A Simple Algorithm for Calculating Adomian Polynomials

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 20, A Simple Algorithm for Calculating Aomian Polynomials J. Biazar an S. M. Shafiof Department of Mathematics, Faculty of Science University of Guilan, P.O. Box 94, Rasht, Iran biazar@guilan.ac.ir, shafiof@yahoo.com Abstract In this paper, we introuce a new Algorithm for calculating Aomian polynomials an present some example to show the simplicity of the new Algorithm. The new Algorithm requires less computations in comparison with the previous methos, an can be extene to calculate Aomian polynomials for nonlinear functional with several variables. Keywors: Aomian ecomposition metho, Aomian polynomials, Nonlinearity of several variables Introuction The Aomian ecomposition metho[-2] is a technique for solving functional equations in the form: u = f + G) () In some functional space, say F. The solution u is consiere as the summation of a series, say; u = u n (2) n=0 an G) as the summation of a series, say; G) = A n 0,u,...,u n ) (3) n=0 Where A n s, calle Aomian polynomials, has been introuce by the Aomian himself by the formula A n 0,u,...,u n )= n n! λ [G( u n i λ i )] λ=0 (4) Corresponing author i=0

2 976 J. Biazar an S. M. Shafiof Other authors have suggeste ifferent Algorithm for computing Aomian polynomials [3-6]. Using the Algorithm presente in Ref.[3], nees classification of the terms u i, i =0,, 2,..., n, which is very complicate for large n s. Applying the Algorithm presente in [4], which uses Taylor expansion of the functional G), will be very complicate especially when the unknown function u, appears in the enominator. Calculation of n th Aomian polynomials using the technique in Ref.[5], requires to computing the n th orer erivative, that will be complicate for large values of n. In spite of ifficulties in applying the use metho in [6], it coul not be applie for functionals with several variables. Here we suggest a new Algorithm, which seems to be more easier than other methos introuce so far. This Algorithm can also be extene for calculating Aomian polynomials in several variables. Some examples, will illustrate how easy is the new Algorithm for implementation. 2 The new Algorithm The following is the Algorithm for calculating A 0,A,..., A n : Step : Input nonlinear term G) an n, the number of Aomian polynomials neee. Step 2: Set A 0 = G 0 ) Step 3: For k =0ton o A k 0,..., u k ):=A k 0 + u λ,..., u k +(k +)u k+ λ) {in A k : u i u i +(i +)u i+ λ for i =0,,..., k} Step 4:Taking the first orer erivative of A k, with respect to λ, an then let λ =0: λ A k λ=0 =(k +)A k+. En o Step 5: Output A 0,A,..., A n. Accoring to the above Algorithm, Aomian polynomials will be compute as follows: A 0 = G 0 ),

3 Aomian polynomials 977 A = λ G 0 + u λ) λ=0 = u G 0 ), A 2 = ( 2 λ +2u 2 λ)g 0 + u λ)) λ=0 = u 2 G 0 )+ u2 2! G u 0 ), A 3 = ( 3 λ 2 +3u 3 λ)g 0 + u λ)+ 2! +2u 2 λ) 2 G 0 + u λ)) λ=0 = u 3 G 0 )+u u 2 G 0 )+ u3 3! G 0 ), A 4 = ( 4 λ 3 +4u 4 λ)g 0 + u λ)+ +2u 2 λ) 2 +3u 3 λ)g 0 + u λ)+ 3! +2u 2 λ) 3 G 0 + u λ)) λ=0 = u 4 G 0 )+ u 3 + u2 2 2 )G 0 )+ u 2 u 2 2 G 0 )+ u4 4! G4 0 ), Continuing this course, we can get the other Aomian polynomials. 3 Application of the new Algorithm Here we apply the above Algorithm to some nonlinear operators functionals. Example :(The case of nonlinear polynomials) G) =u 3 + u 4 Now by applying the new Algorithm we get A 0 = u u4 0, A = λ ( 0 + u λ) u λ) 4 ) λ=0 =3u u u u 3 0, A 2 = (3 2 λ +2u 2 λ) 0 + u λ) u 2 λ) 0 + u λ) 3 ) λ=0 = 3u 2 u u 0 u 2 +4u 2 u u 2 0u 2, A 3 = (3 3 λ 2 +3u 3 λ) 0 + u λ) u λ) +2u 2 λ) u 3 λ) 0 + u λ) u λ) 2 +2u 2 λ) 2 ) λ=0 =3u 3 u u 0 u u 2 +4u 3 u u 2 0 u u 2 +4u 0 u 3 + u3, While by corresponing metho presente in [3], one must calculate 0 + u + u 2 + u ) 3, 0 + u + u 2 + u ) 4 an reorer the terms in inex, to obtain Aomian polynomials. Example 2:( Case of nonlinear erivatives) Consier G) =u 2 u x, Accoring to the new Algorithm : A 0 =G 0 )=u 2 0u 0x,

4 978 J. Biazar an S. M. Shafiof A = λ ( 0 + u λ) 2 + 0x + u x λ)) λ=0 =2u 0 u u 0x + u 2 0 u x, A 2 = (2 2 λ 0 + u λ) +2u 2 λ) 0x + u x λ)+ 0 + u λ) 2 x +2u 2x λ)) λ=0 = u 2 u 0x +2u 0 u 2 u 0x +2u 0 u u x + u 2 0 u 2x, A 3 = ( 3 λ +2u 2 λ) 2 0x + u x λ) u λ) 0x + u x λ) 2 +3u 3 λ) u λ) +2u 2 λ) x +2u 2x λ)+ 0 + u λ) 2 2x +3u 3x λ)) λ=0 = 2u u 2 u 0x +2u 0 u 3 u 0x +2u 0 u 2 u x +2u 0 u u 2x + u 2 u x + u 2 0u 3x, A 4 = u 2 u 2x+2u u 2 u x +2u u 3 u 0x +2u 0 u 4 u 0x +2u 0 u 3 u x +2u 0 u u 3x +2u 0 u 2 u 2x + u 2 2u 0x + u 2 0u 4x, Example 3:( Case of hyperbolic an trigonometric nonlinearity) G) = sinh u + sin 2 u cos 2 u By using the new Algorithm,the Aomian polynomials for G) are given as: A 0 =G 0 ) = sinh u 0 + sin 2 u 0 cos 2 u 0, A = (sinh λ 0 + u λ)+sin u λ)cos u λ)) λ=0 = u cosh u 0 + 2u sin u 0 cos 3 u 0 2u sin 3 u 0 cos u 0, A 2 = ( 2 λ +2u 2 λ) cosh 0 +u λ)+2 +2u 2 λ) sin 0 +u λ) cos 3 0 +u λ) 2 +2u 2 λ) sin u λ) cos 0 + u λ)) λ=0 = u 2 cosh u 0 + u2 2! sinh u 0 + u 2 cos 4 u 0 6u 2 sin 2 u 0 cos 2 u 0 +2u 2 sin u 0 cos 3 u 0 + u 2 sin4 u 0 2u 2 sin 3 u 0 cos u 0, A 3 = ( 3 λ 2 +3u 3 λ) cosh 0 + u λ)+ +2u 2 λ) 2 sinh 2! 0 + u λ)+ +2u 2 λ) 2 cos u λ) 6 +2u 2 λ) 2 sin u λ) cos u λ) u 3 λ) sin 0 + u λ) cos u λ)+ +2u 2 λ) 2 sin u λ) u 3 λ) sin u λ) cos 0 + u λ)) λ=0 = u 3 cosh u 0 + u u 2 sinh u 0 + u3 cosh u 3! u3 cos 3 u 0 sin u 0 +2u u 2 cos 4 u u3 sin3 u 0 cos u 0 2u u 2 sin 2 u 0 cos 2 u 0 +2u 3 sin u 0 cos 3 u 0 +2u u 2 sin 4 u 0 2u 3 sin 3 u 0 cosu 0, Example 4:(Case of exponential an logarithmic nonlinearity) G) =e u + lnu By using the new Algorithm, for calculating the Aomian polynomials for G) =e u + lnu, we have:

5 Aomian polynomials 979 A 0 = G 0 )=e u 0 + lnu 0, A = λ (e 0+u λ) + ln 0 + u λ)) λ=0 = u e u 0 + u u 0, A 2 = ( 2 λ +2u 2 λ)e 0+u λ) + +2u 2 λ) ) 0 +u λ) λ=0 = u 2 e u u2 eu 0 + u 2 A 3 = ( 3 λ 2+3u 3 λ)e 0+u λ) u 2 λ) 2 e 0+u λ) + 2+3u 3 λ) u 3 e u 0 + u u 2 e u 0 + 3! u3 eu 0 + u 3 u 0 u u 2 u u 3 3 u 3 0, 0 +u λ) 2 4. Extension of the new Algorithm for calculating Aomian polynomials in several variables: u 0 u 2 2 u u 2 λ) 2 0 +u λ) 2 ) λ=0 = Now, this new approach for computing Aomian polynomials can be applie to functionals with several variables. Let we are concerne with a system of nonlinear equations u i = f i + N i i =, 2,..., p Where N i,u 2,..., u p ) are nonlinear operators an u i,n i are represente as the summation of the following series: u i = u ij λ j j=0 N i,u 2,..., u p )= A ij 0,..., u j,u 20,..., u 2j,..., u p0,..., u pj )λ j, j=0 The following Algorithm calculates Aomian polynomials A i0,a i,..., A in : For i =, 2,..., p o Step : Input nonlinear term N i,u 2,..., u p ) an n, the number of Aomian polynomials neee. Step 2: Set A i0 = N i 0,u 20,..., u p0 ) Step 3: For k =0ton o A ik 0,..., u k,..., u p0,..., u pk ):= A ik 0 + u λ,..., u k +(k +)u k+ λ,..., u p0 + u p λ,..., u pk +(k +)u pk+ λ) {in A ik : u ij u ij +(j +)u ij+ λ for j =0,,..., k } Step 4: By taking, first orer erivative of A ik, with respect to λ, an let λ = 0, we have :, En o En o λ A ik λ=0 =(k +)A ik+.

6 980 J. Biazar an S. M. Shafiof Step 5: Output A i0,a i,..., A in. 4. Two variables (p=2): By using the new Algorithm, the Aomian polynomials for the following system of two functional equations can be compute: { u = f + N A i0 = N i 0,u 20 ), u 2 = f 2 + N 2 A i = λ (N i 0 + u λ, u 20 + u 2 λ)) λ=0 = u i u 0,u 20 )+u 2 i u 2 0,u 20 ), A i2 = ( 2 λ +2u 2 λ) i u 0 + u λ, u 20 + u 2 λ)+ 2 +2u 22 λ) i u u λ, u 20 + u 2 λ)) λ=0 = u i 2 u 0,u 20 )+u i 22 u 2 0,u 20 )+ 2! u2 2 N i u 2 0,u 20 )+ 2! u2 2 2 N i u 2 0,u 20 )+u u 2 N i 2 2 u u 2 0,u 20 ), A i3 = ( 3 λ 2 +3u 3 λ) i u 0 + u λ, u 20 + u 2 λ) u 23 λ) i u u λ, u 20 + u 2 λ)+ 2! +2u 2 λ) 2 2 N i u u λ, u 20 + u 2 λ)+ 2! 2 +2u 22 λ) 2 2 N i u u λ, u 20 + u 2 λ)+ +2u 2 λ) 2 +2u 22 λ) 2 2 N i u u u λ, u 20 + u 2 λ)) λ=0 = u i 3 u 0,u 20 )+u i 23 u 2 0,u 20 )+u u 2 N i 2 u 2 0,u 20 )+ u 2 u 2 N i 22 0,u 20 )+u 2 u 2 N i 2 u u 2 0,u 20 )+u u 2 N i 22 u 2 u 0,u 20 )+ u 2 2 2! u2 u 2 N i 2 3! u3 2 3 N i u 3 2 u 0,u 2 u 2 20 )+ 2! u2 2u 0,u 20 ), 2 N i u u 2 2 0,u 20 )+ 3! u3 3 N i u 3 0,u 20 )+ To illustrate the application of the metho for two variables, here an example is presente. Example 6: Let N, v) =uvv x, by applying the new Algorithm we get A 0 = N 0,v 0 )=u 0 v 0 v 0x, A = λ ( 0 + u λ)(v 0 + v λ)(v 0x + v x λ)) λ=0 = u v 0 v 0x + u 0 v v 0x + u 0 v 0 v x, A 2 = ( 2 λ +2u 2 λ)(v 0 +v λ)(v 0x +v x λ)+ 0 +u λ)(v +2v 2 λ)(v 0x +v x λ)+ 0 + u λ)(v 0 + v λ)(v x +2v 2x λ)) λ=0 = u 2 v 0 v 0x + u v v 0x + u 0 v 2 v 0x + u v 0 v x + u 0 v v x + u 0 v 0 v 2x, A 3 = ( 3 λ 2+3u 3 λ)(v 0 +v λ)(v 0x +v x λ)+ +2u 2 λ)(v +2v 2 λ)(v 0x +v x λ)+ 0 + u λ)(v 2 +3v 3 λ)(v 0x + v x λ)+ +2u 2 λ)(v 0 + v λ)(v x +2v 2x λ)+

7 Aomian polynomials u λ)(v +2v 2 λ)(v x +2v 2x λ)+ 0 +u λ)(v 0 +v λ)(v 2x +3v 3x λ)) λ=0 = u 3 v 0 v 0x + u 2 v v 0x + u 0 v 3 v 0x + u 2 v 0 v x + u v v x + u 0 v 2 v x + u v 0 v 2x + u 0 v v 2x + u 0 v 0 v 3x + u v 2 v 0x, 4.2 Three variables (p=3): Using the new Algorithm, the Aomian polynomials for N i,u 2,u 3 ) Where u,u 2,u 3,N i are represente by series: are thus given by A i0 = N i 0,u 20,u 30 ), u i = u ij λ j for i =, 2, 3, j=0 N i = A ij 0,..., u j,u 20,..., u 2j,u 30,..., u 3j )λ j j=0 A i = (N λ i 0 + u λ, u 20 + u 2 λ, u 30 + u 3 λ)) λ=0 = u i u 0,u 20,u 30 )+ u i 2 u 2 0,u 20,u 30 )+u i 3 u 3 0,u 20,u 30 ), A i2 = ( 2 λ +2u 2 λ) i u 0 + u λ, u 20 + u 2 λ, u 30 + u 3 λ)+ 2 +2u 22 λ) i u u λ, u 20 + u 2 λ, u 30 + u 3 λ)+ 3 +2u 32 λ) i u u λ, u 20 + u 2 λ, u 30 + u 3 λ)) λ=0 = u i 2 u 0,u 20,u 30 )+ u 22 i u 2 0,u 20,u 30 )+u i 32 u 3 0,u 20,u 30 )+ 2! u2 2 N i u 2 0,u 20,u 30 )+ 2! u2 2 2 N i u 2 0,u 20,u 30 )+ 2 2! u2 3 2 N i u 2 0,u 20,u 30 )+ 3 u 2! u 2 N i 2 u u 2 0,u 20,u 30 )+ u 2! u 2 N i 3 u u 3 0,u 20,u 30 )+ u 2! 2u 2 N i 3 u 2 u 3 0,u 20,u 30 ), Continue this course, we can get the Aomian polynomials for any p- variables functional operator. 4 Discussion The Aomian ecomposition metho is a powerful evice for solving a large class of nonlinear problems in sciences.the main part of this metho is calculating Aomian polynomials for nonlinear polynomials. In this paper, we introuce a new Algorithm for calculating Aomian polynomials. The main specification of this Algorithm is no requirement to complicate calculations.

8 982 J. Biazar an S. M. Shafiof This Algorithm is very easy to implement, because it uses first orer erivative, once. In each step to obtain the Aomian polynomials an will be extene to calculate Aomian polynomials for nonlinear functionals of several variables. 5 References [] G. Aomian, G.E. Aomian, A global metho for solution of complex systems,math. Moel. 5 (984) [2] G. Aomian, Solving Frontier Problems of Physics: The Decomposition Metho, Kluwer Acaemic Publishers, Dorecht, 994. [3] A. M. Wazwaz, A new algorithm for calculating Aomian polynomials for non-linear Operators, Appl. Math. Comput. (2000) [4] J. Biazar, E. Babolian, A. Nouri, R. Islam, An alternate algorithm for computing Aomin polynomials in special cases, App. Math. Comp.38 (2003) [5] Yonggui Zhu, Qianshun Chang, Shengchang Wu, A new algorithm for calculating Aomian polynomials Appl. Math. Comput. 69 (2005) [6] E. Babolian, Sh. Javai, New metho for calculating Aomian polynomials, Appl. Math. Comput.53 (2004) Receive: November 8, 2006