IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn:2319-765x. Volume 1, Issue 1Ver. I. (Jan. 214), PP 8-12 Adomian Decomposition Method for Solving the Kuramoto Sivashinsky Equation Saad A. Manaa 1, Fadhil H. Easif 2, Majeed A. Yousif 3 1,2,3 Department of Mathematics, Faculty of Science, University of Zakho,Duhok, Kurdistan Region, Iraq Abstract: The approimate solutions for the Kuramoto Sivashinsky Equation are obtained by using the Adomian Decomposition method (ADM). The numerical eample show that the approimate solution comparing with the eact solution is accurate and effective and suitable for this kind of problem. Keywords: Adomian decomposition method (ADM); Kuramoto Sivashinsky Equation. I. INTRODUCTION The ADM was first introduced by Adomian in the beginning of 198's. The method is useful obtaining both a closed form and the eplicit solution and numerical approimations of linear or nonlinear differential equations and it is also quite straight forward to write computer codes. This method has been applied to obtain a formal solution to a wide class of stochastic and deterministic problems in science and engineering involving algebraic, differential, integro-differential, differential delay, integral and partial differential equations. In the present study, Adomian decomposition method (ADM) has been applied to solve the Kuramoto Sivashinsky equations. The numerical results are compared with the eact solutions. It is shown that the errors are very small. [1] II. II.1 Mathematical Model The Kuramoto Sivashinsky equation is a non-linear evolution equation and has many applications in a variety of physical phenomena such as reaction diffusion systems (Kuramoto and Tsuzuki, 1976)[2], long waves on the interface between two viscous fluids (Hooper and Grimshaw, 1985)[3], and thin hydrodynamics films (Sivashinsky, 1983)[4]. The Kuramoto-Sivashinsky equation has been studied numerically by many authors (Akrivis and Smyrlis, 24; Manickam et al.,1998; Uddin et al., 29)[5-7]. Consider the Kuramoto Sivashinsky equation u t uu α u γ u β u = (1) Subject to the initial condition u, = f() a b. 2 And boundary conditions u a, t = g 1 t, u b, t = g 2 t t > 2 u 2 = 3 1, at = a and = b where 1. II.2 Basic idea of Adomian Decomposition Method (ADM) Consider the differential equation:- Lu Ru Nu = g (4) u, = f (5) where L is the operator of the highest-ordered derivatives and R is the remainder of the linear operator. The nonlinear term is represented byn(u). Thus we get: Lu = g Ru Nu (6) L = t, DefineL 1 t =. dt (7) operating with the operator L 1 on both sides of Eq. (6) we have u = f L 1 Ru L 1 Nu, wherel 1 g = f (8) The standard Adomian decomposition method define the solution u, t an infinite series of the form: 8 Page
u, t = u k (, t) (9) k= u = f (1) And u 1, u 2, aredetermined by u k1 = L 1 Ru k L 1 Nu k, k (11) and the nonlinear operator N u can be decomposed by an infinite series ofpolynomials given by N u = A k (12) k= d k dλi F λi k i= u i λ=, k =,1, (13) A k = 1 k! It is now well known in the literature that these polynomials can be constructed for all classes of nonlinearity according to algorithms set by Adomian [8,9] and recently developed by an alternative approach in [8-1]. II.3 Derivative of (ADM) for Kuramoto-Sivashinsky equation We consider Kuramoto-Sivashinsky equation u t uu α u γ u β u =, 14 with the initial condition of u, = f() 15 Applying the operator L 1 on both sides of Eq. (14) and using the initial condition we find u, t = f L 1 uu α u γ u β u () L = t, DefineL 1 t =. dt (17) k= u k (, t) = f L 1 u, t = u k (, t) (18) k= A k k= α u k γ u k β u k k= k= k= (19) Identifying the zeros component u (, t)by f(), the remaining componentsk 1 can be determined by using the recurrence relation u, t = f (2) u k1, t = L 1 A k α u k γ u k β u k, k, (21) wherea k are Adomian polynomials that represent the nonlinear term (uu ) and given by d k A k = 1 k! dλ i F λi k i= u i When F u = uu, (23) Then d A = 1! dλ [F(λ u )] λ= = F u = u u (24) A 1 = 1 d 1! dλ [F u λu 1 ] λ= = d dλ [(u λu 1 )(u λu 1 ) ] λ= λ= = d dλ [(u u o u u 1 λ u 1 u λ u 1 u 1 λ 2 )] λ=, k =,1, (22) = u u 1 u 1 u (25) A 2 = 1 2! dλ 2 [F(u u 1 λ u 2 λ 2 )] λ= = 1 2 dλ 2 [(u u 1 λ u 2 λ 2 )(u u 1 λ u 2 λ 2 ) ] λ= = 1 2 dλ 2 [(u u u u 1 λ u u 2 λ 2 u 1 u λ u 1 u 1 λ 2 u 1 u 2 λ 3 u 2 u λ 2 u 2 u 1 λ 3 u 2 u 2 λ 4 )] λ= = u u 2 u 1 u 1 u 2 u (26) 9 P a g e
Other polynomials can be generated in a similar way. The first few components of u k, t follows immediately upon setting u, t = f (27) u 1, t = L 1 A α u γ u β u (28) u 2, t = L 1 A 1 α u 1 γ u 1 β u 1 (29) u 3, t = L 1 A 2 α u 2 γ u 2 β u 2 (3) Then u, t = k= u k (, t) = u, t u 1, t u 2, t u 3, t (31) III. Figures And Tables III.1 Numerical Eample In this section, we apply the technique discussed in the previous section to find numerical solution of the Kuramoto Sivashinsky equations and compare our results with eact solutions. Eample: [11] u t uu u u =,,32π, 32 With the initial condition of u, = cos 1 sin ; 33 Eact solution of problem is given by: u, t = cos t 1 sin t (34) A = u u = cos 1 sin sin 1 sin cos 2 (35) u 1, t = t cos 92 sin 8192 cos2 sin 12288 cos2 8447 65536 (36) A 1 = u u 1 u 1 u = t cos 35294 sin 11872 cos 2 sin 49152 cos 2 sin 148576 134861 cos2 124 cos 4 35294 (37) u 2, t = t 2 cos 5437696 sin 5177344 cos2 sin 21326592 cos4 sin 8589934592 5797688 cos 2 419434 cos4 4855297 8589934592 (38) A 2 = u u 2 u 1 u 1 u 2 u = t 2 cos 97256964 sin 682338992 cos2 sin 943968 cos 4 sin 2147483648 cos6 sin 734897891 cos 2 12242182144 cos4 575458588 cos6 971435586 (39) 1 P a g e
u 3, t = t 3 cos 485935914912 sin 375881134592 cos2 8884986263936 sin 3863859953664 cos4 sin 8796932228 cos6 8884986263936 sin 3371821131776 cos2 52566188621824 cos4 23577852448 cos6 4826353967871 8884986263936 (4) Then approimation solution of Eq.(32) is u, t = u u 1 u 2 u 3 with third-order approimation. π t EXACT SOLUTION APPROXIMATION ABSOLUTE ERROR 1. 1..2.999799985333.999988278219568 1.88298214234e-4.4.9995999242668.9999765564818 3.766364391323274e-4.6.999399821445.999964834786718 5.6514642713798e-4.8.9991996834135.999953113134345 7.534327929941131e-4.1.998999566678.9999413915247 9.41898579912344e-4.629962521184.629962521184.2.63265525986.6292429951888 3.37575574976372e-4 6.4.4.63613531113795.629384431289 6.75918255741e-4.6.63965397251127.6295285759426 1.1254649171349e-3.8.64317239455.62967262136334 1.3499417681749e-3.1.646689514459.629874253 1.68727687845974e-3-1.284545252522524-1.284545252522524.2-1.284489444647476-1.2845518271811 6.23766333447943e-5 12.8.4-1.28443352832249-1.2845583882576 1.24859928267992e-4.6-1.2843775354176-1.28456495514284 1.87451599273e-4.8-1.2843213729945-1.284571521381397 2.5151819914454e-4.1-1.284265128591897-1.28457886973377 3.12958381484882e-4 -.333488736227371 -.333488736227371.2 -.333668118536193 -.333484463982 1.839538963727683e-4 19.2.4 -.333847544554259 -.333479593175463 3.679513787964717e-4.6 -.3342714266963 -.33347521834294 5.51992432669129e-4.8 -.33426527659685 -.3334745638 7.3677433769148e-4.1 -.33438684717798 -.33346587952152 9.2251962963198e-4.15124368228711.15124368228711.2.159597765235.15123677789543 2.7713719346869e-5 25.6.4.1567621719845.1512298735399 5.536563414442614e-5.6.1539341296.151222969225 8.29965914345494e-5.8.151113713235.15126493723 1.1592784877852e-4.1.14982762528.15129699 1.381544664819934e-4 1. 1..2.999799985333.999988278219567 1.88298214234657e-4 32.4.9995999242668.9999765564818 3.7663643913223e-4.6.999399821445.999964834786718 5.65146427135239e-4.8.9991996834135.999953113134344 7.534327929941131e-4.1.998999566678.999941391524699 9.41898579915675e-4 Table (1) comparison eact with Adomian Decomposition Method (ADM) 11 P a g e
Now we compare eact solution with Adomian Decomposition Method (ADM) solution in Fig.1,Fig.2. Fig.(1) Eact solution of Kuramoto-Sivashinsky equation Fig.(2) (ADM) solution of Kuramoto-Sivashinsky equation IV. Conclusion The (ADM) applied to Kuramoto-Sivashinsky equation and by comparing with the eact solution Fig.(1), Fig.(2) and Table(1) shows that the absolute error is so small and the approimate solution is so closed to the eact solution. References [1] S.T. Mohyud-Din, On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He's Homotopy Perturbation Method,J. Applications nd Applied Mathematics,1932-9466, 21, 1-11. [2] Kuramoto, Y., Tsuzuki, T., Persistent propagation of concentrationwaves in dissipative media far from thermal equilibrium,pron. Theor. Phys.,55, 1976, 356. [3] Hooper, A.P., Grimshaw, R., Nonlinear instability at theinterface between two viscous fluids. Phys. Fluids,28, 1985, 37 54. [4] Sivashinsky, G.L., Instabilities, pattern-formation, and turbulencein flames. Ann. Rev. Fluid Mech.,15, 1983, 179 199. [5] Akrivis, Georgios, Smyrlis, Yiorgos-sokratis, Implicit-eplicitBDF methods for the Kuramoto- Sivashinsky equation,appl.numer. Math. 51, 24, 151 9. [6] Manickam, A.V., Moudgalya, K.M., Pani, A.K., Second-ordersplitting combined with orthogonal cubic spline collocation methodfor the Kuramoto Sivashinsky equation, Comput. Math. Appl. 35, 1998, 5 25. [7] Uddin, M., Haq, S., Siraj-ul-Islam, A mesh-free numericalmethod for solution of the family of Kuramoto Sivashinskyequations, Appl. Math. Comput. 212, 29, 458 469. [8] G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method,( Kluwer, Boston, 1994). [9] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135, 1988,51 544. [1] G. Adomian, New approach to nonlinear partial differential equations, J. Math. Anal. Appl. 12,1984, 42 434. [11] Ling lin, High Order Collocation Software for the Numerical Solution of Fourth Order Parabolic PDEs., Master Thesis,Halifa, Nova.Scotia, University of Saint Mary's, 29. 12 P a g e