Int J Contemp Math Sciences Vol 7 212 no 37 1839-1852 Variational Iteration Method for Solving Nonlinear Coupled Equations in 2-Dimensional Space in Fluid Mechanics A A Hemeda Department of Mathematics Faculty of Science Tanta University Tanta Egypt aahemeda@yahoocom Abstract In this paper we shall use the variational iteration method to introduce a framework for analytic treatment of linearized Burgers equation and coupled Burgers equations in 2- dimensional space For these equations the exact solutions are obtained The paper confirms the power and efficiency of the method in reducing the size of the calculation also the simplicity of the method compared with the other methods Keywords: Linearized Burgers equation; Coupled Burgers equations; Variational iteration method; Lagrange multipliers 1 Introduction The variational iteration method proposed by Ji-Huan He 1-13] in 1999 is used widely by many authors to solve linear and nonlinear physical and mathematical problems The efficiency of the method and the reduction in the size of computational domain gave this method a wider applicability 14-16] In the past several decades many more of the methods such as finite difference methods perturbation methods and inverse scattering method were used to solve these problems but with high expensive hard work and low accuracy Recently the variational iteration method is used excellency in place of these methods with low expensive simple work and high accuracy The numerical solution of Burgers equation is of great importance due to the equation s application in the approximate theory of flow through a shock wave traveling in a viscous fluid 17] and is obtained by several researchers for more details see 16] For simplicity the linearized Burgers equation is often used in place of Burgers equation The coupled Burgers equations derived
184 A A Hemeda by Esipov 18] is a simple model of evolution of scaled volume concentration of two kinds of particles in fluid suspensions or colloids under the effect of gravity 19] In this paper we shall solve the linearized Burgers equation in addition to the coupled homogenous and inhomogenous Burgers equations in 2- dimensional space 162] The paper shows the efficiency and the simplicity of the variational iteration method comparable with the other methods The paper is organized as: in section 2 we describe the proposed method In section 3 numerical problems are used to show the efficiency of the method Finally in section 4 conclusions are given 2 The variational iteration method To illustrate the basic concepts of the variational iteration method we consider the following differential equation: Lu + Nu = g(x (1 where L is a linear operator N a nonlinear operator and g(x an inhomogenous term According to the variational iteration method we can construct the correction functional for (1 as: u n+1 (x =u n (x+ x λ Lu n (ξ+nũ n (ξ] dξ (2 where λ is a general Lagrange multiplier 1-13] which can be identified optimally via the variational theory the subscript n denotes the nth order approximation and ũ n is considered as a restricted variation 1-13] ie δũ n = To illustrate the powerful of the variational iteration method three problems of special interest are discussed in details in the following section 3 Application problems Problem 31 2- dimensional linearized Burgers equation For the purpose of illustration of the variational iteration method for solving the linearized Burgers equation in 2- dimensional space we will consider the following equation:
Variational iteration method 1841 with the initial conditions: u t + c u = μ 2 u (3 u(yt = exp( 2μk 2 t sin k(y 2ct (4a u x (yt=k exp( 2μk 2 t cos k(y 2ct (4b where c is a free parameter If μ = the first- order wave equation in 2- dimension is obtained If c = the 2-dimensional heat equation is obtained 2] The correction functional for (3 in x- direction is: u n+1 (x y t =u n (x y t+ x ( ũn (ξ y t ũn (ξ y t λ + c + ũ n(ξ y t t ξ y The stationary conditions for (5 can be obtained as follows: ( ] 2 u n (ξ y t μ ξ 2 + 2 ũ n (ξ y t dξ y 2 (5 μλ (ξ = 1+μλ (ξ ξ=x = μλ(ξ ξ=x = (6 The Lagrange multiplier therefore can be identified as: λ = x ξ μ (7 Substituting this value into the functional (5 gives the iteration formula: u n+1 (x y t =u n (x y t+ 1 μ x un (ξ y t (x ξ + t
1842 A A Hemeda ( un (ξyt c + u ( ] n(ξ y t 2 u n (ξyt μ ξ y ξ 2 + 2 u n (ξyt dξ y 2 (8 Starting with the zeroth approximation: u = A + Bx where A and B are functions in y ant t to be determined by using the initial conditions (4 thus we can obtain: u (x y t = exp( 2μk 2 t(sin k(y 2ct+kx cos k(y 2ct (9 Using (9 into (8 we obtain the first approximation: u 1 (x y t =u (x y t+ exp( 2μk2 t μ x (x ξ ck 2 sin k(y 2ctξ μk 3 cos k(y 2ctξ μk 2 sin k(y 2ct ] dξ = u (x y t+ exp( 2μk2 t μ x ck 2 x sin k(y 2ctξ ck 2 sin k(y 2ctξ 2 μk 3 x cos k(y 2ctξ+ μk 3 cos k(y 2ctξ 2 μk 2 x sin k(y 2ct+μk 2 sin k(y 2ctξ ] dξ = exp( 2μk 2 t (1 k2 x 2 + ck2 x 3 sin k(y 2ct+ 2 6μ (kx k3 x 3 6 ] cos k(y 2ct (1 In the same manner we can obtain using the Maple Package the following successive approximations: u (x y t = exp( 2μk 2 t(sin k(y 2ct+kx cos k(y 2ct
Variational iteration method 1843 u 1 (x y t = exp( 2μk 2 t (1 (kx2 + ck2 x 3 sin k(y 2ct+ 2! 6μ ] (kx (kx3 cos k(y 2ct 3! u 2 (x y t = exp( 2μk 2 t (1 (kx2 + (kx4 ck4 x 5 2! 4! 6μ + c2 k 2 x 4 24μ 2 sin k(y 2ct+ (kx (kx3 3! + (kx5 5! ] c2 k 3 x 5 cos k(y 2ct 12μ 2 u n (x y t = exp( 2μk 2 t (1 (kx2 + (kx4 (kx6 + sin k(y 2ct+ 2! 4! 6! (kx (kx3 3! + (kx5 5! (kx7 7! ] + cos k(y 2ct (11 In closed form the solution u(x y t takes the form: u(x y t = exp( 2μk 2 t sin k(x + y 2ct (12 which satisfies exactly the linearized Burgers equation (3 and the initial conditions (4 Problem 32 2- dimensional homogenous coupled Burgers equations For the purpose of illustration of the variational iteration method for solving the homogenous form of coupled Burgers equations in 2- dimensional space we will consider the following system of equations 162]: u t 2 u 2u u +(uv x +(uv y = (13
1844 A A Hemeda with the initial conditions: v t 2 v 2v v +(uv x +(uv y = (14 u(x y = cos(x + y v(x y = cos(x + y (15 The correction functional for equations (13 and (14 in t- direction are: u n+1 (x y t =u n (x y t+ λ 1 (un τ 2 ũ n 2ũ n ũ n +(ũ n v n x +(ũ n v n y ] dτ (16 v n+1 (x y t =v n (x y t+ λ 2 (vn τ 2 v n 2v n v n +(ũ nv n x +(ũ n v n y] dτ (17 The stationary conditions for equations (16 and (17 can be obtained as follows: λ 1(τ = 1+λ 1 (τ τ=t =; λ 2(τ = 1+λ 2 (τ τ=t = (18 The Lagrange multipliers therefore can be identified as: λ 1 = λ 2 = 1 (19 Substituting (19 into the functionals (16 and (17 give the iteration formulae: u n+1 (x y t =u n (x y t (un τ 2 u n
Variational iteration method 1845 2u n u n +(u n v n x +(u n v n y ] dτ (2 v n+1 (x y t =v n (x y t (vn τ 2 v n 2v n v n +(u n v n x +(u n v n y ] dτ (21 Starting with the zeroth approximations: u (x y t =u(x y = cos(x + y and v (x y t =v(x y = cos(x + y we can obtain the following first approximations: u 1 (x y t =u (x y t + 2 cos(x + y + 4 cos(x + y sin(x + y 4 cos(x + y sin(x + y] dτ = u (x y t 2 cos(x + y dτ = cos(x + y(1 2t (22 v 1 (x y t =v (x y t + 2 cos(x + y + 4 cos(x + y sin(x + y 4 cos(x + y sin(x + y] dτ = u (x y t 2 cos(x + y dτ = cos(x + y(1 2t (23
1846 A A Hemeda In the same manner we can obtain using the Maple Package the following successive approximations: u (x y t = cos(x + y v (x y t = cos(x + y u 1 (x y t = cos(x + y(1 2t v 1 (x y t = cos(x + y(1 2t u 2 (x y t = cos(x + y (1 2t + (2t2 2! v 2 (x y t = cos(x + y (1 2t + (2t2 2! u 3 (x y t = cos(x + y (1 2t + (2t2 2! v 3 (x y t = cos(x + y (1 2t + (2t2 2! (2t3 3! (2t3 3! u n (x y t = cos(x + y (1 2t + (2t2 (2t3 + (24 2! 3! v n (x y t = cos(x + y (1 2t + (2t2 (2t3 + (25 2! 3! In closed form the solutions u(x y t and v(x y t take the forms: u(x y t = exp( 2t cos(x + y (26 v(x y t = exp( 2t cos(x + y (27
Variational iteration method 1847 which are the exact solutions satisfies the homogenous coupled Burgers equations (13 and (14 and the initial conditions (15 Problem 33 2- dimensional inhomogenous coupled Burgers equations For the purpose of illustration of the variational iteration method for solving the inhomogenous form of coupled Burgers s equations in 2- dimensional space we will consider the following system of equations 162]: u t 2 u 2u u +(uv x +(uv y = f(x y t (28 with the initial conditions: v t 2 v 2v v +(uv x +(uv y = g(x y t (29 u(x y = exp( x y v(x y = exp( x y (3 The correction functional for equations (28 and (29 in t- direction are: u n+1 (x y t =u n (x y t+ λ 1 (un τ 2 ũ n 2ũ n ũ n + (ũ n v n x +(ũ n v n y f(x y τ] dτ = (31 v n+1 (x y t =v n (x y t+ λ 2 (vn τ 2 v n 2v n v n + (ũ n v n x +(ũ n v n y g(x y τ] dτ = (32 Following the same procedure as problem 32 the Lagrange multipliers are: λ 1 = λ 2 = 1 (33 Substituting these values into the functionals (31 and (32 give the iteration formulae:
1848 A A Hemeda u n+1 (x y t =u n (x y t (un τ 2 u n 2u n u n + (u n v n x +(u n v n y f(x y τ] dτ = (34 v n+1 (x y t =v n (x y t (vn τ 2 v n 2v n v n + (u n v n x +(u n v n y g(x y τ] dτ = (35 Starting with the zeroth approximations: u (x y t =u(x y = exp( x y and v (x y t =v(x y = exp( x y we can obtain the following first approximations: u 1 (x y t =u (x y t 2e x y +4e 2x 2y 4e 2x 2y + e x y sin τ +2e x y cos τ ] dτ = u (x y t exp( x y sin τ + cos τ 2] dτ = exp( x y cos t sin t +2t] (36 v 1 (x y t =v (x y t 2e x y +4e 2x 2y 4e 2x 2y + e x y sin τ +2e x y cos τ ] dτ = v (x y t exp( x y
Variational iteration method 1849 sin τ + cos τ 2] dτ = exp( x y cos t sin t +2t] (37 In the same manner we can obtain using the Maple Package the following successive approximations: u (x y t = exp( x y v (x y t = exp( x y u 1 (x y t = exp( x y cos t +2(t sin t] v 1 (x y t = exp( x y cos t +2(t sin t] u 2 (x y t = exp( x y 5 cos t 4 v 2 (x y t = exp( x y u 3 (x y t = exp( x y cos t +8 v 3 (x y t = exp( x y u 4 (x y t = exp( x y 16 v 4 (x y t = exp( x y 5 cos t 4 cos t +8 16 u 5 (x y t = exp( x y cos t +32 v 5 (x y t = exp( x y cos t +32 ] (1 t2 2! ] (1 t2 2! ( ] sin t (t t3 3! ( ] sin t (t t3 3! ] (1 t2 15 cos t 4! ] (1 t2 15 cos t 4! ] ((t t3 sin t 5! ] ((t t3 sin t 5!
185 A A Hemeda u 6 (x y t = exp( x y 65 cos t 64 v 6 (x y t = exp( x y 65 cos t 64 u 7 (x y t = exp( x y cos t + 128 v 7 (x y t = exp( x y u 8 (x y t = exp( x y 256 v 8 (x y t = exp( x y cos t + 128 256 (1 t2 (1 t2 6! (1 t2 6! ( sin t (t t3 ( sin t 6! + t8 8! (1 t2 6! + t8 8! ] ] 5! t7 7! (t t3 5! t7 7! ] ] ] 255 cos t ] 255 cos t ] u 4n+1 (x y t = exp( x y cos t +2 ((t n+1 t3 5! t7 7! + sin t (38a ] u 4n+2 (x y y = exp( x y (2 n+2 + 1 cos t 2 (1 n+2 t2 6! + (38b ( ] u 4n+3 (x y t = exp( x y cos t +2 n+3 sin t (t t3 5! t7 7! + (38c ] u 4n+4 (x y t = exp( x y 2 (1 n+4 t2 6! + (2 n+4 1 cos t (38d ] v 4n+1 (x y t = exp( x y cos t +2 ((t n+1 t3 5! t7 7! + sin t (39a
Variational iteration method 1851 ] v 4n+2 (x y y = exp( x y (2 n+2 + 1 cos t 2 (1 n+2 t2 6! + (39b ( ] v 4n+3 (x y t = exp( x y cos t +2 n+3 sin t (t t3 5! t7 7! + (39c ] v 4n+4 (x y t = exp( x y 2 (1 n+4 t2 6! + (2 n+4 1 cos t (39d where in both (38 and (39 n = 1 2 3 When n = we obtain the successive approximations u 1 u 2 u 3 u 4 v 1 v 2 v 3 and v 4 and so on In closed form the solutions u(x y t and v(x y t take the forms: u(x y t = exp( x y cos t (4 v(x y t = exp( x y cos t (41 which are the exact solutions satisfies the inhomogenous coupled Burgers equations (28 and (29 and the initial conditions (3 4 Conclusion In this paper the variational iteration method has been successfully used to finding the solution of a linearized Burgers equation homogenous coupled Burgers equations and inhomogenous coupled Burgers equations in 2- dimensional space The solution obtained by the mentioned method is an infinite series for appropriate initial conditions which can in turn be expressed in a closed form the exact solution The obtained results show that the variational iteration method is a powerful method to solving linearized Burgers and coupled Burgers equations in 2-dimensional space it is also a promising method to solve other equations References 1] JH He Commun Nonlinear Sci Numer Simul 2(4 (1997 23
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