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Homotopy perturbation method for (2+1) -dimensional coupled Burgers system
Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938 www.elsevier.com/locate/nonrwa
Homotopy perturbation method for (2 + 1)-dimensional coupled Burgers system E. Hızel a,∗ , S. K¨uc¸u¨ karslan b a Department of Mathematics, Istanbul ˙ ˙ Technical University, Istanbul, Turkey b Department of Engineering Sciences, Istanbul ˙ ˙ Technical University, Istanbul, Turkey
Received 19 December 2007; accepted 28 February 2008
Abstract In this paper, a numerical solution of the (2 + 1)-dimensional coupled Burgers system is studied by using the Homotopy Perturbation Method (HPM). For this purpose, the available analytical solutions obtained by tanh method will be compared to show the validity and accuracy of the proposed numerical algorithm. The results approve the convergence and accuracy of the Homotopy Perturbation Method for numerically analyzed (2 + 1) coupled Burgers system. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Homotopy Perturbation Method (HPM); Coupled Burgers system; Numerical analysis of nonlinear PDEs
1. Introduction The exact and numerical solutions of the nonlinear partial differential equations play an important role in the physical sciences and in the engineering fields. The exact or the numerical solutions of the coupled nonlinear partial differential equation systems are also taken a great attention of researches in these fields. Tanh method is successfully applied to get exact solutions in many applications [1–6]. The further extended tanh method to the coupled Burgers equation [7,8] was applied in [9]. The model equation for the (2 + 1) Burgers system is given in the following form u t = u x x + u yy + 2uu x + 2vu y , vt = vx x + v yy + 2uvx + 2vvx .
(1)
A numerical method called Homotopy Perturbation Method (HPM) is proposed by He [10] and further developed by himself in [11–14] for better accuracy. It yields a fast convergence for most of the selected problems. It also showed a high accuracy and a rapid convergence to solutions of the nonlinear partial differential equations. Recently, homotopy perturbation technique was successfully applied to (1 + 1)-dimensional generalized Hirota–Satsuma coupled KdV equation by Abbasbandy [15]. ∗ Corresponding author. Tel.: +90 212 2853345; fax: +90 212 285 6386.
E-mail address: [email protected] (E. Hızel). c 2008 Elsevier Ltd. All rights reserved. 1468-1218/$ - see front matter doi:10.1016/j.nonrwa.2008.02.033
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
1933
In this paper, as an alternative to the exact solutions the HPM will be applied for numerical solutions of (2 + 1)dimensional coupled Burgers system. The accuracy and convergence of the numerical results will be compared with the analytical ones. 2. Homotopy Perturbation Method (HPM) One considers the following nonlinear differential equation to represent the procedure of this method, Au − f (r ) = 0,
r ∈Ω
(2)
with the boundary conditions of ∂u = 0, r ∈ Γ B u, ∂n
(3)
where A and B are general differential operator and boundary operator, respectively. Γ is the boundary of the domain Ω , and f (r ) is a given analytical function. After dividing the general operator into linear part (L) and nonlinear part (N ), one can rewrite the Eq. (2) as Lu + N u − f (r ) = 0.
(4)
By constructing the homotopy technique to Eq. (4), one can write a homotopy in the form H (V, p) = (1 − p)[L(V ) − L(u 0 )] + p[A(V ) − f (r )] = 0,
p ∈ [0, 1],
r ∈Ω
(5)
where p ∈ [0, 1] is an embedding parameter, u 0 is an initial approximation of the Eq. (2) which satisfies the Eq. (3). In HPM, one can use the embedding parameter as a small parameter. Therefore, the solution of Eq. (5) can be written as a power series of p in the form, V = Vo + pV1 + p 2 V2 + · · · .
(6)
By setting p = 1, one can get an approximate solution of the Eq. (2) as, u = lim V = Vo + V1 + V2 + · · · . p→1
(7)
The combination of a small parameter (perturbation parameter) with a homotopy is called homotopy perturbation method as presented in the final equation (7). 3. Application of the HPM to (2 + 1)-dimensional coupled Burgers system In this section, two different solutions of coupled Burgers system [9] will be investigated by using the HPM. 3.1. Solitary solution to 2 + 1 coupled Burgers system The exact solutions to Eq. (1) are given by √ √ u = f − α F −δ tanh( −δ(−α F x + (α f + β)y + 2β Ft + γ )), √ √ v = F + (α f + β) −δ tanh( −δ(−α F x + (α f + β)y + 2β Ft + γ ))
(8)
where f, F, δ, α, β, γ are any arbitrary functions. For simplicity, f = 1, F = 1, δ = −1, α = 1, β = 1 and γ = 1 are used for arbitrary variables. Then, the Eq. (8) takes the following form u = 1 − tanh(−x + 2y + 2t + 1), v = 1 + 2 tanh(−x + 2y + 2t + 1).
(9)
By using an initial condition to Eq. (9), u 0 = u 0 (x, y, 0) = 1 − tanh(−x + 2y + 1) and v0 = v0 (x, y, 0) = 1 + 2 tanh(−x + 2y + 1) and then substituting Eq. (9) into Eq. (5) and later by substituting the Eq. (6) to obtained
1934
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
one, one gets a system of equation with n + 1 terms that needs to be solved simultaneously. Since computations are dependent on the value of u o and vo , a minor modification gives flexibility to choose the initial u o and vo . For this purpose, the following homotopies are constructed as in Refai et al. [16] (1 − p)(wt − u 0t ) + p(wt − wx x − w yy − 2wwx − 2r w y ) = 0 (1 − p)(rt − vot ) + p(rt − r x x − r yy − 2wr x − 2rr y ) = 0 p2 w
(10)
where w = w0 + pw1 + 2 + · · · and r = r0 + pr1 + 2 + · · ·. The variables u and v can be obtained similar to Eq. (7) as p2r
u = lim w = wo + w1 + w2 + · · · p→1
v = lim r = ro + r1 + r2 + · · · . p→1
(11)
After expanding the Eq. (10) for n = 3, i.e. the third power of p, one obtains p0 :
w0t = u 0t , r0t = v0t
p1 :
w1t − w0x x − w0yy − 2w0 w0x − 2r0 w0y = 0 r1t − r0x x − r0yy − 2w0r0x − 2r0r0y = 0
p2 :
w2t − w1x x − w1yy − 2w0 w1x − 2r0 w1y − 2w1 w0x − 2r1 w0y = 0
(12)
r2t − r1x x − r1yy − 2w0r1x − 2w1r0x − 2r0r1y − 2r1r0y = 0 p3 :
w3t − w2x x − w2yy − 2w0 w2x − 2w1 w1x − 2w2 w0x − 2r0 w2y − 2r1 w1y − 2r2 w0y = 0 r3t − r2x x − r2yy − 2w0r2x − 2w1r1x − 2w2r0x − 2r0r2y − 2r1r1y − 2r2r0y = 0.
The solution of the Eq. (12) can be obtained as, w1 = 2 tSech(1 − x + 2 y)2 (1 − tanh(1 − x + 2 y)) + 10 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y) − 4 Sech(1 − x + 2 y)2 (1 + 2 tanh(1 − x + 2 y)) r1 = −4t Sech(1 − x + 2 y)2 (1 − tanh(1 − x + 2 y)) − 20 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y) + 8 Sech(1 − x + 2 y)2 (1 + 2 tanh(1 − x + 2 y)) w2 = t 2 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y)
(13)
r2 = −8 t 2 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y) −8 3 t (−2 + cosh(2 − 2 x + 4 y)) Sech(1 − x + 2 y)4 w3 = 3 16 3 r3 = t (−2 + cosh(2 − 2 x + 4 y)) Sech(1 − x + 2 y)4 . 3 Finally, the approximated solutions are given by u∼ = w0 + w1 + w2 + w3 v∼ = r0 + r1 + r2 + r3 .
(14)
In Fig. 1, the exact distribution of the u(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 is plotted. The approximated distribution of the u(x, y, 0.3) is plotted for the first four terms in Fig. 2 for the same interval. In the Fig. 3, the exact distribution of the v(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 is plotted. The approximated distribution of the v(x, y, 0.3) is plotted for the first four terms in Fig. 4 for the same interval. 3.2. A special solution to 2 + 1 coupled Burgers system The exact solutions to Eq. (1) are given by p √ u = aq − qx −δ tanh( −δq) + b,
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
1935
Fig. 1. Exact solution of u(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
Fig. 2. HPM solution of u(x, y, 0.3) ∼ = w0 + w1 + w2 + w3 for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
Fig. 3. Exact solution of v(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
p √ v = cq + q yx −δ tanh( −δq) + d where q =
−ax−cy−2(cd+ab)t+aβ 2(c2 +a 2 )t+aα
and a, b, c, d, α, β are any arbitrary constants.
(15)
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E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
Fig. 4. HPM solution of v(x, y, 0.3) ∼ = r0 + r1 + r2 + r3 for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
For simplicity, a = b = c = α = β = 1 and d = 0 are used for arbitrary variables. Then, the Eq. (15) takes the following form −x − y + 2 1 −x − y − 4t + 1 u= − tanh , 4t + 1 4t + 1 4t + 1 1 −x − y − 4t + 1 −x − y − 4t + 1 − tanh v= . (16) 4t + 1 4t + 1 4t + 1 By using an initial condition to Eq. (16), u 0 = u 0 (x, y, 0) = (−x − y + 2) − tanh[−x − y + 1] and v0 = v0 (x, y, 0) = (−x − y + 1) − tanh[−x − y + 1] and by following the same procedure as in the previous solution, one obtains the following approximations w1 = r1 = −2t ( (−1 + Sech(1 − x − y)2 ) (1 − x − y − tanh(1 − x − y)) −1 + Sech(1 − x − y)2 × (2 − x − y − tanh(1 − x − y)) + 2Sech(1 − x − y)2 tanh(1 − x − y)) w2 = r2 = −2 t 2 Sech(1 − x − y)3 (−3 cosh(3 − 3 x − 3 y) + 2 x cosh(3 − 3 x − 3 y) + 2 y cosh(3 − 3 x − 3 y) + 15 cosh(1 − x − y) − 10 x cosh(1 − x − y) − 10 y cosh(1 − x − y) + 2 sinh(3 − 3 x − 3 y) − 16 sinh(1 − x − y) + 24 x sinh(1 − x − y) − 8 x 2 sinh(1 − x − y) + 24 y sinh(1 − x − y) − 16 x y sinh(1 − x − y) − 8 y 2 sinh(1 − x − y)) 4 w3 = r3 = t 3 Sech(1 − x − y)4 (−27 + 162 x − 144 x 2 + 32 x 3 + 162 y − 288 x y + 96 x 2 y 3 − 144 y 2 + 96 x y 2 + 32 y 3 − 9 cosh(4 − 4 x − 4 y) + 6 x cosh(4 − 4 x − 4 y)
(17)
+ 6 y cosh(4 − 4 x − 4 y) + 126 cosh(2 − 2 x − 2 y) − 156 x cosh(2 − 2 x − 2 y) + 72 x 2 cosh(2 − 2 x − 2 y) − 16 x 3 cosh(2 − 2 x − 2 y) − 156 y cosh(2 − 2 x − 2 y) + 144 x y cosh(2 − 2 x − 2 y) − 48 x 2 y cosh(2 − 2 x − 2 y) + 72 y 2 cosh(2 − 2 x − 2 y) − 48 x y 2 cosh(2 − 2 x − 2 y) − 16 y 3 cosh(2 − 2 x − 2 y) + 6 sinh(4 − 4 x − 4 y) − 150 sinh(2 − 2 x − 2 y) + 216 x sinh(2 − 2 x − 2 y) − 72 x 2 sinh(2 − 2 x − 2 y) + 216 y sinh(2 − 2 x − 2 y) − 144 x y sinh(2 − 2 x − 2 y) − 72 y 2 sinh(2 − 2 x − 2 y)). Finally, the approximated solutions are given by u v
∼ = w0 + w1 + w2 + w3 ∼ = r0 + r1 + r2 + r3 .
(18)
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
1937
Fig. 5. Exact solution of u(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
Fig. 6. HPM solution of u(x, y, 0.1) ∼ = w0 + w1 + w2 + w3 for 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
Fig. 7. Exact solution of v(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
In Fig. 5, the exact distribution of the u(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3 is plotted. The approximated distribution of the u(x, y, 0.1) is plotted for the first four terms in Fig. 6 for the same interval. In Fig. 7, the exact distribution of the v(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3 is plotted. The approximated distribution of the v(x, y, 0.1) is plotted for the first four terms in Fig. 8 for the same interval.
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E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
Fig. 8. HPM solution of v(x, y, 0.1) ∼ = r0 + r1 + r2 + r3 for 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
4. Conclusions The iterative numerical solution of the (2 + 1)-dimensional coupled Burgers system was studied by using the Homotopy Perturbation Method (HPM) and the obtained numerical results were compared with the available tanh method results. The results showed that the convergence and accuracy of the Homotopy Perturbation Method for numerically analyzed (2 + 1) Burgers system was in a good agreement with the analytical solutions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
L. Huibin, W. Kelin, J. Phys. A 23 (1990) 4097. W. Malfliet, Am. J. Phys. 60 (1992) 650. W.X. Ma, Int. J. Non-Linear Mech. 31 (1996) 329. E.G. Fan, H.Q. Zhang, Phys. Lett. A 246 (1998) 403. E.G. Fan, Appl. Math. J. Chin. Univ. Ser. B 16 (2001) 149. Y.T. Gao, B. Tian, Comput. Math. Appl. 33 (1997) 115. K.M. Tamizhmani, P. Punithavathi, Int. J. Non-Linear Mech. 26 (1991) 427. M. Salerno, Phys. Lett. A 121 (1987) 15. Z. L¨u, H. Zhang, Phys. Lett. A 324 (2004) 293. J.H. He, Commun. Nonlinear Sci. Numer. Simul. 3 (2) (1998) 92. J.H. He, Comput. Methods Appl. Mech. Engrg. 178 (3–4) (1999) 257. J.H. He, Int. J. Nonlinear Mech. 35 (1) (2000) 37. J.H. He, Appl. Math. Comput. 135 (2003) 73. J.H. He, Appl. Math. Comput. 156 (2004) 527. S. Abbasbandy, Phys. Lett. A 361 (2007) 478. M. Refai, D.D. Ganji, H.R M. Daniali, H. Pashaei, Phys. Lett. A 364 (2007) 1.
Homotopy perturbation method for (2 + 1)-dimensional coupled Burgers system E. Hızel a,∗ , S. K¨uc¸u¨ karslan b a Department of Mathematics, Istanbul ˙ ˙ Technical University, Istanbul, Turkey b Department of Engineering Sciences, Istanbul ˙ ˙ Technical University, Istanbul, Turkey
Received 19 December 2007; accepted 28 February 2008
Abstract In this paper, a numerical solution of the (2 + 1)-dimensional coupled Burgers system is studied by using the Homotopy Perturbation Method (HPM). For this purpose, the available analytical solutions obtained by tanh method will be compared to show the validity and accuracy of the proposed numerical algorithm. The results approve the convergence and accuracy of the Homotopy Perturbation Method for numerically analyzed (2 + 1) coupled Burgers system. c 2008 Elsevier Ltd. All rights reserved.
Keywords: Homotopy Perturbation Method (HPM); Coupled Burgers system; Numerical analysis of nonlinear PDEs
1. Introduction The exact and numerical solutions of the nonlinear partial differential equations play an important role in the physical sciences and in the engineering fields. The exact or the numerical solutions of the coupled nonlinear partial differential equation systems are also taken a great attention of researches in these fields. Tanh method is successfully applied to get exact solutions in many applications [1–6]. The further extended tanh method to the coupled Burgers equation [7,8] was applied in [9]. The model equation for the (2 + 1) Burgers system is given in the following form u t = u x x + u yy + 2uu x + 2vu y , vt = vx x + v yy + 2uvx + 2vvx .
(1)
A numerical method called Homotopy Perturbation Method (HPM) is proposed by He [10] and further developed by himself in [11–14] for better accuracy. It yields a fast convergence for most of the selected problems. It also showed a high accuracy and a rapid convergence to solutions of the nonlinear partial differential equations. Recently, homotopy perturbation technique was successfully applied to (1 + 1)-dimensional generalized Hirota–Satsuma coupled KdV equation by Abbasbandy [15]. ∗ Corresponding author. Tel.: +90 212 2853345; fax: +90 212 285 6386.
E-mail address: [email protected] (E. Hızel). c 2008 Elsevier Ltd. All rights reserved. 1468-1218/$ - see front matter doi:10.1016/j.nonrwa.2008.02.033
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
1933
In this paper, as an alternative to the exact solutions the HPM will be applied for numerical solutions of (2 + 1)dimensional coupled Burgers system. The accuracy and convergence of the numerical results will be compared with the analytical ones. 2. Homotopy Perturbation Method (HPM) One considers the following nonlinear differential equation to represent the procedure of this method, Au − f (r ) = 0,
r ∈Ω
(2)
with the boundary conditions of ∂u = 0, r ∈ Γ B u, ∂n
(3)
where A and B are general differential operator and boundary operator, respectively. Γ is the boundary of the domain Ω , and f (r ) is a given analytical function. After dividing the general operator into linear part (L) and nonlinear part (N ), one can rewrite the Eq. (2) as Lu + N u − f (r ) = 0.
(4)
By constructing the homotopy technique to Eq. (4), one can write a homotopy in the form H (V, p) = (1 − p)[L(V ) − L(u 0 )] + p[A(V ) − f (r )] = 0,
p ∈ [0, 1],
r ∈Ω
(5)
where p ∈ [0, 1] is an embedding parameter, u 0 is an initial approximation of the Eq. (2) which satisfies the Eq. (3). In HPM, one can use the embedding parameter as a small parameter. Therefore, the solution of Eq. (5) can be written as a power series of p in the form, V = Vo + pV1 + p 2 V2 + · · · .
(6)
By setting p = 1, one can get an approximate solution of the Eq. (2) as, u = lim V = Vo + V1 + V2 + · · · . p→1
(7)
The combination of a small parameter (perturbation parameter) with a homotopy is called homotopy perturbation method as presented in the final equation (7). 3. Application of the HPM to (2 + 1)-dimensional coupled Burgers system In this section, two different solutions of coupled Burgers system [9] will be investigated by using the HPM. 3.1. Solitary solution to 2 + 1 coupled Burgers system The exact solutions to Eq. (1) are given by √ √ u = f − α F −δ tanh( −δ(−α F x + (α f + β)y + 2β Ft + γ )), √ √ v = F + (α f + β) −δ tanh( −δ(−α F x + (α f + β)y + 2β Ft + γ ))
(8)
where f, F, δ, α, β, γ are any arbitrary functions. For simplicity, f = 1, F = 1, δ = −1, α = 1, β = 1 and γ = 1 are used for arbitrary variables. Then, the Eq. (8) takes the following form u = 1 − tanh(−x + 2y + 2t + 1), v = 1 + 2 tanh(−x + 2y + 2t + 1).
(9)
By using an initial condition to Eq. (9), u 0 = u 0 (x, y, 0) = 1 − tanh(−x + 2y + 1) and v0 = v0 (x, y, 0) = 1 + 2 tanh(−x + 2y + 1) and then substituting Eq. (9) into Eq. (5) and later by substituting the Eq. (6) to obtained
1934
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
one, one gets a system of equation with n + 1 terms that needs to be solved simultaneously. Since computations are dependent on the value of u o and vo , a minor modification gives flexibility to choose the initial u o and vo . For this purpose, the following homotopies are constructed as in Refai et al. [16] (1 − p)(wt − u 0t ) + p(wt − wx x − w yy − 2wwx − 2r w y ) = 0 (1 − p)(rt − vot ) + p(rt − r x x − r yy − 2wr x − 2rr y ) = 0 p2 w
(10)
where w = w0 + pw1 + 2 + · · · and r = r0 + pr1 + 2 + · · ·. The variables u and v can be obtained similar to Eq. (7) as p2r
u = lim w = wo + w1 + w2 + · · · p→1
v = lim r = ro + r1 + r2 + · · · . p→1
(11)
After expanding the Eq. (10) for n = 3, i.e. the third power of p, one obtains p0 :
w0t = u 0t , r0t = v0t
p1 :
w1t − w0x x − w0yy − 2w0 w0x − 2r0 w0y = 0 r1t − r0x x − r0yy − 2w0r0x − 2r0r0y = 0
p2 :
w2t − w1x x − w1yy − 2w0 w1x − 2r0 w1y − 2w1 w0x − 2r1 w0y = 0
(12)
r2t − r1x x − r1yy − 2w0r1x − 2w1r0x − 2r0r1y − 2r1r0y = 0 p3 :
w3t − w2x x − w2yy − 2w0 w2x − 2w1 w1x − 2w2 w0x − 2r0 w2y − 2r1 w1y − 2r2 w0y = 0 r3t − r2x x − r2yy − 2w0r2x − 2w1r1x − 2w2r0x − 2r0r2y − 2r1r1y − 2r2r0y = 0.
The solution of the Eq. (12) can be obtained as, w1 = 2 tSech(1 − x + 2 y)2 (1 − tanh(1 − x + 2 y)) + 10 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y) − 4 Sech(1 − x + 2 y)2 (1 + 2 tanh(1 − x + 2 y)) r1 = −4t Sech(1 − x + 2 y)2 (1 − tanh(1 − x + 2 y)) − 20 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y) + 8 Sech(1 − x + 2 y)2 (1 + 2 tanh(1 − x + 2 y)) w2 = t 2 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y)
(13)
r2 = −8 t 2 Sech(1 − x + 2 y)2 tanh(1 − x + 2 y) −8 3 t (−2 + cosh(2 − 2 x + 4 y)) Sech(1 − x + 2 y)4 w3 = 3 16 3 r3 = t (−2 + cosh(2 − 2 x + 4 y)) Sech(1 − x + 2 y)4 . 3 Finally, the approximated solutions are given by u∼ = w0 + w1 + w2 + w3 v∼ = r0 + r1 + r2 + r3 .
(14)
In Fig. 1, the exact distribution of the u(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 is plotted. The approximated distribution of the u(x, y, 0.3) is plotted for the first four terms in Fig. 2 for the same interval. In the Fig. 3, the exact distribution of the v(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 is plotted. The approximated distribution of the v(x, y, 0.3) is plotted for the first four terms in Fig. 4 for the same interval. 3.2. A special solution to 2 + 1 coupled Burgers system The exact solutions to Eq. (1) are given by p √ u = aq − qx −δ tanh( −δq) + b,
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
1935
Fig. 1. Exact solution of u(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
Fig. 2. HPM solution of u(x, y, 0.3) ∼ = w0 + w1 + w2 + w3 for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
Fig. 3. Exact solution of v(x, y, 0.3) for the intervals 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
p √ v = cq + q yx −δ tanh( −δq) + d where q =
−ax−cy−2(cd+ab)t+aβ 2(c2 +a 2 )t+aα
and a, b, c, d, α, β are any arbitrary constants.
(15)
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E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
Fig. 4. HPM solution of v(x, y, 0.3) ∼ = r0 + r1 + r2 + r3 for 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3.
For simplicity, a = b = c = α = β = 1 and d = 0 are used for arbitrary variables. Then, the Eq. (15) takes the following form −x − y + 2 1 −x − y − 4t + 1 u= − tanh , 4t + 1 4t + 1 4t + 1 1 −x − y − 4t + 1 −x − y − 4t + 1 − tanh v= . (16) 4t + 1 4t + 1 4t + 1 By using an initial condition to Eq. (16), u 0 = u 0 (x, y, 0) = (−x − y + 2) − tanh[−x − y + 1] and v0 = v0 (x, y, 0) = (−x − y + 1) − tanh[−x − y + 1] and by following the same procedure as in the previous solution, one obtains the following approximations w1 = r1 = −2t ( (−1 + Sech(1 − x − y)2 ) (1 − x − y − tanh(1 − x − y)) −1 + Sech(1 − x − y)2 × (2 − x − y − tanh(1 − x − y)) + 2Sech(1 − x − y)2 tanh(1 − x − y)) w2 = r2 = −2 t 2 Sech(1 − x − y)3 (−3 cosh(3 − 3 x − 3 y) + 2 x cosh(3 − 3 x − 3 y) + 2 y cosh(3 − 3 x − 3 y) + 15 cosh(1 − x − y) − 10 x cosh(1 − x − y) − 10 y cosh(1 − x − y) + 2 sinh(3 − 3 x − 3 y) − 16 sinh(1 − x − y) + 24 x sinh(1 − x − y) − 8 x 2 sinh(1 − x − y) + 24 y sinh(1 − x − y) − 16 x y sinh(1 − x − y) − 8 y 2 sinh(1 − x − y)) 4 w3 = r3 = t 3 Sech(1 − x − y)4 (−27 + 162 x − 144 x 2 + 32 x 3 + 162 y − 288 x y + 96 x 2 y 3 − 144 y 2 + 96 x y 2 + 32 y 3 − 9 cosh(4 − 4 x − 4 y) + 6 x cosh(4 − 4 x − 4 y)
(17)
+ 6 y cosh(4 − 4 x − 4 y) + 126 cosh(2 − 2 x − 2 y) − 156 x cosh(2 − 2 x − 2 y) + 72 x 2 cosh(2 − 2 x − 2 y) − 16 x 3 cosh(2 − 2 x − 2 y) − 156 y cosh(2 − 2 x − 2 y) + 144 x y cosh(2 − 2 x − 2 y) − 48 x 2 y cosh(2 − 2 x − 2 y) + 72 y 2 cosh(2 − 2 x − 2 y) − 48 x y 2 cosh(2 − 2 x − 2 y) − 16 y 3 cosh(2 − 2 x − 2 y) + 6 sinh(4 − 4 x − 4 y) − 150 sinh(2 − 2 x − 2 y) + 216 x sinh(2 − 2 x − 2 y) − 72 x 2 sinh(2 − 2 x − 2 y) + 216 y sinh(2 − 2 x − 2 y) − 144 x y sinh(2 − 2 x − 2 y) − 72 y 2 sinh(2 − 2 x − 2 y)). Finally, the approximated solutions are given by u v
∼ = w0 + w1 + w2 + w3 ∼ = r0 + r1 + r2 + r3 .
(18)
E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
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Fig. 5. Exact solution of u(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
Fig. 6. HPM solution of u(x, y, 0.1) ∼ = w0 + w1 + w2 + w3 for 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
Fig. 7. Exact solution of v(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
In Fig. 5, the exact distribution of the u(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3 is plotted. The approximated distribution of the u(x, y, 0.1) is plotted for the first four terms in Fig. 6 for the same interval. In Fig. 7, the exact distribution of the v(x, y, 0.1) for the intervals 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3 is plotted. The approximated distribution of the v(x, y, 0.1) is plotted for the first four terms in Fig. 8 for the same interval.
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E. Hızel, S. K¨uc¸u¨ karslan / Nonlinear Analysis: Real World Applications 10 (2009) 1932–1938
Fig. 8. HPM solution of v(x, y, 0.1) ∼ = r0 + r1 + r2 + r3 for 1 ≤ x ≤ 3 and 1 ≤ y ≤ 3.
4. Conclusions The iterative numerical solution of the (2 + 1)-dimensional coupled Burgers system was studied by using the Homotopy Perturbation Method (HPM) and the obtained numerical results were compared with the available tanh method results. The results showed that the convergence and accuracy of the Homotopy Perturbation Method for numerically analyzed (2 + 1) Burgers system was in a good agreement with the analytical solutions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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