Numerical Solution for Non-linear Boussinesq System Using the Haar Wavelet Method

101 Numerical Solution for Non-linear Boussinesq System Using the Haar Wavelet Method Ekhlass S. Al-Rawi drekhlass-alrawi@uomosul.edu.iq Ahmed F. Qasim ahmednumerical@yahoo.com College of Computer Sciences and Mathematics University of Mosul, Mosul, Iraq Received on: 02/09/2012 Accepted on: 30/01/2013 ABSTRACT In this paper, an operational matrix of integrations based on the Haar wavelet method is applied for finding the numerical solution of non-linear third-order boussinesq system and the numerical results were compared with the exact solution. The accuracy of the obtained 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 as shown by solving an example. We have been reduced the boundary conditions in the solution by using the finite differences method with respect to time. Also we have reduced the order of boundary conditions used in the numerical solution by using the boundary condition at x=L instead of the derivatives of order two with respect to space.

Using the operational matrix of an orthogonal function to perform integration for solving, identifying and optimizing a linear dynamic system has several advantages: (i) the method is computer oriented, thus solving higher order differential equation becomes a matter of dimension increasing; (ii) the solution is a multi-resolution type and (iii) the answer is convergent, even the size of increment is very large .
The main characteristic of the operational method is to convert a differential equation into an algebraic one, and the core is the operational matrix for integration. We are the discrete representation of the basic functions which are orthogonal on the interval [0,1) and  Q is the operational matrix for integration of ( ) t  [9].
Lepik [6] studied the application of the Haar wavelet transform to solve integral and differential equations, he demonstrated that the Haar wavelet method is a powerful tool for solving different types of integral equations and partial differential equations. AL-Rawi and Qasem [1] found the numerical solution for nonlinear Murray equation by the operational matrices of Haar wavelet method and compared the results of this method with the exact solution, they transformed the nonlinear Murray equation into a linear algebraic equations that can be solved by Gauss-Jordan method. Hariharan and Kannan [5] developed an accurate and efficient Haar transform or Haar wavelet method for some of the well-known non-linear parabolic partial differential equations. The equations include the Nowell-whitehead equation, Cahn-Allen equation, FitzHugh-Nagumo equation, and other equations.
Many authors have studied the solution for non-linear boussinesq systems. Bona et al [3] derived a four-parameter family of boussinesq systems from the two-dimensional Euler equations for free-surface flow and formulated criteria to help decide which of these equations one might choose in a given modeling situation. The analysis of the systems according to these criteria is initiated. Bona et al. [4] are investigating numerically generalized solitary wave solutions of two coupled Kdv systems of boussinesq type. We present numerical experiments describing the generation and evolution of such waves, their interactions, the resolution of general initial profiles into sequences of such waves, and their stability under small perturbations. Anotonopoulos et al. [2] are studing three initial-boundary-value problems for Bona-Smith family of Boussinesq systems corresponding, respectively, to nonhomogeneous Dirichlet, reflection, and periodic boundary conditions posed at the endpoints of a finite spatial interval, and establish existence and uniqueness of their solutions. He proved that the initial-boundary-value problem with Dirichlet boundary conditions is well posed in appropriate spaces locally in time, while the analogous problems with reflection and periodic boundary conditions are globally well posed under mild restrictions on the initial data. Micu et al. [8] are studing the internal controllability and stabilizability of a family of boussinesq systems. The space of the controllable data for the associated linear system is determined for all values of the four parameters. As an application of this newly established exact controllability, some simple feedback controls are constructed such that the resulting closed-loop systems are exponentially stable.
In this paper, we study the numerical solution for non-linear third-order boussinesq system by the operational matrices of Haar wavelet method and we compare the results of this method with the exact solution.
We organized our paper as follows. In section 2, the Haar wavelet is introduced and an operational matrix is established. Section 3 function approximation is presented. In section 4 Haar wavelets used to solve nonlinear boussinesq system. Section 5 deals with the reducing of the order of boundary conditions used in the numerical solution. Numerical results are presented in section 6. Concluding remarks are given in section 7.

Haar wavelet
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 ( [5] , [6] and [7] ): indicates the level of the wavelet; k=0,1,2,..,m-1 is the translation parameter. Maximal level of resolution is J. The indix i is calculated according to the formula i=m+k+1; in the case of minimal values. m=1,k=0 we have i=2, the maximal value of i is The operational matrix of integration P, which is a 2M square matrix, is defined by the equation: [6] These integrals can be evaluated using equation (2) are given below: We also introduce the following notations:

Function Approximation
Any square integrable function u(x) in the interval [0,1] can be expanded by a Haar series of infinite terms : where the Haar coefficients i c are determined as follows: Such that the following integral square error  is minimized: Usually, the series expansion of (10) contains infinite terms for smooth u(x). If u(x) is piecewise constant by itself, or may be approximation as piecewise constant during each subinterval, then u(x) will be terminated at finite m terms, that is:

. Mathematical Model
In this section, we will consider the following nonhomogeneous system [8]: where f and g are given functions. For …(12) and periodic boundary conditions : where the parameters a, b, c, and d are constant, but not completely independent . they are in fact required to fulfill the relations : the values of q0 depend on the parameters c and d in a similar way . Since the Haar wavelets are defined for x[0,1], we must first normalize system (11) and initial-boundary condition (12) and (13) in regard to x .
We change the variables [7]: x x x then the system (11) and the initial -boundary conditions (12) become: with initial -boundary conditions: Now, let us divide the interval (0,T] into N equal parts of length ∆t = T /n and denote to ts = (s-1) ∆t , s = 1,2,…..n.
We assume that ) can be expanded in terms of Haar wavelets as follows: . Integrating (15) with respect to t from s t to t and third with respect to x from 0 to x , we obtain: Now, the differential of equation (21) with respect t , we get: We can use the backward finite different formula in equation (22) , we get :

Numerical Solution for Non-linear Boussinesq System Using the Haar Wavelet Method
107 Now, the integrating equation (17) with respect to x from 0 to x , we get : then, using the backward finite difference formula : Similarly, we assume that ) can be expanded in terms of Haar wavelets as follows: Now, we can substitute the equations (18)-(25) respect to u(x,t) and the equations (27) -(33) respect to v(x,t) in the system (14) , we get : are found by using the boundary condition (16) and the functions ) are found by using initial conditions (15) in the first step .

Reducing of the Order of Boundary Conditions:
We can reduce the order of the boundary conditions used in equations (18)-(25) by using the boundary condition at x=1 and notation (8) Now, the different equation (41) with respect t , we get: Similarly, we can reduce the order of the boundary conditions used in equations (27)-(33) by using the boundary condition at x=1 and notation (8) . When x = 1 from equation (29), we get : Now, we can substitute the equations (38) -(44) with respect to u(x,t) and the equations (46) -(52) with respect to v(x,t) in the system (14) , we get :

111
The functions are found by using the boundary conditions (69), and the functions ) are found by using initial conditions (69) in the first step .
The Haar coefficient vectors T m C and T m D are calculated from the solution of the linear system (53a) and (53b)

Conclusions
In this paper, solving the non-linear third-order Boussinesq system by using Haar wavelet method was discussed. The fundamental idea of Haar wavelet method is to convert the third-order Boussinesq system into a group of algebra equations which involves a finite number of variables.
We found that Haar wavelet had good approximation effect by comparing with exact solution of Boussinesq system at the same time. The bigger resolution J is obtained more accurate approximation in the solution, as note in tables (1) and (2) when m=16. Also, when m=32 , m=64 , m=128 , …, we can obtain the results closer to the exact values as noted in table (5).
We have also been reducing the boundary conditions used in the solution by using the finite different method with respect to time and by using the notation (8) when x=L respect to space and the results were of a high resolution as note in tables (3) and (4) and Figure (2). Matlab language is used in finding the results and figure draw, it's a characteristic at high accuracy and large speed.