Numerical Solution for Non-linear Korteweg-de Vries-Burger ' s Equation Using the Haar Wavelet Method

In this paper, an operational matrix of integrations based on the Haar wavelet method is applied for finding numerical solution of nonlinear third-order korteweg-de Vries-Burger's equation, we compared this numerical results 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 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. يددعلا لحلا ةلداعمل korteweg-de Vries-Burger's مادختساب ةیطخلا ریغ جیوم ةقیرط ة Haar صخلملا ثحبلا اذه يف ، ىلع دمتعت يتلا تلاماكتلل لماوعلا ةفوفصم قیبطت مت جیوم ة Haar داجیلإ ةلداعمل يددعلا لحلا korteweg-de Vries-Burger's دقو ةثلاثلا ةبترلا نم ةیطخلا ریغ تنروق طوبضملا لحلا عم جئاتنلا . لولحلا ةقد نإ ةبوسحملا ةكبشلا طاقن ددع ناك اذإ ىتح ةیلاع اهیلع انلصح يتلا لایلق ً تدا ز املكو نم كلذ حیضوت مت دقو صقانتی أطخلاو دادزت ةقدلا ناف ةبوسحملا ةكبشلا طاقن ددع لاثم لح للاخ . لطملا ةیدودحلا طورشلا ةبتر ضیفخت اضیأ مت دقل لحلا يف ةبو * Assist. Prof. \ College of Computers Sciences and Math.\ University of Mosul ** College of Computers Sciences and Math.\ University of Mosul Received:1/10 /2011 ____________________Accepted: 21 /12 / 2011 Numerical Solution for Non-linear Korteweg-de ] 94 [ ةیهتنملا تاقورفلا ةقیرط مادختساب كلذو يددعلا ضیفخت مت كلذكو نمزلل ةبسنلاب دعبلل ةبسنلاب ةیدودحلا طورشلا ةبتر ةیاهن دنع ةیدودحلا طورشلا مادختساب كلذو ةرتفلا x=L ةیناثلا ةقتشملل ةیدودحلا طورشلا مادختسا نم لادب . 1.Introduction: As a powerful mathematical tool, Wavelet analysis has been widely used in image digital processing, quantum field theory, numerical analysis and many other field in recent years. Haar wavelets have been applied extensively for signal processing in communications and physics research, and more mathematically focused on differential equations and even nonlinear problems. After discrediting the differential equation in a convential way like the finite difference approximation, wavelets can be used for algebraic manipulations in the system of equations obtained which may lead to better condition number of the resulting system [11]. Using the operational matrix of an orthogonal function to perform integration for solving, identifying and optimizing a linear dynamic system has several advantages: (1) the method is computer oriented, thus solving higher order differential equation becomes a matter of dimension increasing; (2) the solution is a multi-resolution type and (3) the answer is convergent, even the size of increment is very large [10]. 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 start with the integral property of the basic orthonormal matrix, ( ) t f by write the following approximation: ( )( ) ( ) t Q dt t k k k t t t t f f f @ ò ò ò ò 43 42 10 0 0 0 ...... ...(1) Where ( ) ( ) ( ) ( ) [ ] 1 m 1 0 t t t t j j j = f r K r r in which the elements ( ) ( ) ( ) t , , t , t 1 m 1 0 j j j r K r r are the discrete representation of the basis functions which are orthogonal on the interval [0,1) and f Q is the operational matrix for integration of ( ) t f [10]. Many authors have studied the solution for nonlinear third-order korteweg-de vries-burger's (KdVB) equation. EL-Danaf T. (2002) is discuss the solution of the modified (KDVB) equation by using the collocation method with quintic splines and comparison between the numerical and exact solution, also he discuss the stability analysis of this method. Darvishia M. T. , Khanib F. and Kheybari S. (2007) are using the spectral collocation method to solve the KDVB equation numerically, and The Fourth Scientific Conference of the College of Computer Science & Mathematics ] 95 [ to reduce round off error, they are use central left and right darvishi's preconditioning. Lepik and Tamme (2007) derived the solution of nonlinear Fredholm integral equations via the Haar wavelet method, they are find that the main benefits of the Haar wavelet method are sparse representation, fast transformation, and possibility of implementation of fast algorithms especially if matrix representation is used. Lepik Uio (2007) 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. The method with far less degrees of freedom and with smaller CPU time provides better solutions then classical ones. Zhi S. LI-Y. and Qing-J. C. (2007) are establishes a clear procedure for finite-length beam problem and convection-diffusion equation solution via Haar wavelet technique, The main advantages of this method is its simplicity and small computation costs. Bhatta D. (2008) is studied the modified Bernstein polynomials for solve korteweg-de veries-burger's equation over the spatial domain. Bpolynomials are used to expand the desired solution requiring discreitization with only the time variable. AL-Rawi Ekhlass S. and Qasem A. F. (2010) 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. G. Hariharan · K. Kannan (2010) are develop an accurate and efficient Haar transform or Haar wavelet method for some of the wellknown nonlinear parabolic partial differential equations. The equations include the Nowell-whitehead equation, Cahn-Allen equation, FitzHughNagumo equation, and other equations. In this paper, we study the numerical solution for nonlinear thirdorder korteweg-de vries-burger's equation 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. Section 4 we use Haar wavelets to solve nonlinear KdVB equation. Section 5 Reducing of the order boundary conditions used in the numerical solution is presented .Section 6 numerical results are presented. Concluding remarks are given in section 7. Numerical Solution for Non-linear Korteweg-de ] 96 [ 2. 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 [6]:


1.Introduction:
As a powerful mathematical tool, Wavelet analysis has been widely used in image digital processing, quantum field theory, numerical analysis and many other field in recent years.
Haar wavelets have been applied extensively for signal processing in communications and physics research, and more mathematically focused on differential equations and even nonlinear problems.After discrediting the differential equation in a convential way like the finite difference approximation, wavelets can be used for algebraic manipulations in the system of equations obtained which may lead to better condition number of the resulting system [11].
Using the operational matrix of an orthogonal function to perform integration for solving, identifying and optimizing a linear dynamic system has several advantages: (1) the method is computer oriented, thus solving higher order differential equation becomes a matter of dimension increasing; (2) the solution is a multi-resolution type and (3) the answer is convergent, even the size of increment is very large [10].
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 start with the integral property of the basic orthonormal matrix, ( ) t f by write the following approximation: are the discrete representation of the basis functions which are orthogonal on the interval [0,1) and f Q is the operational matrix for integration of ( ) t f [10].Many authors have studied the solution for nonlinear third-order korteweg-de vries-burger's (KdVB) equation.
EL-Danaf T. ( 2002) is discuss the solution of the modified (KDVB) equation by using the collocation method with quintic splines and comparison between the numerical and exact solution, also he discuss the stability analysis of this method.
Darvishia M. T. , Khanib F. and Kheybari S. (2007) are using the spectral collocation method to solve the KDVB equation numerically, and

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 [6]: 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 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 . It is assumed that the value i=1 corresponds to the scaling function for which and discredits the Haar function ; in this way we get the coefficient matrix , which has the dimension 2M*2M.
The operational matrix of integration P, which is a 2M square matrix, is defined by the equation: [8] These integrals can be evaluated using equation (2) and first four of them are given:

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: Such that the following integral square error e 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: Where the coefficients [ ] Where T means transpose.[6] Numerical Solution for Non-linear Korteweg-de ] 98 [

Mathematical Model
Let us consider the nonlinear third-order korteweg-de Vries-Burger's (KdVB) equation which has the form [5]: With the initial and boundary conditions: where ε, ν and μ are positive parameters.ε is the coefficient of nonlinear t e r m s , ν i s t h e v i s c o si t y c o ef f i ci en t an d μ i s t h e co ef f i ci en t o f t h e dispersive term., we must first normalize equation (11) and initial-boundary conditions (12) in regard to x.We changing the variables [9]: Then equation ( 11) and (12) becomes: With the initial and boundary conditions: Where the row vector . Integrating (15) with respect to ( t) from ) ( s t to (t) and third with respect to (x) from (0) to (x) , we obtain: We can be reduce the boundary condition in equation (20) by using the finite difference method , we get:

Reducing of the order boundary conditions:
We can be reducing of order boundary conditions used in equations ( 16)-(21) by using the boundary condition at x=1 and notation (8) 16)-(21), we get: The Haar coefficients vector ) (m c is calculated from the system of linear equations (30).The solution of the problem is found according to (28).

Numerical results
In this section, we have solved Kdv-Burger's equation ( 13) with the initial-boundary conditions (14) by using two formula: a-) we have solved equation ( 13) with the initial-boundary conditions (14) by using the equation ( 22) such that [5]

…(32)
Now, by substitute the boundary condition (31) in equations ( 16)-( 22) we get: This process is started with: The exact solution of KdVB equation (13) in a closed form is given by [5]: The value of the constant E is large to be in the neighborhood of the boundary conditions [5].
Results of the computer simulation are presented in table (1)

Conclusions
In this paper, solving the nonlinear third-order KdV-Burger's equation by using Haar wavelet method was discussed.The fundamental idea of Haar wavelet method is to convert the differential equation 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 KdV-Burger's equation at the same time.The bigger resolution J is obtained more accurate approximation in the solution, as note in table (1) when m=16 and the table (2) when m=32.Also when m=64 , m=128 , …, we can obtain the results closer to the exact values.Figure (2) shows the Comparison between numerical solution when m=8 and m=16 with exact solution.
We have also been reducing the boundary conditions used in the solution by using the finite differences method with respect to time and by using the notation (9) when x=L respect to space and the results were a high resolution as note in table (3) and Figure (3).Matlab language is using in find the results and figure draw, it's characteristic at high accuracy and large speed.

Fig. 1 .
Fig.1.First eight Haar functions[6] the system of linear equations (22).The solution of the problem is found according to (19).The Fourth Scientific Conference of the College of Computer Science & Mathematics ] 101 [ substitute equation (23) in equations ( a positive constant.

Fig. ( 3 )
Fig. (3) Comparison of the numerical solutions and the exact solution When m=16.
instead of the derivatives