to the horizontal velocity of the fluid at some height. Specifically, we have the so-called BBM-BBM system corresponds to 6 1 , 3 , we change the variables: * Then, the system (25) becomes: With the initial and boundary conditions: The solution by the Haar wavelets method is started by dividing the interval (0,T] into N equal parts of length can be expanded in terms of Haar wavelets as follows: wewell focus on the function Integrating (31b) with respect to ( t) from ) ( s t to (t) and double integrating with respect to (x) from (0) to (x) , and double integrating with respect to (y) from (0) to (y) ,we obtain: We can reduce the order of boundary conditions used in equations (34)-(36) by using the boundary condition at x=1 and notation (6) is defined in equation (6).By substituting equation (37) Similarly, and by using the boundary condition at y=1 and notation (6), we get: By substituting equation (40) in equation (39), we get: Now, the derivatives of equation (41) with respect (t),(x) and (y), we get:

,
We can write system (46) by the form: Also, we get: The system (48) is the Lyapunov matrix equations which can be solved by one of the packages [4] or by using MATLAB Language: X=Lyap(A,B,C) To solve the equation AX+XB+C=0 ,such that the matrices A,B and C must have compatible dimensions but need not be square.Finally, The solution of the problem is found according to (41). Now, we use the Legendre wavelets to solve the system, that is, we can replace the Legendre wavelets instead of Haar wavelets in equations (41)-(45) and by substituting in the system (28), we obtain:

Numerical Experiments
In this section, we present the results of two-dimensional BBM-BBM system (28) which solved numerically by using the wavelets technique.
In the first example, we took zero Dirichlet homogenous boundary conditions for v and u ,  on the whole boundary in the square      (2), the corresponding right hand side in order to obtain the 2 L norm of the error between the exact solution and the numerical solution by using the Haar wavelets and Legendre wavelets, respectively.  In the second example, we consider the numerical solution of 2D BBM-BBM system (28) with initial and homogeneous Dirichlet boundary conditions [15]:

Conclusions
In this paper, we develop an accurate and efficient the wavelets methods for solving non-linear two-dimensional BBM-BBM system by convert the partial differential equation into a simple Lyapunov matrix equation.
The benefits of the wavelets approach are sparse matrices of representation, fast transformation and possibility of implementation of fast algorithms. It's worth mentioning that the wavelets solution provides excellent results even for small values of (2M) as noted in table (1). Also, when 2M=64 , 2M=128 , …, we can obtain the results closer to the exact values. We have also been reducing the boundary conditions used in the solution by using the notation (6) when x=L respect to space and the results were a high resolution. Matlab language is used in finding the results and figure draw, its characteristic at high accuracy and large speed.
Also, we compared between the wavelet methods in the numerical solution for non-linear BBM-BBM system and we have found that the Legendre wavelets method is better and closer to the exact solution of the Haar wavelets method as shown in table (1).