CMES. Computer Modeling in Engineering & Sciences. Tech Science Press. Reprinted from. Founder and Editor-in-Chief: Satya N.

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1 Reprinted fro CMES Coputer Modeling in Engineering & Sciences Founder and Editor-in-Chief: Satya N. Atluri ISSN: print ISSN: on-line Tech Science Press

2 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 A Developed New Algorith for Evaluating Adoian Polynoials M. Azreg-Aïnou 1 Abstract: Adoian polynoials AP s are expressed in ters of new objects called reduced polynoials RP s. These new objects, which carry two subscripts, are independent of the for of the nonlinear operator. Apart fro the well-known two properties of AP s, curiously enough no further properties are discussed in the literature. We derive and discuss in full detail the properties of the RP s and AP s. We focus on the case where the nonlinear operator depends on one variable and construct the ost general analytical expressions of the RP s for sall values of the difference of their subscripts. It is shown that each RP depends on a nuber of functions equal to the difference of its subscripts plus one. These new properties lead to ipleent a draatically siple and copact Matheatica progra for the derivation of individual RP s and AP s in their general fors and provide useful hints for elegant hand calculations of AP s. Application of the progra is considered. Keywords: Adoian decoposition ethod, Adoian polynoials, diophantine equations, Matheatica. 1 Introduction Decoposition ethods continue to develop and gain ground in applied atheatics and integral ethods. They have been reviewed, odified and applied to different fields of science and engineering. The idea behind a decoposition ethod, which is used to solve differential, integro-differential, algebraic equations and their cobinations, is to obtain by easily handled successive iterations or recursions approxiate solutions within a predefined accuracy. Fro this point of view, the ost applied ethods are a variety of doain decoposition ethods, which solve boundary value probles [Davies, Crann, Kane, and Lai 007; Han, and Atluri 007; Huang, Zhang, Ma, and Wang 008; Patrício, Mattheij, and de With 008; Takei, Yoshiura, and Kanayaa 008; Vodička, Mantič, and 1 Başkent University, Departent of Matheatics, Bağlıca Capus, Ankara, Turkey.

3 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 París 007], and Adoian decoposition ethod and its odifications, which tackle both initial and boundary values probles [Adoian 1994; Adoian, and Rach 199; Cochran 1968; El-Tawil, Bahnasawi, and Abdel-Naby 004; Hosseini 006; Lai, Chen, and Hsu 008; Wazwaz 000; Wazwaz and El- Sayed 001; Zhang 005]. Adoian decoposition ethod and soe of its odifications and extensions are atheatical tools providing analytical and rapidly convergent if not exact solutions to a variety of probles in nonlinear science. A proble in nonlinear science is odeled by Lu Ru Nu = g, 1 where the unknown scalar, vector or atrix function ux is subject to soe initial and/or boundary conditions. The function gx is a source ter, L R is a linear operator and N includes nonlinear operator ters, if there are any. The original idea due to Adoian is to split the linear operator L R into two linear ters, L and R, where L represents the highest order derivative and is easily invertible and R groups the reaining lower order derivatives. Integrating 1 leads to a Volterra integral equation [Cochran 1968] u = f L 1 Ru L 1 Nu, where f is the su of L 1 g and the ters arising fro the application of the initial and/or boundary conditions to u. It sees that a part of the ethod originated in the work of Cochran [Cochran 1968]. The other new part of the ethod consists in replacing u by su of coponents u u = u =0 leading to =0 u = f L 1 R =0 u L N 1 =0 u, 3 Now, in order to solve 1 all one needs is to fix the coponents u using 3. This can be done recursively, however, upon splitting first the action of N on =0 u into su of ters =0 A in such a way that the first ter A 0 depends only on u 0, which will be the first coponent of u to be fixed, and A 1 depends only on u 0,u 1,..., and A depends only on u 0,u 1,...,u. With that said, the action of N on u reads as N =0 u = =0 A u 0,u 1,...,u, 4

4 A developed new algorith 3 leading to = f L =0u 1 Ru L 1 A. 5 =0 =0 There is no unequivocal way to fix u 0 [Wazwaz and El-Sayed 001], however, a straightforward way to do it is to identify u 0 x with f x. Siilarly and in a straightforward way the other coponents are chosen to be u 1 = L 1 Ru L 1 A, 0, with u 0 = f. 6 If the action of N on u is represented by soe function Fu, then A u 0,u 1,...,u, for > 0, is expressed as a su of ters, each ter includes a derivative of order k 1 k of F at u 0, F k u 0, ties a polynoial in u 1,u,...,u and A 0 = Fu 0. The first five so called Adoian polynoials", A 0,...,A 4, available in the literature are A 0 = Fu 0, A 1 = u 1 F u 0, A = u F u 0 u 1! F u 0, 7 A 3 = u 3 F u 0 u 1 u F u 0 u3 1 3! F3 u 0, u A 4 = u 4 F u 0! u 1u 3 F u 0 u 1 u! F3 u 0 u4 1 4! F4 u 0. Adoian decoposition ethod ADM is known as a reliable atheatical tool for solving algebraic, differential or integral linear and nonlinear equations [El- Tawil, Bahnasawi, and Abdel-Naby 004; Hosseini 006; Kao, Jiang, and Yu 005; Lai, Chen, and Hsu 008; Wang, and Bajaj 008; Wazwaz and El- Sayed 001; Zhang 005] to ention but a few. The ethod solves nonlinear equations appealing neither to linearization nor to perturbation approaches. ADM provides in any cases the exact solution, if it exists in closed for, or a seianalytical solution showing the ain features of the exact one. However, a fastidious task in ADM is the evaluation of AP s which necessitates a big deal of calculations. A first foral forula to evaluate AP s is that introduced by Adoian hiself [Adoian 1994] followed by a series of soe other alternative techniques developed in [Rach 1984; Seng, Abbaoui, and Cherruault 1996] and soe other papers aiing to reduce the size of calculations, however, they have fallen short of their ai. Based on the property that in each A the su of subscripts of

5 4 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 the coponents u 0,u 1,...,u is equal to, a new algorith has been developed to evaluate AP s without using Adoian forula [Wazwaz 000]. Besides the above entioned property, AP s have soe other interesting properties, never discussed in the literature, which will be discussed and derived in the next section. The ai of this paper is to use these new properties as well as the above entioned one to derive new reliable techniques for hand and coputer evaluations of AP s. Fro the latter point of view a very siple and copact Matheatica progra to copute any polynoial A in its ost general for as a function of F k u 0, 1 k, and of u 1,u,...,u is ipleented. When the function F is given, the progra derived below allows iediate evaluation of any individual polynoial A without evaluating the polynoials with subscripts saller than and without using Adoian forula. Furtherore, the use of the new properties allows for elegant hand calculations of AP s. In section we introduce the reduced polynoials Z,k and derive and discuss in detail their properties as well as the properties of A. In section 3 we apply our results to derive general analytical expressions of A for sall values of k, which are valid for any F. In section 4 we discuss in detail the ipleentation of the Matheatica progra and apply it to solve the proble of the pendulu in section 5. We conclude in section 6. Properties of Adoian polynoials The polynoials A, > 0, possess the following properties Property 1. A depends by construction only on the vector u 1,u,..., u and does not depend on u n with n > ; Property. A is the su of ters of the for A = k=1 Z,k u 1,u,...,u F k u 0 8 where Z,k are called here the reduced polynoials and depend on u 1,u,..., u. We will see below that the properties of A are encoded in Z,k ; Property 3. In each reduced polynoial Z,k, the coponents of the vector u 1,u,..., u appear in such a way that the su of their subscripts is equal to [Wazwaz 000]. A consequence of this property is that each reduced polynoial Z,k depends only on few coponents of the vector u 1,u,...,u. This is obvious since the su of the subscripts of the coponents of the vector u 1,u,...,u exceeds ;

6 A developed new algorith 5 Property 4. Another quit iportant property, curiously enough it has never been discussed in the literature to our knowledge, is that in each Z,k the su of the powers of the coponents u n, 1 n, is equal to k the order of the derivative in 8. Hence, both subscripts of Z,k have a direct eaning: one is the su of the subscripts and the other is the su of the powers of the coponents of the vector u 1,u,...,u which appear in the expression of Z,k. The other properties are discussed below. Using the Taylor expansion for Fu 0 ε in powers of ε, one obtains F Fu = F u 0 u i = Fu 0 }{{} k u 0 u 1 u k. 9 k! k=1 A 0 Now, substituting the ultinoial forula u 1 u k j=1 u n j j = k! n 1 n =k j=1 n j! into Eq. 9 and using the properties 3. and 4. of Z,k > 0, one obtains Z,k = S u n 1 1 un un n 1!n! n! { with S = n 1,...,n N n i = k and } in i =. 10 In the above expression of Z,k, if soe power n i = 0 we have then u n i i = 1 and the corresponding coponent u i will not appear in the expression of Z,k. Now, since Z,k depends only on few coponents of the vector u 1,u,...,u as entioned earlier, soe of the powers n i in 10 ust vanish. The suation in 10 being restricted to the set S, the powers n i are solutions to the syste of diophantine equations n 1 n n 3 n = k and n 1 n 3n 3 n =, 11 where we are interested to find solutions in N. Subtracting the first equation in 11 fro the second one, one obtains n n 3 3n 4 kn k1 1n = k. 1 Since all the coefficients in 1 and the powers n i are positive integers or zero, Eq. 1, and consequently the syste 11, holds only if all the coponents n i,

7 6 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 i > k 1, vanish identically for all and 1 k. Said otherwise, all possible solutions to the syste 11 are such that: n k = n k3 = n k4 = = n 0, 13 eaning that Z,k does never depend on u k,u k3,...,u. We have thus established the proof that Z,k depends at ost on u 1,u,...,u k1, reducing Eq. 10 to n u 1 1 Z,k = un un k1 k1 n 1!n! n k1! S R = S R with { n 1,...,n k1 N k1 k1 n i = k and k1 } in i =. 14 Said otherwise, the syste 11 is equivalent to the following reduced syste of diophantine equations n 1 n n k1 = k and n 1 n k 1n k1 =, 15 where Eq. 1 becoes n n 3 3n 4 kn k1 = k. 16 Does Z,k, as provided by 14, depend on the whole vector u 1,u,..., u k1? We will shortly show that this is the case if 1 < k, however, if k = 1, Z,1 does not depend on the whole vector u 1,u,..., u ; rather, it depends only on u. Assue that 1 < k. It is straightforward to show that Z,k depends on u 1 and linearly on u k1 for 1 < k <. In fact, if we set n k1 = 1 in 16 then the reaining powers in 16 ust vanish and this leads using the first equation in 11 to n 1 = k 1. Furtherore, the solution n 1 = k 1 and n k1 = 1 all the other powers are zero is the unique solution to 15 for which n k1 0. Consequently, Z,k will always have the ter u k 1 1 u k1 k 1! 17 in its expansion. For k =, the ter 17 reduces to u 1 /!. To show that Z,k depends on u and u k we first rewrite the nuerator in 17 as u k 1 u 1 u k1.

8 A developed new algorith 7 Now, rising the subscript of u 1 by 1 and lowering the subscript of u k1 by 1, in the above expression, while keeping u k 1 unchanged, leads to u k 1 u 1 u k1 u k 1 u u k. Notice that such a transforation does not affect both the su of the powers and the su of the subscripts. Hence, n 1 = k, n = 1 and n k = 1 is a solution to 15 or 11 and u k 1 u u k k! 18 is a ter in the expansion of Z,k. Continuing so by rising one subscript and lowering the other subscript in 18 while keeping u k 1 unchanged, leads to u k 1 u i u k i k! i i ax 19 where i ax = k / if k is even or i ax = k 1/ if k is odd. Ters of the for 19 all appear in the expansion of Z,k. Now, as i runs fro to i ax, the coponents u i and u k i span the whole vector u,u 3,..., u k eaning that Z,k depends on u,u 3,..., u k and consequently, taking into account 17, it depends explicitly on the whole vector u 1,u,..., u k1. Assue k = 1. The syste of equations 15 reduces to n 1 n n = 1 and n 1 n n =. 0 Subtracting the second equation fro the first one, one obtains 1n 1 n in i n n 1 = 0, where the left-hand side is a su of positive nubers with positive coefficients. Hence, the unique solution to the previous equation is n 1 = n = = n 1 = 0. Substituting into 0 leads to n = 1. We have then shown that the unique solution to 0 is n = 1 and the other powers are zero, so Z,1 depends only on u and is given by Z,1 = u. 1 Fro the above discussion one can now write a ore or less developed expression for A > 0 A = Z,1 u F u 0 Z,k u 1,u,...,u k1 F k u 0 Z, 3 u 1,u,u 3,u 4 F 3 u 0 Z, u 1,u,u 3 F u 0 Z, 1 u 1,u F 1 u 0 Z, u 1 F u 0.

9 8 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 If one reads the forula fro the right to the left one sees that u 1 appears for the first tie in Z,, that u appears for the first tie in Z, 1, that u 3 appears for the first tie in Z,,..., u 1 appears for the first tie in Z, and u appears for the first tie in Z,1. One can still derive further properties of Z,k fro 16 and 15 or 1 and 11, however, for the analysis that will follow in the next section we will rely only on the properties discussed here. We can now apply the previous results to derive soe expressions for Z,k, being fixed, which will serve as saples for the ipleentation the Matheatica progra in the next section. 3 Applications and further properties According to or 14, Z, depends only on u 1 and to obtain the value of the power n 1 we solve 15 or 11: n 1 =. Hence Z, = u 1!. 3 Siilarly, Z, 1 depends on u 1 and u. Solving 15 which reduces to: n 1 n = 1 and n 1 n = leads to n = 1 and n 1 =. Hence Z, 1 = u 1 u!, 4 which is of the for 17. Z, depends on u 1,u,u 3. The powers n 1,n,n 3 satisfy the syste of equations n 1 n n 3 = and n 1 n 3n 3 =, 5 which leads upon subtracting the first equation fro the second one to n n 3 =. This last equation has two possible solutions in N: n = 0,n 3 = 1 or n =,n 3 = 0. We obtain the corresponding values of n 1 fro the first equation in 5: n 1 = 3 or n 1 = 4, respectively. Hence Z, = u 3 1 u 0 u1 3 3!0!1! u 4 1 u u0 3 4!!0! = u 3 1 u 3 3! u 4 1 u 4!!. 6

10 A developed new algorith 9 Siilarly, one obtains Z, 3 = u 4 1 u 4 4! u 5 1 u u 3 5! u 6 1 u 3 6!3! ; Z, 4 = u 5 1 u 5 5! u 6 1 u u 4 6! u 6 1 u 3 6!! u 7 1 u u 3 7!! u 8 1 u 4 8!4! ; Z, 5 = u 6 1 u 6 6! u 7 1 u 3 u 4 7! u 7 1 u u 5 7! u 8 1 u u 3 8!! u 8 1 u u 4 8!! u 9 1 u 3 u 3 9!3! u 10 1 u 5 10!5! ; 7 Z,1 = u, this has been derived in 1. We have also obtained the expression of Z, 6 which is the su of 11 ters. One can use the above forulas to evaluate the AP s A 1 to A 7. For instance, if = 7 one obtains Z 7, fro the expression of Z, 5 dropping all negative powers of u 1 since they are not in N. Hence, Z 7, = u 1 u 6 u 3 u 4 u u 5. Siilarly, one obtains the other Z 7,k 1 k 7 fro the above forulas. A final conclusion, which can be read fro 4, 6 and 7 or fro 14, is that each reduced polynoial Z,k is a su of soe nuber of ters, each ter is a product of k factors and each factor represents one of the coponents of the vector u 1,u,..., u k1. For instance, Z 7, depends on 7 1 = 6 functions: u 1,u,..., u 6. Z 7, is a su of soe ters 3 ters, each ter is a product of k = factors and each factor is one of the six functions u 1,u,..., u 6 such that the su of the powers is and the su of the subscripts is 7. This reark helps, in fact, iproving hand calculations of A. 4 Ipleentation of the Matheatica progra The Matheatica Reduce function helps solving diophantine equations. For instance, Eqs. 15 for = 4 and k = write as 3 n i = and 3 Matheatica solves 8 as follows: Reduce[ 3 n i == && in i = && {n 1,n,n 3 } Integers] in i == 4 && n 1 0 && n 0 && n 3 0 n 1 == 0 && n == && n 3 == 0 n 1 == 1 && n == 0 && n 3 == 1

11 10 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 There are two possible solutions and Z 4, = u /! u 1u 3 which is the coefficient of F u 0 in the fifth line of 7. In the above codes, the conditions n 1 0 && n 0 && n 3 0 and {n 1,n,n 3 } Integers depend on and are not convenient for large values of. We replace the by the equivalent expressions 3 n i Abs[n i ] == 3 n i and Table[n i,{i,3}] Integers, respectively. These new expressions are very iportant for the atheatica progra we will develop in subsection 4.3 where only and F reain as free paraeters. In the following two subsections, we provide two further applications of Reduce to solve 15 and 11 for = 4 and 1 k 4, respectively, and introduce and discuss in detail the codes needed for the Matheatica progra. 4.1 Solving Eqs. 15 for = 4 In the following Matheatica codes we solve 15 for = 4 and 1 k 4, however, the codes are valid for any value of : changing leads to new results. In the codes P represents the list of all possible power solutions. It is a list of four objects eleents where each object is ade of one or any sublists. For anipulating lists and sublists see [Don 000]. = 4; P = Table[0,{ j,}]; k = 0; While[k = k 1, P[[k]] = Reduce[ && P k1 Abs[n i ] == k1 k1 {n 4 == 1&&n 3 == 0&&n == 0&&n 1 == 0, n i == k && k1 in i == n i && Table[n i,{i, k 1}] Integers]] n 1 == 0&&n == &&n 3 == 0 n 1 == 1&&n == 0&&n 3 == 1, n 1 == &&n == 1, n 1 == 4} One sees P as a list of four objects, each object appears in one line. The objects P[[1]], P[[3]] and P[[4]] are each ade of one sublist, while the object P[[]] is ade of two sublists separated by. The object P[[]] corresponds to = 4 and k =, which we know has two solutions. A drawback of solving 15 is that the different sublists P[[1]], P[[, 1]], P[[, ]], P[[3]] and P[[4]] have different lengths. For instance, Length[P[[1]]] = 4, Length[P[[,1]]] = 3 and Length[P[[3]]] =. We will see in the next subsection that solving 11 will always lead to unifor sublists with the sae length.

12 A developed new algorith Solving Eqs. 11 for = 4 In the following Matheatica codes we solve 11 for = 4 and 1 k 4, however, the codes are valid for any value of : changing leads to new results. = 4; P = Table[0,{ j,}]; k = 0; While[k = k 1, P[[k]] = Reduce[ && P Abs[n i ] == n i == k && n i && Table[n i,{i,}] Integers]] {n 4 == 1&&n 3 == 0&&n == 0&&n 1 == 0, n 4 == 0&&n 3 == 0&&n == &&n 1 == 0 n 4 == 0&&n 3 == 1&&n == 0&&n 1 == 1, n 4 == 0&&n 3 == 0&&n == 1&&n 1 ==, n 4 == 0&&n 3 == 0&&n == 0&&n 1 == 4} in i == Notice that the sublists P[[1]], P[[, 1]], P[[, ]], P[[3]] and P[[4]] have the sae length 4 =. In order to be able to extract the values of Z 4,k 1 k 4 and evaluate A 4 one needs to generate the corresponding list, U, of the functions u i. This is done by the code line U = P /. {n u} which replaces n by u in the list of powers P. The list U has the sae properties as the list P. Now, if the object U[[k]] of U or the object P[[k]] of P has only one sublist these are the objects U[[1]], U[[3]] and U[[4]] in our exaple then Z,k = j=1u[[k, j,1]]p[[k, j,]] j=1 P[[k, j,]]!, 9 and if the object U[[k]] of U has any sublists this is the object U[[]] in our exaple, the nuber of sublists being Length[U[[k]]] leads to Length[U[[k]]] Z,k = 4.3 The Matheatica progra j=1u[[k,i, j,1]]p[[k,i, j,]] j=1 P[[k,i, j,]]!. 30 In order to distinguish between the case where U[[k]] has only one sublist and the case where it has any sublists we use the condition U[[k,1,1,1]] == 0 which

13 1 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 holds if U[[k]] has only one sublist with > 1. The Matheatica progra is readily written for the case = 7 which has been chosen for illustration: = 7;P = Table[0,{ j,}]; If[ > 1,k = 0; While[k = k 1, P[[k]] = Reduce[ n i == k && in i == && Abs[n i ] == n i && Table[n i,{i,}] Integers]]; U = P/.{n u}; A,1 = 0;A, = 0; Do[If[Length[U[[k, 1, 1, 1]]] == 0, P[[k, j,]] U[[k, j,1]] j=1 A,1 = A,1 P[[k, j,]]! A, = A, j=1 Length[P[[k]]] j=1 Derivative[k][F][u 0 ], U[[k,i, j,1]] P[[k,i, j,]] P[[k,i, j,]]! j=1 {k,1,}]; A = A,1 A, ] If[==1,A = u 1 Derivative[1][F][u 0 ]] If[==0,A = F[u 0 ]] Derivative[k][F][u 0 ]], A 7 = u 7 F u 0 u 3 u 4 u u 5 u 1 u 6 F u 0 u u 3 u 1u 3 u 1u u 4 u 1 u 5 u1 u 3 6 u 1 u u 3 u3 1 u 4 6 F 3 u 0 F 4 u 0 u 3 1 u 1 u4 1 u 3 F 5 u 0 4 u5 1 u 10 F6 u 0 u F7 u The constants that appear in the above expression are products of factorials: 5040 = 7!, 1 = 3!!, etc. In case of applications, the functions F and u 0 have to be defined earlier.

14 A developed new algorith 13 5 Application of the progra: the pendulu A solution for the proble of the pendulu can be written down explicitly using elliptic functions, however, the pendulu is used here for ere illustration of the progra since it is known to all workers. In the case where there are no friction and applied forces, the differential equation describing the otion of a pendulu is u tt b sinu = 0, 3 where ut is the angular displaceent and b is a positive geoetric" constant. The initial conditions are u0 = a and u t 0 = 0, Fu sinu and the operator L 1 takes the for t q L 1 [ f t] = [ f p]dp dq Applying L 1 to both sides of 3 and using the prescribed initial conditions, one obtains ut = a L 1 [bsinu]. 34 The proble of the pendulu is solved upon choosing u 0 = a and u 1 = L 1 [ba ], 0, 35 where A are evaluated using the Matheatica progra developed in subsection 4.3. We have used ten polynoials A : fro A 0 to A 9 to evaluate eleven coponents of ut: fro u 0 t to u 10 t. The results of calculations are shown below as produced by Matheatica u = Collect[ 10 u i,t] a bt Sin[a] t 6 b 3 Sin[a] 360 i=0 b t 4 Cos[a]Sin[a] 4 b3 Sin[3a] 70 t 8 b4 Sin[a] 5040 t 10 b5 Sin[a] b5 Sin[3a] 31b5 Sin[5a] b t 1 6 Sin[a] b6 Sin[4a] b6 Sin[6a] b4 Sin[4a] b t 14 7 Sin[a] b7 Sin[3a] b7 Sin[5a] b7 Sin[7a]

15 14 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 t b8 Sin[a] b8 Sin[4a] b8 Sin[6a] b8 Sin[8a] t 18 47b9 Sin[a] b9 Sin[3a] b 9 Sin[5a] b9 Sin[7a] b9 Sin[9a] b t 0 10 Sin[a] b10 Sin[4a] b10 Sin[6a] b 10 Sin[8a] b10 Sin[10a] The special case with a = π/ reduces to ut = π bt b3 t 6 40 b5 t b7 t b9 t One ay wonder whether the expression of ut for arbitrary a reduces to the linear case for sall values of a. A series expansion of ut,a about the point a = 0 and for an arbitrary value of t reads as produced by Matheatica Series[u,{a, 0, 3}] 1 bt b t 4 4 b3 t 6 70 b4 t b5 t b6 t b 7 t b 8 t b 9 t b 10 t 0 a bt 1 b t b3 t b4 t b5 t b6 t b7 t b 8 t b9 t b 10 t 0 a 3 O[a] It is straightforward to check that the coefficient of a in the above expression is cos bt. Hence, to the first order of approxiation the solution is ut = a cos bt which is the exact solution to the linear proble: u tt bu = 0 with the sae initial conditions. 6 Conclusion For a nonlinear operator N depending on one variable, it was shown that the reduced polynoial Z,k depends explicitly on k 1 functions u 1,u,..., u k1 if k > 1 and that Z,1 depends only on u. The AP A splits into a su of ters

16 A developed new algorith 15 where each ter is the product of Z,k, which is independent of N, with a derivative factor, F k u 0, depending on N. General expressions for Z,k have been derived for sall values of k, hand calculations have been discussed briefly and a siple and copact Matheatica progra has been ipleented and applied. For the purpose of hand calculations and analytical studies, we coplete the lists 7 and 31 providing general expressions for A, which is not available in the literature, by adding the polynoials A 5, A 6, A 8 to A 10 using our progra. A 5 u 5 F [u 0 ] u u 3 u 1 u 4 F u1 u [u 0 ] u 1 u 3 u F5 [u 0 ]; A 6 u u 6 F [u 0 ] 3 u u 4 u 1 u 5 u 1 u 4 u3 1 u 3 6 A 8 u 4 4 u 1u u 3 u 1 u 3 F 3 [u 0 ] u3 1 u 6 F4 [u 0 ] u F 3 [u 0 ] 6 u 1u u 3 u 1 u 4 F 4 [u 0 ] u4 1 u 4 F5 [u 0 ] u F6 [u 0 ]; u u 8 F [u 0 ] 4 u 3u 5 u u 6 u 1 u 7 F [u 0 ] u u 3 u u 4 u 1u 3 u 4 u 1 u u 5 u 1 u 6 F 3 [u 0 ] F 4 [u 0 ] u 1 u 3 1 u3 1 u u 3 6 u 4 1 u 48 u5 1 u u 1 u u 4 F 5 [u 0 ] u4 1 u 4 4 u3 1 u 5 6 F 6 [u 0 ] u6 1 u 70 F7 [u 0 ] u F8 [u 0 ]; F 3 [u 0 ] A 9 u 9 F [u 0 ] u 4 u 5 u 3 u 6 u u 7 u 1 u 8 F [u 0 ] u u u 3 u 4 u 1u 4 u u 5 u 1u 3 u 5 u 1 u u 6 u 1 u 7 F 3 [u 0 ] u 3 u 3 6 u 1u u 3 u 1u u 4 u 1 u 3u 4 u 1 u u 5 u3 1 u 6 F 4 [u 0 ] 6

17 16 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 u1 u 4 4 u 1 u u 3 4 u 3 1 u 3 36 u4 1 u u 3 4 u 5 1 u 40 u6 1 u 3 70 A 10 u3 1 u 3 1 u3 1 u u 4 6 F 6 [u 0 ] u5 1 u 4 10 u4 1 u 5 F 5 [u 0 ] 4 F 7 [u 0 ] u7 1 u 5040 F8 [u 0 ] u F9 [u 0 ]; u u 10 F [u 0 ] 5 u 4u 6 u 3 u 7 u u 8 u 1 u 9 F [u 0 ] u 3 u 4 u u 4 u u 3 u 5 u 1 u 4 u 5 u u 6 u 1u 3 u 6 u 1 u u 7 u 1 u 8 u u 3 4 u 1u u3 u 4 u 1u u 5 u 1 u 3u 5 6 u 1u u 3 u 4 u 1 u 4 4 u 1 u u 6 u3 1 u 7 6 u 5 10 u 1u 3 u 3 u 1 u u 3 u 1 u u u 1 u 4 48 u3 1 u u 3 1 u4 1 u 3 u 4 1 u u5 1 u u 3 10 u6 1 u 4 70 F 4 [u 0 ] u3 1 u 3u u4 1 u u 4 u5 1 u 5 F 6 [u 0 ] 4 10 u F 7 6 [u 0 ] 1 u 1440 u7 1 u u8 1 u 4030 F9 [u 0 ] u F10 [u 0 ]. References u3 1 u u 5 6 F 3 [u 0 ] u4 1 u 6 F 5 [u 0 ] 4 F 8 [u 0 ] Adoian, G. 1994: Solving Frontier Probles of Physics: The Decoposition Method Kluwer, Boston, MA. Adoian, G.; Rach, R. 199: Modified decoposition solution of nonlinear partial differential equations. Coput. Math. Applic., vol. 3, pp Cochran, J. A. 1968: On the uniqueness of solutions of linear differential equations. J. Math. Anal. Appl., vol., pp Davies, A. J.; Crann, D.; Kane, S. J.; Lai, C. -H. 007: A Hybrid Laplace Transfor/Finite Difference Boundary Eleent Method for Diffusion Probles. CMES: Coputer Modeling in Engineering & Sciences, vol. 18, no., pp Don, E. 000: Theory and Probles of Matheatica McGraw-Hill, Blacklick, OH.

18 A developed new algorith 17 El-Tawil, M. A.; Bahnasawi, A. A.; Abdel-Naby, A 004: Solving Reccati differential equation usingg Adoian s decoposition ethod. Appl. Math. Coput., vol. 157, pp Hosseini, M. M. 006: Adoian decoposition ethod for solution of nonlinear differential algebraic equations. Appl. Math. Coput., vol. 181, pp Han, Z. D.; Atluri, S. N. 007: A Systeatic Approach for the Developent of Weakly Singular BIEs. CMES: Coputer Modeling in Engineering & Sciences, vol. 1, no. 1, pp Huang, P.; Zhang, X.; Ma, S.; Wang, H. K. 008: Shared Meory OpenMP Parallelization of Explicit MPM and Its Application to Hypervelocity Ipact. CMES: Coputer Modeling in Engineering & Sciences, vol. 38, no., pp Kao, Y.-M.; Jiang, T. F.; Yu, I. A. 005: Adoian s decoposition ethod for electroagnetically induced transparency. Phys. Rev. E, vol. 7, pp Lai, H. -Y.; Chen, C. K.; Hsu, J. -C. 008: Free Vibration of Non-Unifor Euler-Bernoulli Beas by the Adoian Modified Decoposition Method. CMES: Coputer Modeling in Engineering & Sciences, vol. 34, no. 1, pp Patrício, M.; Mattheij, R.; de With, G. 008: Solutions for periodically distributed aterials with localised iperfections. CMES: Coputer Modeling in Engineering & Sciences, vol. 38, no., pp Rach, R. 1984: A convenient coputational for for Adoian polynoials. J. Math. Anal. Appl., vol. 10, pp Seng, V.; Abbaoui, K.; Cherruault, Y. 1996: Adoian s polynoials for nonlinear operators. Math. Coput. Modelling, vol. 4, pp Takei, A.; Yoshiura, S.; Kanayaa, H. 008: Large-Scale Parallel Finite Eleent Analyses of High Frequency Electroagnetic Field in Couter Trains. CMES: Coputer Modeling in Engineering & Sciences, vol. 31, no. 1, pp Vodička, R.; Mantič, V.; París, F. 007: Syetric Variational Forulation of BIE for Doain Decoposition Probles in Elasticity An SGBEM Approach for Nonconforing Discretizations of Curved Interfaces. CMES: Coputer Modeling in Engineering & Sciences, vol. 17, no. 3, pp Wang, F.; Bajaj, A. K. 008: Adoian decoposition ethod applied to nonlinear noral odes of an inertially coupled conservative syste. J. Vib. Control, vol. 14, pp

19 18 Copyright 009 Tech Science Press CMES, vol.4, no.1, pp.1-18, 009 Wazwaz, A. M. 000: A new algorith for calculating Adoian polynoials for nonlinear operators. Appl. Math. Coput., vol. 111, pp Wazwaz, A. M.; El-Sayed, S. M. 001: A new odification of the Adoian decoposition ethod for linear and nonlinear operators. Appl. Math. Coput., vol. 1, pp Zhang, X. 005: A odification of the Adoian decoposition ethod for a class of nonlinear singular boundary value probles. J. Coput. Appl. Math., vol. 180, pp

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