A note on the convergence of the Adomian decomposition method
aa r X i v : . [ m a t h . G M ] J un A NOTE ON THE CONVERGENCE OF THE ADOMIANDECOMPOSITION METHOD
HICHAM ZOUBEIRThis modest work is dedicated to the memory of our beloved master Ahmed Intissar(1951-2017), a distinguished professor, a brilliant mathematician, a man with a golden heart.
Abstract.
In this note we obtain a new convergence result for the Adomiandecomposition method. Introduction
The Adomian decomposition method (ADM) was developped in the 1980’s bythe American physicist G. Adomian (1923 − u of a vector nonlinear equation u = f + N ( u ) where N is an analyticoperator and f a given vector, into a series u = + ∞ P n =0 u n such that the term u n +1 is determined from the terms u , u , ..., u n by a reccurence relation involving apolynomial A n generated by the Taylor expansion of the operator N. Many paperson the applications of the ADM to the problems arising from different areas of pureand applied sciences have been published ([5]-[7], [9], [15], [16], [23]). Many workshave been also devoted to the convergence of the ADM ([8], [10]-[14]). However letus pointwise that some recent works on the application of the ADM to nonlinearproblems avoid the theoretical treatment of the question of the convergence of themethod. On the other hand we observe that the current convergence criteria of theADM give rise to small regions of convergence (that is to small set of vectors f forwhich the vector series P u n is convergent according to the criteria of convergence)which limits the application of this method. Our purpose in this note is to obtain anew convergence result for the ADM, which we believe will contribute to strenghtingthe method by enlarging the field of its applications.2. Preliminary notes
Main definitions.
Along this paper ( X, k . k ) is a given real or complex Ba-nach space.The following definitions were introduced in ([22]). Definition 2.1. L n ( X ) denotes for each n ∈ N ∗ a set of continuous symetric n -multilinear mappings from X n to X . It is well known that L n ( X ) is a Banach Mathematics Subject Classification..
Key words and phrases.
Adomian decomposition method, Analytic operators. space for the norm || . || [ n ] defined by the relation : || f || [ n ] := sup ( x ,...,x n ) ∈ ( X −{ } ) n (cid:18) || f ( x , . . . , x n ) |||| x || . . . || x n || (cid:19) Let x ∈ X and F ∈ L n ( X ). We will denote F ( x, . . . , x ) by F.x [ n ] . Given x , ..., x p ∈ X, n , ..., n p ∈ N such that n + ... + n p = n, we will write F. (cid:16) x [ n ]1 · · · x [ n p ] p (cid:17) for F x , ..., x | {z } n times , . . . , x p , ..., x p | {z } n p times . For convenience we set L ( X ) := X and || . || [0] := || . || . Definition 2.2.
A power series in x with values in X ([22]) is a series of the form P ∞ n =0 F n .x [ n ] , where F ∈ X and F n ∈ L n ( X ) for n ≥ Definition 2.3.
A mapping F : X → X is called an analytic operator on X ([22])if there exists for each vector a ∈ X a sequence A n ( a ) ∈ L n ( X ) ( n ≥
1) and a openneighborhood U a such that the real series P n ! k A n ( a ) k [ n ] k x − a k n is convergentwhen x ∈ U a and the power series P n ! A n ( a )( x − a ) [ n ] is convergent to F ( x ) − F ( a )for all x ∈ U a . Definition 2.4.
The radius of the power series P n ! A n ( a )( x − a ) [ n ] is the radiusof convergence of the complex variable power series P n ! k A n ( a ) k [ n ] t n , that is thenumber : R ( F, a ) := 1lim sup n → + ∞ (cid:16) n ! k A n ( a ) k [ n ] (cid:17) n with the conventions that = + ∞ . Definition 2.5.
Let F : X → X be an analytic operator on the Banach space X and a ∈ X. We say that F has an infinite radius of convergence at the point a if R ( F, a ) = + ∞ . Remark 2.6.
If a mapping F : X → X is an analytic operator on X then theoperator F is of class C ∞ on X and the following relations hold for every a ∈ X and n ∈ N : F ( n ) ( a ) = 1 n ! A n ( a ) Remark 2.7.
If an analytic operator F has an infinite radius of convergence atsome point a ∈ X on the Banach space X , then thanks to ([22]) F will have thesame property at every point of X . Thence we will say, without loss of precision,that F has an infinite radius of convergence.3. Abstract presentation of the ADM
Let f ∈ X and N : X → X an analytic operator on X with infinite radius ofconvergence. We consider the vector equation :(3.1) u = N ( u ) + f The ADM for solving the vector equation (3.1) consists in writing the unknownvector u in the form of an absolutely convergent vector series u = P + ∞ n =0 u n and NOTE ON THE CONVERGENCE OF THE ADOMIAN DECOMPOSITION METHOD 3 in splitting the nonlinear term N ( u ) into an absolutely convergent vector series N ( u ) = P + ∞ n =0 A n where the term A n is obtained for all n ∈ N by the formula : A n := 1 n ! d n dε n N + ∞ X j =0 ε j u j ε =0 Thence the equation (3.1) becomes : + ∞ X n =0 u n = f + + ∞ X n =0 A n Then we set by formal identification : (cid:26) u = fu n +1 = A n and the following relation holds for every n ∈ N :(3.2) A n := 1 n ! d n dε n N n X j =0 ε j u j ε =0 We denote by A dm ( N ; f ) the vector series P u n if it is well-defined. In the sequelwe will continue to denote by u n the general term of the vector series A dm ( N ; f ) . Theorem 1. ([ ? ]) Let f ∈ X, N : X → X be an analytic mapping on X which has an infinite radiusof convergence. Then the following formula holds for every n ∈ N :(3.3) A n = X j +2 j + ... + nj n = n j ! ...j n ! N ( j + ... + j n ) ( f ) . (cid:16) u [ j ]1 · · · u [ j n ] n (cid:17) Proof.
Since N is analytic, we can write for all n ∈ N and ε > N n X j =0 ε j u j = N ( u ) + + ∞ X p =0 p ! N ( p ) ( u ) . n X j =0 ε j u j [ p ] = N ( f ) ++ + ∞ X p =0 p ! X j + j + ... + j n = p p ! j ! ...j n ! ε j +2 j + ... + nj n N ( p ) ( f ) . (cid:16) u [ j ]1 · · · u [ j n ] n (cid:17) = N ( f ) ++ + ∞ X q =0 ε q X j +2 j + ... + nj n = q j ! ...j n ! N ( j + j + ... + j n ) ( u ) . (cid:16) u [ j ]1 · · · u [ j n ] n (cid:17) It follows then from the relation (3.2) that the following relation holds for each n ∈ N : A n = X j +2 j + ... + nj n = n j ! ...j n ! N ( j + j + ... + j n ) ( f ) . (cid:16) u [ j ]1 · · · u [ j n ] n (cid:17) (cid:3) HICHAM ZOUBEIR Statement of the main result
Our main result in this paper is the following.
Theorem 2.
Let f ∈ X, N : X → X be an analytic mapping on X on some open neighborhoodof the vector f and such that the following estimates hold :(4.1) || N ( n ) ( f ) || [ n ] ≤ M a n n ! , n ∈ N where the constants M, a > satisfy the condition :(4.2) M a ≤ . Then the vector series A dm ( N ; f ) is absolutely convergent in the Banach space X to a vector u which is a solution of the equation (3.1). . If M a = , then the vector u fullfiles the following estimates : k u k ≤ k f k + (cid:16) √ π (cid:17) M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − n P j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M √ π √ n , n ∈ N ∗ . If M a < then the vector u fullfiles the following estimates : k u k ≤ k f k + (cid:16) M a √ π (1 − Ma ) (cid:17) M (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − n P j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M a √ π (1 − Ma ) (4 Ma ) n ( n +1) √ n +1 , n ∈ N ∗ Proof of the main result
Relying on the formula (3.3) we can easily prove, under the assumption (4.1) ofthe main result, that :(5.1) || u n || ≤ w n M n a n − , n ∈ N ∗ where ( w n ) n ≥ is the sequence of positive coefficient defined by the relations :(5.2) ( w = 1 w n +1 = P j +2 j + ... + nj n = n ( j + ... + j n )! j ! ...j n ! ( w ) j ... ( w n ) j n , n ∈ N ∗ It follows that : X n ≥ w n +1 T n +1 = T X n ≥ X j +2 j + ... + nj n = n ( j + ... + j n )! j ! ...j n ! ( w T ) j ... ( w n T n ) j n = T X p ≥ X n ≥ w n T n p Let us denote by S the formal series P n ≥ w n T n ∈ C [[ T ]] . Then we can write : S − T = T (cid:18) S − S (cid:19) NOTE ON THE CONVERGENCE OF THE ADOMIAN DECOMPOSITION METHOD 5
It follows that : S = T (cid:18) − S (cid:19) It follows from the well known Lagrange inversion formula for formal series ([17])that the following equalities hold for all n ≥ T n ] S = 1 n ! (cid:18) d n − dt n − (cid:20)(cid:18) − t (cid:19) n (cid:21)(cid:19) t =0 = 1 n ! X j + ... + j n = n − ( n − j ! ...j n ! ·· (cid:18) d j dt j (cid:20)(cid:18) − t (cid:19)(cid:21)(cid:19) t =0 ... (cid:18) d j n dt j n (cid:20)(cid:18) − t (cid:19)(cid:21)(cid:19) t =0 = 1 n ! X j + ... + j n = n − ( n − j ! ...j n ! j ! ...j n ! = 1 n X j + ... + j n = n −
1= (2 n − n !) Thence we have for all n ≥ w n = (2 n − n !) The inequalities (5.1) become : || u n || ≤ (2 n − n !) M n a n − , n ∈ N ∗ But we have by virtue of the well known Stirling formula :(2 n − n !) M n a n − ∼ n → + ∞ √ πa n √ n (4 M a ) n It follows then from the assumption (4.2) that the vector series A dm ( N ; f ) is abso-lutely convergent in the Banach space X. Let us then set for all ε ∈ [0 ,
1] : U ( ε ) := + ∞ X n =0 ε n u n , u := + ∞ X n =0 u n = U (1)Since N is analytic on E, it follows that we have for all ε ∈ [0 ,
1] : N ( U ( ε )) = + ∞ X n =0 ε n A n If we choose ε = 1 , we will then obtain the relations : N ( u ) = + ∞ X n =0 A n = + ∞ X n =0 u n +1 = u − f HICHAM ZOUBEIR
It follows that u is a solution of the equation (3.1). On the other hand we have foreach n ∈ N ∗ : (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − n X j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ + ∞ X j = n +1 k u j k ≤ + ∞ X j = n +1 (2 j − j !) M j a j − ≤ M + ∞ X j = n (2 j )! (2 j + 1)( j !) ( j + 1) ( M a ) j But, according to ([ ? ]) the double inequality holds for all integers j ≥ √ πj j + e − j ≤ j ! ≤ s jj − √ πj j + e − j It follows then, by easy computations, that we have for each j ∈ N ∗ \ { } :(2 j )! (2 j + 1)( j !) ( j + 1) ρ j ≤ √ πj √ j (4 M a ) j • First case : 4
M a = 1In this case the following estimate holds for all n ∈ N ∗ : (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − n X j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M √ π + ∞ X j = n +1 j √ j ≤ M √ π + ∞ Z n s √ s ds ≤ M √ π √ n It follows that : k u k ≤ k u k + k u k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − X j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k f k + M + 8 M √ π ≤ k f k + (cid:18) √ π (cid:19) M • Second case : 4
M a < n ∈ N ∗ : (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − n X j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M √ π + ∞ X j = n +1 j √ j (4 M a ) j ≤ M a √ π (1 − M a ) (4
M a ) n ( n + 1) √ n + 1 NOTE ON THE CONVERGENCE OF THE ADOMIAN DECOMPOSITION METHOD 7
It follows that : k u k ≤ k u k + k u k + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) u − X j =0 u j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ k f k + M + 64 M a √ π (1 − M a ) ≤ k f k + (cid:18) M a √ π (1 − M a ) (cid:19) M Thence the proof of the main result is complete. (cid:3)
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E-mail address : [email protected] Ibn Tofail University, Department of Mathematics,, Faculty of Sciences, P. O. B : ,,