Numerical solution for the systems of variable-coefficient coupled Burgers equation by two-dimensional Legendre wavelets method

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Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 939466 Vol. 9 Issue (June 04) pp.

1 Available at Appl. Appl. Math. ISSN: Vol. 9 Issue (June 04) pp Applications and Applied Mathematics: An International Journal (AAM) Numerical solution for the systems of variablecoefficient coupled Burgers equation by twodimensional Legendre wavelets method Hossein Aminikhah and Sakineh Moradian Department of Applied Mathematics University of Guilan P.O. Bo 94 P.C Rasht Iran aminikhah@guilan.ac.ir; s.moradian6@yahoo.com Received: March 04; Accepted: May 6 04 Abstract In this paper a numerical method for solving the systems of variablecoefficient coupled Burgers equation is proposed. he method is based on twodimensional Legendre wavelets. wodimensional operational matrices of integration are introduced and then employed to find a solution to the systems of variablecoefficient coupled Burgers equation. wo eamples are presented to illustrate the capability of the method. It is shown that the numerical results are in good agreement with the eact solutions for each problem. Keywords: variablecoefficient coupled Burgers equation; twodimensional Legendre wavelets; operational matri integration MSC (00) No.: A35 4C40. Introduction he Burgers equation retains the nonlinear aspects of the governing equations in many applications such as the mathematical model of turbulence heat conduction and the approimate theory of flow through a shock wave traveling in a viscous fluid [Burger (948); Cole (95); Rashidi and Erfani (009)]. he study to coupled Burgers equations is very 34

2 AAM: Intern. J. Vol. 9 Issue (June 04) 343 significant in that the system is a simple model of sedimentation or evolution of scaled volume concentrations of two kinds of particles in fluid suspensions or colloids under the effect of gravity [Nee and Duan (998)]. It has been studied by many authors using different methods [Esipov (995); Biazar and Aminikhah (009); Abbasbandy and Darvishi (005); Na (00) and Aminikhah (03)]. In the present work a numerical algorithm based on the twodimensional Legendre wavelets is proposed and then applied to the nonlinear systems of variable coefficient coupled Burgers equation that can be written in the following basic form [Na (00) and Aminikhah (03)] u + r () t u + s () t uu + p ()( t uv) = 0 t v + r () t v + s () t vv + p ()( t uv) = 0 t () subject to the initial conditions: u (0) = f ()(0) v = g () and the boundary conditions: u(0 t) = f ( t) u (0 t) = f ( t) v(0 t) = g ( t) v (0 t) = g ( t) where the subscripts r () t r () t s () t s () t p () t and p( t ) are arbitrary smooth functions of t. Wavelet theory is a relatively new and an emerging area in mathematical research. Wavelets analysis possesses several useful properties such as orthogonality compact support eact representation of polynomials to a certain degree and multiresolution (MRA) [Yousefi (0)]. Moreover wavelets establish a connection with fast numerical algorithms [Beilkin et al. (99)]. herefore the wavelet is successfully used in many fields. he fundamental idea of the Legendre wavelet method is using the operational matrices the nonlinear system of variablecoefficient coupled Burgers equation which satisfies the boundary and initial conditions that can be converted into a set of algebraic equations. he article is summarized as follows. In the section we introduce the twodimensional Legendre wavelets and we introduced operational matrices of integration in section 3. Section 4 is devoted to the solutions of () that utilize the aforementioned matrices and the D Legendre wavelets. In Section 5 by considering numerical eamples reported in our work the accuracy of the proposed scheme is demonstrated.. wodimensional Legendre wavelets wo dimensional Legendre wavelets in L ( ) over interval [ 0] [ 0] defined as [Parsian (005)]

3 344 H. Aminikhah and S. Moradian y nmn m k k ìï + æ öæ ö m m + + ç ç è øè ø k k n n ( y) Pm( n ) Pm ( y n = ï + + ) í k k ïï n n y ; k k ï0 otherwise ïî () and k k n =... n =... m = 0... M m = 0... M. he coefficient ( m )( m ) + + is for orthonormality. Here Pm () are the wellknown Legendre polynomials of order m which are defined on the interval [ ] and can be determined with the aid of the following recurrence formulae: P ( ) = P( ) = 0 m + m ( ) ( ) Pm+ ( ) = Pm( ) Pm( ) m =... m + m + where the twodimensional Legendre wavelets are an orthonormal set over [ 0] [ 0] òò yn m n m ( y) yn m n m( ) yddy= dnn d mm d n nd m m d ij = í (3) 0 0 ìï ï0 ïî i i = ¹ j j he function u( y) Î L ( ) defined over [ 0] [ 0] may be epanded as å (4) uy ( ) XYy () () c y ( y). nmn m nmn m n= m= 0n = m = 0 If the infinite series (4) is truncated then can be written as k k M M uy ( ) XYy () () c y ( y) å (5) nmn m nmn m n= m= 0n = m = 0 where

4 AAM: Intern. J. Vol. 9 Issue (June 04) 345 nmn m ynmn m 0 0 c = òò X( ) Y( y) ( y) ddy. (6) he equation (5) may be epressed in the form where C uy ( ) = C Y ( y) (7) and Y ( y ) are the coefficient matri and the wavelet vector respectively. he k k MM dimensions of those are and given by [Beilkin Coifman and Rokhlin (99)] in the form and C = [ c00... c0 M c00... c0 M... c k... c 0 k 0 0 M c0... c M c0... c M... c k... c k... 0 M c M 0... c M M c M 0... c M M... c k... M 0 c k c00... c 0 M c00... c0... M M M c k... c k... c M 0... c M M c M 0... M c M M... c k... c k M 0 M c M k c k 0 k k M c c 0 M... c k k c k k... c k... c k c k... 0 M M0 M M M0 c k... c k k... c k ] k M M M 0 M M Y ( y ) = [ y00... y0 M y00... y0 M... y k... y k M y... y y... y... y k... 0 M 0 M 0 y k... y M 0... y M M y 0... M M y M M... y k... y k y M M M y0 M y00... y0 M... y k... y... 0 k 0 0 M y M0... y M M y M0... y M M... y k... y k... y k y k M M M 0 M y k y k 0 M... y k k... y k k M y k... y k y k... y k... y M0 M M M0 M M k k... y k k ] M 0 M M (8) (9) he integration of the product of the two Legendre wavelet function vectors is obtained as òò Y( y ) Y ( y ) d dy = I (0) 0 0

5 346 H. Aminikhah and S. Moradian where I is identity matri. 3. wodimensional operational matri of integration 3.. Operational matri of integration for the variable he integration matri for the variable defined by where ò Y ( y) d = P Y( y) () 0 k k k k P is the MM MM operational matri for integration given by [Beilkin et al. (99)] as P = M k+ k él F F F Fù O L F F F O O L F F O O O L F êo O O O Lú ë û () and FLand O are k k MM MM matrices that defined as below: F L éd O O O ù O O O O = O O O O O O O O ê ë úû é ù D D O O 3 3 D O D O = 5 O D O O 5 3 O O O O ê ë úû

6 AAM: Intern. J. Vol. 9 Issue (June 04) 347 and O éo O O O ù O O O O = O O O O O O O O ê ë úû k k where D is the M M matri defined below as: é ù D = ê ú ë û k k O is M M zero matri. 3.. Operational matri of integration for y variable he integration matri for the y variable defined as: 0 y ( y ) dy P ( y) ò Y = yy. (3) Here P y = M k ép P P P Pù P P P P P P P P P P P P P P P êp P P P Pú ë û (4) k k k k P is MM MM k k matri P is M M matri and defined as: y

7 348 H. Aminikhah and S. Moradian P = k él F F F Fù O L F F F O O L F F O O O L F êo O O O O Lú ë û where OL and F are M M matrices. O is the zero matri and LF are defined as: F é 0 0ù = ê0 0 0ú ë û and L é ù = ê ë úû 4. wodimensional Legendre wavelets applied to the systems of variablecoefficient coupled Burgers' Equation Consider the nonlinear systems of variablecoefficient coupled Burgers Equation (). Let and ' denote differentiation with respect to and t respectively. In order to using Legendre wavelets to approimate u ( t) and v ( t) we have u ( t)» C Y( t) (5) v ( t)» C Y( t). (6) Integrating Equation (5) with respect to t once from 0 to t and with respect to twice from 0 to we obtain

8 AAM: Intern. J. Vol. 9 Issue (June 04) 349 u ( t)» C P Y ( t) + u ( 0) t t 0 = C P Y ( t) + U Y( t) ( )» t Y ( ) + ( 0) ( 00) + ( 0 ) = C P P Y ( t) + U Y( t) u t C P P t u u u t t ( )» t Y ( ) + ( 0 ) ( 0 0 ) ( 0 0 ) + ( 0 ) + ( 0 ) = C P P Y ( t) + U Y( t). u t C P P t u u u u t u t t (7) (8) (9) Integrating Equation (5) with respect to twice from 0 to we obtain u ( t)» C P Y ( t) + u ( 0 t) (0) u ( t)» C P Y ( t) + u ( 0 t) + u ( 0 t) 3 = C P Y ( t) + U Y( t). () Similarly integrating Equation (6) with respect to t once from 0 to t and with respect to twice from 0 to we obtain v ( t)» C P Y ( t) + v ( 0) t t 0 = C P Y ( t) + V Y( t) v ( t)» C P P Y ( t) + v ( 0) v ( 00) + v ( 0 t) t t = C P P Y ( t) + V Y( t) v( t)» C P P Y ( t) + v( 0 )v( 0 0 ) v ( 0 0 ) + v ( 0 t) + v( 0 t) t t = C P P Y ( t) + V Y( t). () (3) (4) Also integrating Equation (6) with respect to twice from 0 to we obtain v ( t)» C P Y ( t) + v ( 0 t) (5) v ( t)» C P Y ( t) + v ( 0 t) + v ( 0 t) 3 = C P Y ( t) + V Y( t) where the coefficients U0 U U U 3 and V0 V V V 3 are known and obtained from the initial and boundary conditions. P and Pt are defined similarly in Equations () and (4). Now consider the following approimations (6)

9 350 H. Aminikhah and S. Moradian r () t u» Y Y( t) s () t uu» Y Y( t) 3 p ()( t uv) = p ( t)( u v + v u)» Y Y( t) 4 r () t v» Y Y( t) 5 s () t vv» Y Y( t) 6 p ()( t uv) = p () t ( u v + v u)» Y Y( t) (7) where YYYYY 3 4 5and Y 6 are column vectors with the entries of the vectors C and C. Substitution of approimations () (6) and (7) in to the systems () results in 3 3 C P Y ( t ) + U Y ( t ) + Y Y ( t ) + Y Y ( t ) + Y Y ( t ) = C P Y ( t ) + V Y ( t ) + Y Y ( t ) + Y Y ( t ) + Y Y ( t ) = 0. (8) From the simplified system (8) the nonlinear system of the entries of C C is obtained 3 3 C P + U + Y + Y + Y = 0 C P + V + Y + Y + Y = (9) he elements of the vector functions C and C can be computed by solving systems (9). 5. Numerical eamples In this section two eamples of systems of the variablecoefficient coupled Burgers equation are considered and will be solved by the method proposed. Eample. Consider the following variable coefficient coupled Burgers equation u u t u ( uv) = ( ) ( ) + e sin t u sin t t v v t v ( uv) = ( ) ( ) e cos t v + cos t t (30) subject to the initial conditions: u( 0 ) = v( 0 ) = e

10 AAM: Intern. J. Vol. 9 Issue (June 04) 35 and the boundary conditions: t u t u( 0 t) = e ( 0 t) = e. t v t v( 0 t) = e ( 0 t) = e. t he eact solution of the equation is u( t) = e t and v( t) = e +. We solve the system (30) by introduced method in the paper with k = k = and M = M = 5. Let s consider the following approimations: u ( t)» C Y( t) v ( t)» C Y( t) u ( t)» C P Y ( t) + u ( 0) t t 0 = C P Y ( t) + U Y( t) u ( t)» C P P Y ( t) + u ( 0) u ( 00) + u ( 0 t) t t = C P P Y ( t) + U Y( t) u( t)» C P P Y ( t) + u( 0 )u( 0 0 ) u ( 0 0 ) + u ( 0 t) + u( 0 t) t t = C P P Y ( t) + U Y( t) u ( t)» C P Y ( t) + u ( 0 t) + u ( 0 t) 3 = C P Y ( t) + U Y( t) v ( t)» C P Y ( t) + v ( 0) t t 0 = C P Y ( t) + V Y( t) v ( t)» C P P Y ( t) + v ( 0) v ( 00) + v ( 0 t) t t = C P P Y ( t) + V Y( t) v( t)» C P P Y ( t) + v( 0 )v( 0 0 ) v ( 0 0 ) + v ( 0 t) + v( 0 t) t t = C P P Y ( t) + V Y( t)

11 35 H. Aminikhah and S. Moradian v ( t)» C P Y ( t) + v ( 0 t) + v ( 0 t) 3 = C P Y ( t) + V Y( t) t u e sin( t) u» Y Y ( t) ( uv ) u v sin( t) = sin( t)( v + u )» Y Y( t) t v e cos( t) v» Y3 Y ( t) and ( uv ) u v cos( t) = cos( t)( v + u )» Y4 Y( t). Substitution into the systems (30) and simplifying we obtain: ( t ) + 3 = C P U C P U Y Y + 3 = t C P V C P V Y Y. (3) Solving system (3) elements of the vector functions C and C can be obtained via the Maple package as follows: and C = [ ] C = [

12 AAM: Intern. J. Vol. 9 Issue (June 04) ]. Resulting in the following solutions will result: and t u t» C P P + U Y t ( ) ( ) ( ) 4 3 = ( t t t t ) + ( t t t t ) + ( t t t t ) ( t t t t ) t t t t t v t» C P P + V Y t ( ) ( ) ( ) 4 3 = ( t t t t ) + ( t t t t ) + ( t t t t ) + ( t t t t ) t t t t Figure and Figure show the numerical solution for Equation (30) obtained by the twodimensional Legendre wavelets method for tî [0]. 4

13 354 H. Aminikhah and S. Moradian Figure.he eact and LWM solution u( t ) of eample Figure : he eact and LWM solution v( t ) of Eample In ables..5 we show the Comparisons between numerical and analytical solutions of Equation (30) in t = 0 t = 0.5 t = 0.5 t = 0.75 and t = for various values of. able.. Numerical results of eample for t = 0 u eact u LWM ueact ulwm veact vlwm veact vlwm

14 AAM: Intern. J. Vol. 9 Issue (June 04) 355 able.. Numerical results of eample for t = 0.5 u eact u LWM ueact u v LWM eact vlwm veact vlwm able.3. Numerical results of eample for t = 0.5 u eact u LWM ueact ulwm veact vlwm veact vlwm able.4. Numerical results of eample for t = 0.75 u eact u LWM ueact ulwm veact vlwm veact vlwm

15 356 H. Aminikhah and S. Moradian able.5. Numerical results of eample for t = u eact u LWM ueact ulwm veact vlwm veact vlwm Eample. Consider the following coupled Burgers Equation u u u ( uv) = + u t v v v ( uv) = + v t (3) subject to the initial conditions: u( 0 ) = v( 0) = sin( ) and the boundary conditions: u( 0 t) = v( 0 t) = 0 u v t ( 0 t) = ( 0 t) = e. he eact solution of the equation is u( t) = v( t) = e sin. We solve the system (3) by the proposed method with k = k = and M = M = 5. he vectors C and C are computed by solving the system of nonlinear equations via the Maple package as follows: C = [ t

16 AAM: Intern. J. Vol. 9 Issue (June 04) ] 7 and C = [ ]. herefore the following solutions will result: and t u t» C P P + U Y t ( ) ( ) ( ) = ( t t t t ) + ( t t t t ) ( t t t t ) + ( t t t t ) t t t t t v t» C P P + V Y t ( ) ( ) ( ) 4 3 = ( t t t t ) + ( t

17 358 H. Aminikhah and S. Moradian t t t ) ( t t t t ) + ( t t t t ) t t t t Figure 3 and Figure 4 show the numerical solution for Equation (30) obtained by twodimensional Legendre wavelets method for tî [0]. Figure 3. Eact and LWM solution u( t ) of eample Figure 4. Eact and LWM solution v( t ) of Eample

18 AAM: Intern. J. Vol. 9 Issue (June 04) 359 In ables..5 we show the Comparisons between numerical and analytical solutions of Equation (3) in t = 0 t = 0.5 t = 0.5 t = 0.75 and t = for various values of. able.. Numerical results of eample for t = 0 u eact u LWM ueact ulwm veact vlwm veact vlwm able.. Numerical results of eample for t = 0.5 u eact u LWM ueact ulwm veact vlwm veact vlwm able.3. Numerical results of eample for t = 0.5 u eact u LWM ueact ulwm veact vlwm veact vlwm

19 360 H. Aminikhah and S. Moradian able.4. Numerical results of eample for t = 0.75 u eact u LWM ueact ulwm veact vlwm veact vlwm able.5. Numerical results of eample for t = u eact u LWM ueact ulwm veact vlwm veact vlwm Conclusion he aim of this paper has been to develop twodimensional Legendre wavelets for obtaining the solutions of systems of variablecoefficient coupled Burgers equation. he illustrative eamples included demonstrate that we have achieved a method is a very effective and useful technique for finding approimate solutions of these systems. he method is fully described possible error and analyzed. he twodimensional operational matrices of integration are used to find the solution of the system of variablecoefficient coupled Burgers equation. In the present method the problem under study reduces to a system of linear or nonlinear algebraic equations. he two eamples presented illustrate the capability and simplicity of the method and the close comparison of the obtained results with those of the eact solutions shows that the proposed method is a highly promising method for various classes of both linear and nonlinear systems of partial differential equations. Here the computations associated with these eamples are performed by the package Maple 3.

20 AAM: Intern. J. Vol. 9 Issue (June 04) 36 Acknowledgments We are very grateful to two anonymous referees for their careful reading and valuable comments which led to the improvement of this paper. REFERENCES Abbasbandy S. and Darvishi M.. (005). A numerical solution of Burgers' equation by time discretization of Adomian's decomposition method Appl. Math. Comput. 70: 950. Aminikhah H. (03). Approimate analytical solution for the systems of variablecoefficient coupled Burgers' Equation Journal of Interpolation and Approimation in Scientific Computing Volume 03: 9. Beylkin G. Coifman R. and Rokhlin V. (99). Fast wavelet transforms and numerical algorithms I Commun. Pure Appl. Math. 44: Biazar J. and Aminikhah H. (009). Eact and numerical solutions for nonlinear Burger's Equation by VIM Math. Comput. Modelling 49: Burger J. M. (948). A mathematical model illustrating the theory of turbulence Adv. Appl. Mech. I: Cole J. D. (95). On a quasilinear parabolic equations occurring in aerodynamics Quart. Appl. Math Esipov S.E. (995). Coupled Burgers equations: a model of polydispersive sedimentation Phys. Rev. E 5: Na Liu (00). Similarity reduction and eplicit solutions the variablecoefficient coupled Burgers Equation Applied Mathematics and Computation 7: Parsian H. (005). wo dimension Legendre wavelets and operational matrices of integration Acta Mathematica Academiae Paedagogicae Ny ıregyh aziensis : Rashidi M.M. and Erfani E. (009). New analytical method for solving Burgers' and nonlinear heat transfer equations and comparison with HAM Comput. Phys. Commun. 80: Yousefi S. A. (0). Numerical Solution of a Model Describing Biological Species Living ogether by Using Legendre Multiwavelet Method International Journal of Nonlinear Science : 093.

Variational Iteration Method for Solving Nonlinear Coupled Equations in 2-Dimensional Space in Fluid Mechanics

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