The use of Adomian decomposition method for

solving a specific nonlinear partial differential

equations

Doǧan Kaya

Abstract
In this paper, by considering the Adomian decomposition method, explicit
solutions are calculated for partial differential equations with initial condi-
tions. The method does not need linearization, weak nonlinearly assumptions
or perturbation theory. The decomposition series analytic solution of the
problem is quickly obtained by observing the existence of the self-cancelling
“noise” terms where sum of components vanishes in the limit.

1 Introduction
The theory of nonlinear problem has recently undergone much study. We do not
attempt to characterize the general form of nonlinear equations [1]. Rather, we solve
a specific equation in the following nonlinear problem by using the Adomian decom-
position method [2-4]. By solving this type of problems, we do not use conventional
transformations which transform a nonlinear problem to an evolution equation and
the reduced to a bilinear form. Some times transformation of the nonlinear problem
might produce an even more complicated problem. Nonlinear phenomena play a
crucial role in applied mathematics and physics. The nonlinear problems are solved
easily and elegantly without linearizing the problem by using the Adomian’s decom-
position method.
Received by the editors A. Bultheel.
Communicated by March 2000.
1991 Mathematics Subject Classification : G.1.8.
Key words and phrases : The Adomian decomposition method, a partial differential equation,
the self-cancelling noise terms.

Bull. Belg. Math. Soc. 9 (2002), 343–349

344 D. Kaya

We now describe how the decomposition method can be used to construct the
solution for the partial differential equation [1], with initial condition

ut = −eu ux + k(ux )2 + kuxx + e−u R(x, t); u(x, 0) = f (x), (1)

where R(x, t) is a given function.

In this paper, we generated an appropriate Adomian’s polynomials for nonlin-
ear terms of equation (1), that will be handled more easily, quickly, and elegantly
by implementing the Adomian’s decomposition method rather than the traditional
methods.

2 The method of solution

Let us consider the nonlinear problem (1) in an operator form

Lt (u) = −(Ku) + k(M u) + kLxx (u) + (N u)R(x, t) (2)

where the notations Ku = eu ux , M u = (ux )2 , and N u = e−u symbolize the non-

∂ ∂2
linear terms. The notation Lt = ∂t and Lxx = ∂x 2 symbolize the linear differential

operators. Assuming that the inverse operator L−1 t exists and it can conveniently
Rt
be
taken as the definite integral with respect to t from 0 to t, i.e., Lt = 0 (.)dt. Thus,
−1

applying the inverse operator L−1t to (2) yields

t Lt (u) = −Lt (Ku) + kLt (M u) + kLt Lxx (u) + Lt ((N u)R(x, t)).
L−1 (3)
−1 −1 −1 −1

Therefore, it follows that

u(x, t) − u(x, 0) = −L−1

t (Ku) + kLt (M u) + kLt Lxx (u) + Lt ((N u)R(x, t)). (4)
−1 −1 −1

We next decompose the unknown function u(x, t) by a sum of components defined

by the decomposition series

X
∞
u(x, t) = un (x, t) (5)
n=0

with u0 identified as u(x, 0). An important part of the method is to express the
P P
Adomian’s polynomials; thus Ku = eu ux = ∞ An , M u = (ux )2 = ∞n=0 Bn , and
P∞ n=0
N u = e = n=0 Cn where the An , Bn , and Cn are the appropriate Adomian’s
−u

polynomials generated as follows:

Adomian decomposition method for solving partial differential equations 345

A0 = (u0 )x eu0
A1 = (u0 )x u1 eu0 + (u1 )x eu0
u2
A2 = (u0 )x ( 1 + u2 )eu0 + u1 (u1 )x eu0 + (u2 )x eu0 (6)
2
u3 u2
A3 = (u0 )x ( 1 + u1 u2 + u3 )eu0 + (u1 )x ( 1 + u2 )eu0 + (u2 )x u1 eu0 + (u3 )x eu0
6 2
..
.

An polynomials can be derived from the general form of Adomian polynomial given
in [2] and an alternative approach can be find in [5],

B0 = (u0x )2
B1 = 2(u0 )x (u1 )x
B2 = (u1x )2 + 2(u0 )x )(u2 )x (7)
B3 = 2(u0 )x (u3 )x + 2(u1 )x (u2 )x
..
.

and

C0 = e−u0
C1 = −u1 e−u0
u2
C2 = ( 1 − u2 )e−u0 (8)
2
u3
C3 = (− 1 + u1 u2 − u3 )e−u0
6
..
.

The remaining components un (x, t), n ≥ 1, can be completely determined such

that each term is computed by using the previous term. Since u0 is known, then

u1 = −L−1
t (A0 ) + kLt (B0 ) + kLt Lxx (u0 ) + Lt (C0 R(x, t))
−1 −1 −1

u2 = −L−1
t (A1 ) + kLt (B1 ) + kLt Lxx (u1 ) + Lt (C1 R(x, t))
−1 −1 −1
(9)
..
.
un = −L−1
t (An−1 ) + kLt (Bn−1 ) + kLt Lxx (un−1 ) + Lt (Cn−1 R(x, t)).
−1 −1 −1

Recently, Wazwaz [6] proposed that the construction of the zeroth component of
the decomposition series can be define in a slightly different way. In [6], the author
assumed that if the zeroth component is u0 = g, the function g is possible to divide
into two parts such as g1 and g2 , then one can formulate the recursive algorithm u0
and (9) general term in a form of the modified recursive scheme as follows:

u0 = g 1 , (10)
u1 = g2 − Lt (A0 ) + kLt (B0 ) + kLt Lxx (u0 ) + Lt (C0 R(x, t)),
−1 −1 −1 −1
(11)
un+1 = −Lt (An ) + kLt (Bn ) + kLt Lxx (un ) + Lt (Cn R(x, t)), n ≥ 1.(12)
−1 −1 −1 −1
346 D. Kaya

This type of modification is giving more flexibility to the modified decomposition

method in order to solve complicate nonlinear differential equations. In many case
the modified scheme avoids the unnecessary computations, especially in calculation
of the Adomian polynomials. Furthermore, sometimes we do not need to evalu-
ate the so-called Adomian polynomials or if we need to evaluate these polynomials
the computation will be reduced very considerably by using the modified recur-
sive scheme. For more details of the modified decomposition method, one can see
references [6]. Illustration purpose we will consider both homogeneous and inhomo-
geneous nonlinear equations in the following section.
For numerical comparisons purposes, based on the decomposition method, one
can constructed the solution u(x, t) as
lim φn = u(x, t) (13)
n→∞

where
X
n
φn (x, t) = uk (x, t), n≥0 (14)
k=0

and the recurrence relation is given as in (9) or (10)-(12). Furthermore, the de-
composition series (5) solutions are generally converge very rapidly in real physical
problems [2]. The convergence of the decomposition series have investigated by sev-
eral authors. The theoretical treatment of convergence of the decomposition method
has been considered by Cherruault [7] and Rèpaci [8]. In [7], Cherruault proposed a
new definition of the technique and then he insisted that it will become possible to
prove the convergence of the decomposition method. Rèpaci [8] shown a convergence
of this method based upon a suitable connection with fixed point techniques. This
is essentially the same conclusion derived by Cherruault [7]. These results have been
improved by Cherruault and Adomian [9], who proposed a new convergence proof of
Adomian’s technique based on properties of convergent series. They obtained some
results about the speed of convergence of this method providing us to solve linear
and nonlinear functional equations.
Adomian and Rach [10] and Wazwaz [11] have investigate the phenomena of
the self-cancelling “noise” terms where sum of noise terms vanishes in the limit. An
important observation was made that “noise” terms appear for inhomogeneous cases
only. Further, it was formally justified that if terms in u0 are cancelled by terms in
u1 , even though u1 includes further terms, then the remaining non cancelled terms
in u1 are cancelled by terms in u2 , and so on. Finally, the exact solution of the
equation is readily found for the inhomogeneous case by determining the first two
or three terms of the solution u(x, t) and by keeping only the non cancelled terms
of u0 .
The solution u(x, t) must satisfy the requirements imposed by the initial condi-
tions. The decomposition method provides a reliable technique that requires less
work if compared with the traditional techniques.
To give a clear overview of the methodology, the following homogeneous nonlinear
example will be consider and the solution of which is to be obtained subject to the
initial condition
ut = −eu ux + (ux )2 + uxx + e−u (1 + x + t), u(x, 0) = ln(x). (15)
Adomian decomposition method for solving partial differential equations 347

To solve this equation, we simply take the equation in an operator form exactly in
the same manner as the form of equation (4). The zeroth component is u0 = ln(x).
The components u1 , u2 , u3 are obtained in succession by using (9). Hence, we find

u1 = −L−1t (A0 ) + Lt (B0 ) + Lt Lxx (u0 ) + Lt (C0 (1 + x + t))

−1 −1 −1

t t2
= + , (16)
x 2x
u2 = −L−1t (A1 ) + Lt (B1 ) + Lt Lxx (u1 ) + Lt (C1 (1 + x + t))
−1 −1 −1

t2 t2 t3 t3 t4
= − − 2− − − , (17)
2x 2x 6x 2x2 8x2
u3 = −L−1t (A2 ) + Lt (B2 ) + Lt Lxx (u2 ) + Lt (C2 (1 + x + t))
−1 −1 −1

t3 t3 t3 t4 t4
= + 2+ + + 3 + ..., (18)
6x 2x 6x 6x 4x
and so on, in this manner the rest of components of the decomposition series were
obtained using Matematica. Substituting (16)-(18) and the other calculated terms
into (5) gives the solution u(x, t) in a series form and in a close form solution by
t t2 t3 t4 t5 t6
u(x, t) = ln(x) + − + − + − +... (19)
x 2 x2 3 x3 4 x4 5 x5 6 x6
or u(x, t) = ln(x + t) which can be verified through substitution.
As an example of an application of the self cancelling phenomena [6,10,11], let us
seek the explicit solution of an inhomogeneous nonlinear equation (1), with initial
condition

ut = F (x, t) − eu ux + k(ux )2 + kuxx + e−u R(x, t); u(x, 0) = 1, (20)

2 2
where F (x, t) = −2t + x2 (1 − 4t2 ) − e−(1+tx ) + 2txe1+tx is a right hand side given
function, R(x, t) = 1 is a given function, and k = 1 is a constant.
To obtain the decomposition solution subject to initial condition given, we first
used (20) in an operator form in the same manner as form (4) and then we used
(10)-(12) to determine the individual terms of the decomposition series, we get
immediately

u0 = 1 + tx2 , (21)
3 2 2
4t x 1 2 (x − 2e ) 2
u1 = −t2 − + 3 {xe−(1+tx ) − + 2e(−1 + tx2 )etx } −
3 x e
−Lt (A0 ) + Lt (B0 ) + Lt Lxx (u0 ) + L−1
−1 −1 −1
t (C0 )
2
1 2e (−2e + x)
= 2
− 3 − , (22)
ex x ex3
un+1 = 0, n ≥ 1. (23)

It is obvious that the “noise” terms appear between the components of u1 , and these
are all cancelled. As seen equation (22), the closed form of the solution can be find
very easily by proper selection of g1 and g2 . In the case of right choice of these
functions, the modified technique accelerate the convergence of the decomposition
series solution by computing just u0 and u1 terms of the series. The term u0 provides
the exact solution as u(x, t) = 1 + tx2 and this can be justifies through substitution.
348 D. Kaya

It may be concluded that, the Adomian methodology is very powerful and ef-
ficient in finding exact solutions for wide classes of problems. With regard to this
application, the decomposition method outlined in the above analysis shows many
of the equations of physics appear to be solvable analytically without linearization,
perturbation or discretization. It is also worth noting that the advantage of the
decomposition methodology is that it displays a fast convergence of the solution.
It may be achieved by observing the self-cancelling “noise” terms. In addition, the
numerical results obtained by this method have illustrated a high degree of accuracy
as discussed in [12,13].
Acknowledgements- This work was supported by Grand No. 346 of the Firat
University of Research Fund (FÜNAF), Turkey. The author thanks the referees for
valuable comments.

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Department of Mathematics,
Firat University, Elazig, 23119, TURKEY.
E-mail: dkaya@firat.edu.tr.