Download Adomian method for solving some coupled systems of two equations

Applied Mathematics and Computation 177 (2006) 553–560
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Adomian method for solving some coupled systems
of two equations
Lazhar Bougoffa
a
a,*
, Smail Bougouffa
b
Faculty of Computer Science and Information, Al-Imam Muhammad Ibn Saud Islamic University, P.O. Box 84880,
Riyadh 11681, Saudi Arabia
b
Department of Physics, Faculty of Science, Taibah University, P.O. Box 344, Madina Mounauara, Saudi Arabia
Abstract
Coupled systems of two linear and nonlinear differential equations for second- and first-orders, respectively, can be
solved with some techniques and Adomian decomposition method. A few simple examples are also studied to show with
analytical results how the (ADM) works efficiently.
Ó 2005 Elsevier Inc. All rights reserved.
Keywords: Adomian’s decomposition method; Coupled system
1. Introduction
Coupled system of two equations for second-order are needed in the formulation of various physical situations [3–9,11]. For comments on their importance, we refer the reader to the above papers. As an example of
such systems is the coupled system of two Schrödinger linear equations which has been given extensive attention in recent years both analytically and numerically [3–8]. The consideration of this system is motivated by a
number of physical problem in various fields. Also, systems of nonlinear differential equations for first-order
are encountered when studying mathematical models for certain natural, physical and biological processes. As
an example of such systems, is the system was proposed for the model of Lotka–Volterra [10]. This mathematical model is based on nonlinear system of two ordinary differential equations.
The plan of the present paper is as follows: In Section 1 we formulate a coupled system of linear equations
for second-order which contains as a special case the system of Schrödinger equations and we prove a result
which governs the separation of a system of two coupled equations. The Adomian decomposition method is
investigated in Section 2. It is shown that (ADM) can be an effective scheme to obtain the analytical and
approximate solutions. In Section 3 we propose a new transformation for solving a system of nonlinear differential equations for first-order and the coupled system can be solved by (ADM) and back-substitution.
*
Corresponding author.
E-mail addresses: bougoff[email protected] (L. Bougoffa), sbougouff[email protected] (S. Bougouffa).
0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2005.07.070
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L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
2. Coupled system of two equations
Consider the following coupled system of two ordinary differential equations for second-order [3–8]:
d2 u
du
þ pðxÞ f1 ðxÞu ¼ b1 ðxÞv þ F 1 ðxÞ;
ð2:1Þ
dx2
dx
d2 v
dv
þ pðxÞ f2 ðxÞv ¼ b2 ðxÞu þ F 2 ðxÞ;
ð2:2Þ
2
dx
dx
du
ð0Þ ¼ b1 ;
ð2:3Þ
uð0Þ ¼ a1 ;
dx
dv
ð0Þ ¼ b2 ;
ð2:4Þ
vð0Þ ¼ a2 ;
dx
where p(x), bi(x), fi and Fi(x), (i = 1, 2) are assumed to be analytic functions.
As announced in the Introduction, we first begin with the following result on the separation of this system
in which the two equations are decoupled.
Lemma 1. The system (2.1)–(2.4) can always be decoupled without increase of the differential equations if and
only if the functions b1(x), b2(x) are proportional to the difference f1(x) f2(x).
Proof. Multiplying Eq. (2.1) by r and adding Eq. (2.2) we get
d2 ðu þ rvÞ
dðu þ rvÞ
b1 þ rf 2
ðf1 þ rb2 Þ u þ
þp
¼ F 1 þ rF 2 .
dx2
dx
f1 þ rb2
ð2:5Þ
It may be shown that the two Eqs. (2.1) and (2.2) are separated if and only if the following condition is
satisfied
b2 ðxÞr2 þ ðf1 ðxÞ f2 ðxÞÞr b1 ðxÞ ¼ 0.
This condition means that r is independent on x such that
"
#12
2
f2 ðxÞ f1 ðxÞ 1 f1 ðxÞ f2 ðxÞ
b1 ðxÞ
r1;2 ¼
þ4
;
2b2 ðxÞ
2
b2 ðxÞ
b2 ðxÞ
ð2:6Þ
ð2:7Þ
and (2.1)–(2.4) is separated in the forms
d2 w1
dw1
þ /1 ðxÞw1 ¼ G1 ðxÞ;
þ pðxÞ
dx2
dx
dw1
ð0Þ ¼ d1 ;
w1 ð0Þ ¼ c1 ;
dx
ð2:8Þ
ð2:9Þ
and
d2 w2
dw2
þ /2 ðxÞw2 ¼ G2 ðxÞ;
þ pðxÞ
dx2
dx
dw2
ð0Þ ¼ d2 ;
w2 ð0Þ ¼ c2 ;
dx
where
wi ðxÞ ¼ uðxÞ þ ri vðxÞ;
i ¼ 1; 2;
h
i12
2
ðf
ðxÞ
f
ðxÞÞ
þ
4b
ðxÞb
ðxÞ
1
2
1
2
f1 ðxÞ þ f2 ðxÞ
þ
;
/1 ðxÞ ¼ 2
2
h
i12
2
ðf1 ðxÞ f2 ðxÞÞ þ 4b1 ðxÞb2 ðxÞ
f1 ðxÞ þ f2 ðxÞ
/2 ðxÞ ¼ ;
2
2
c1 ¼ a1 þ r1 a2 ; d1 ¼ b1 þ r1 b2 ; c2 ¼ a1 þ r2 a2 ; d2 ¼ b1 þ r2 b2 ;
ð2:10Þ
ð2:11Þ
L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
555
and
G1 ðxÞ ¼ F 1 ðxÞ þ r1 F 2 ðxÞ;
G2 ðxÞ ¼ F 1 ðxÞ þ r2 F 2 ðxÞ.
In summary, the coupled system (2.1)–(2.4) can be transformed into an another system in which the equations are decoupled and solvable separately.
In the following section we use Adomian decomposition method to obtain explicit solutions for each of the
two boundary value problems (2.8)–(2.9) and (2.10)–(2.11).
3. Adomian’s decomposition method
Adomian [1,2] has presented and developed a so- called decomposition method for solving linear or nonlinear problems such as ordinary differential equations. It consists of splitting the given equation into linear
and nonlinear parts, inverting the highest-order derivative operator contained in the linear operator on both
sides, identifying the initial and/or boundary conditions and the terms involving the independent variable
alone as initial approximation, decomposing the unknown function into a series whose components are to
be determined, decomposing the nonlinear function in terms of special polynomials called Adomian’s polynomials, and finding the successive terms of the series solution by recurrent relation using Adomian’s polynomials as follows:
Consider the equation
F ðyðxÞÞ ¼ gðxÞ;
ð3:1Þ
where F represents a general nonlinear ordinary differential operator and g is a given function. The linear
terms in Fy are decomposed into Ly + Ry, where L is an easily invertible operator, which is taken as the highest-order derivative and R is the remainder of the linear operator. Thus, Eq. (3.1) can be written as
Ly þ Ry þ Ny ¼ g;
ð3:2Þ
where Ny represents the nonlinear terms in Fy. Solving for Ly,
Ly ¼ g Ry Ny.
ð3:3Þ
Operating L with its inverse L1 yields
L1 Ly ¼ L1 g L1 Ry L1 Ny.
ð3:4Þ
An equivalent expression
y ¼ H þ L1 g L1 Ry L1 Ny;
ð3:5Þ
d2
,
dx2
1
0
then L is a two-fold integration, and H = y(0) + xy (x). Due to
where LH = 0. For example, for L ¼
Adomian [1], the solution y is represented as the infinite sum of series
1
X
yn;
ð3:6Þ
y¼
n¼0
and the nonlinear term Ny assumed to be an analytic function, is decomposed as follows:
1
X
An ;
Ny ¼
ð3:7Þ
n¼0
where the An are Adomian’s polynomials of y0, y1, . . . , yn and are calculated by the formula
!
1
X
1 dn
i
N
k
y
n ¼ 0; 1; 2; . . .
An ¼
i k¼0 ;
n! dkn
i¼0
Putting (3.6) and (3.7) into (3.5) we get
1
1
1
X
X
X
y n ¼ H þ L1 g L1 R
y n L1
An .
n¼0
n¼0
n¼0
ð3:8Þ
ð3:9Þ
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L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
Each term of series (3.6) is given by the recurrent relation
y 0 ¼ H þ L1 g;
ð3:10Þ
1
1
y n ¼ L Ry n1 L An1 ;
n P 1.
ð3:11Þ
Now we apply the Adomian’s decomposition method described above to solve the boundary value problems
(2.8)–(2.9) and (2.10)–(2.11).
A formal inverse of (2.8) can be easily found, we choose it as
Z x Z t
L1 w1 ðxÞ ¼
dt
dyw1 ðyÞ.
ð3:12Þ
0
0
We have
dw1
L1 Lw1 ðxÞ ¼ w1 ðxÞ w1 ð0Þ x
ð0Þ.
dx
P1
1
ð0Þ þ L1 G1 .
Let w1 ðxÞ ¼ n¼0 w1;n ðxÞ and identifying w1;0 ¼ w1 ð0Þ þ x dw
dx
Using the form (3.9) we get the recurrence relation
dw1;n1 ðxÞ
1
þ /1 ðxÞw1;n1 ðxÞ ; n P 1.
w1;n ðxÞ ¼ L
pðxÞ
dx
ð3:13Þ
Similarly, we get the recurrence relation for problem (2.10)–(2.11).
In order to illustrate the possible practical use of this method we apply the above technique to the following
example.
pffiffiffi
pffiffiffi
Example 1. Consider
the system (2.1)–(2.4) with F1(x) = F2(x) = 0, f1 ðxÞ ¼ b= 2xs2 ; f 2 ðxÞ ¼ b= 2xs2 ,
pffiffiffi s2
b1 ðxÞ ¼ b2 ðxÞ ¼ b= 2x and p(x) = 0, where b and s are constants. Therefore the conditions of Lemma 1 are
fulfilled and system (2.1)–(2.4) is separated in the above forms p
(2.8)–(2.9)
and (2.10)–(2.11), straightforward
ffiffiffi
computation yields /1(x) = bxs2, /2(x) = bxs2, r1;2 ¼ 1 2 and G1(x) = G2(x) = 0. We choose ci = 1,
di = 0, i = 1,2.
For the
P first boundary value problem (2.8)–(2.9), we take as usual in the decomposition method
w1 ðxÞ ¼ 1
n¼0 w1;n ðxÞ, where w1,0(x) = w1(0) = 1, and
w1;n ðxÞ ¼ L1 ðbxs2 w1;n1 ðxÞÞ;
where
L1 ðÞ ¼
Z
x
0
Z
n P 1;
x
ðÞ dx dx.
0
We find that
w1;1 ðxÞ ¼
b
xs ;
sðs 1Þ
and by induction
w1;n ðxÞ ¼
bn =n!
xns .
sn ðs 1Þð2s 1Þ ðns 1Þ
In this way we get the well-known solution
w1 ðxÞ ¼ 1 þ
1
X
n¼1
sn ðs
bn =n!
xns .
1Þð2s 1Þ ðns 1Þ
The second solution for (2.10)–(2.11) is found in the same way and is given by
1
X
bn =n!
n
xns .
w2 ðxÞ ¼ 1 þ
ð1Þ n
s ðs 1Þð2s 1Þ ðns 1Þ
n¼1
ð3:14Þ
L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
557
Substituting the values of wi, i = 1, 2 into
wi ðxÞ ¼ uðxÞ þ ri vðxÞ;
i ¼ 1; 2;
we obtain
u¼
r1 w2 ðxÞ r2 w1 ðxÞ
pffiffiffi
;
2 2
and
v¼
w1 ðxÞ w2 ðxÞ
pffiffiffi
.
2 2
4. A system of nonlinear differential equations for first-order
Mathematical modelling of many frontier physical leads to systems of nonlinear ordinary differential equations. Motivated by this, we propose a class of coupled system of two nonlinear differential equations with
first-order similar to model of Lotka–Volterra [10] which may be written in the following form:
du
þ f1 ðxÞu þ b1 ðxÞv ¼ aðxÞðu2 þ uvÞ;
dx
dv
þ f2 ðxÞv þ b2 ðxÞu ¼ aðxÞðv2 þ uvÞ.
dx
ð4:1Þ
ð4:2Þ
Subject to the initial conditions
uð0Þ ¼ u0 ; vð0Þ ¼ v0 ;
ð4:3Þ
where fi, bi, i = 1, 2 and a(x) are being arbitrary analytic functions.
The objective of this Section is to solve analytically the above system.
Firstly we propose this new transformation
u ¼ Xv;
ð4:4Þ
where X(x) is unknown function.
The substitution (4.4) into (4.1) and (4.2) gives a system which we write as
0
dv
X
b1
þ
þ f1 þ
v ¼ aðxÞð1 þ X Þv2 ;
dx
X
X
dv
þ ½f2 þ b2 X v ¼ aðxÞð1 þ X Þ;
dx
ð4:5Þ
ð4:6Þ
. It follows that X satisfies the following ‘‘X equation’’.
where X 0 ¼ dX
dx
dX
þ ðf1 f2 ÞX b2 X 2 ¼ b1
dx
ð4:7Þ
with
X ð0Þ ¼
u0
.
v0
ð4:8Þ
We are now in the position of solving Eq. (4.7) with (4.8). We take as usual in (ADM)
1
X
X n ðxÞ;
X ðxÞ ¼
n¼0
and
X 2 ðxÞ ¼
1
X
An .
n¼0
The first few Adomian polynomials for Nx = X2 are given by
A0 ¼ X 20 ;
A1 ¼ 2X 0 X 1 ;
A2 ¼ X 21 þ 2X 0 X 2 .
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L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
Using the form (3.9) we get the recurrence relation
X 0 ¼ X ð0Þ L1 b1 ðxÞ;
X nþ1 ¼ L1 ðf2 ðxÞ f1 ðxÞÞX n ðxÞ þ L1 b2 ðxÞ
1
X
ð4:9Þ
!
An ;
n P 0.
ð4:10Þ
n¼0
Rx
Here L1 ðÞ ¼ 0 ðÞ dx.
In summary, the direct application of (ADM) to the nonlinear coupled system (4.1)–(4.3) is more difficult
when the functions fi, bi, i = 1, 2 and a(x) are chosen to be complicated. For this, we make the new transformation (4.4), from which follow the ‘‘X equation’’ defined by (4.7) which can be solved by (ADM). Hence, the
coupled system (4.1)–(4.3) is also separable and can be solved.
Two examples with closed form solutions are discussed in order to demonstrate the feasibility and efficiency
of this transformation and the possible practical use of (ADM).
Example 2. Consider system (4.1)–(4.3) with the initial conditions
u0 ¼ 0; v0 ¼ 1;
f1 ðxÞ ¼ f2 ðxÞ ¼ tan x;
b1 ðxÞ ¼ 1; b2 ðxÞ ¼ 1;
and
aðxÞ ¼
cos x
.
cos x þ sin x
Then, Eq. (4.7) can be written as
X 0 ðxÞ ¼ X 2 ðxÞ þ 1;
with the initial condition
X ð0Þ ¼ 0;
which has an explicit solution
X ðxÞ ¼ tan x.
We solve this problem by ADM. Writing X ðxÞ ¼
X 0 ¼ x;
Z
X nþ1 ¼
P1
n¼0 X n ðxÞ,
we express the recurrent scheme of ADM as
x
An dx;
n P 0.
0
Simple calculations lead to
X1 ¼
x3
;
3
X2 ¼
2 5
x ;...
15
and
X ðxÞ ¼ x þ
x3
2
þ x5 þ 3 15
which is the partial sum of the Taylor series of the solution X*(x).
Now, substituting X ðxÞ ¼ tan x into the differential equation (4.6) we obtain
dv
¼ v2 ;
dx
vð0Þ ¼ 1;
which has the exact solution
v ðxÞ ¼
1
.
1x
L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
559
Using (ADM), we get
v0 ¼ 1; v1 ¼ x; v2 ¼ x2 ; . . .
and
vðxÞ ¼ 1 þ x þ x2 þ It can be easily observed that this last solution is equivalent to the exact solution. Hence, it follows from (4.4)
that
uðxÞ ¼
tan x
.
1x
Example 3. Consider system (4.1)–(4.3) with the initial conditions
u0 ¼ 4;
v0 ¼ 2;
2
þ x þ 1;
f1 ðxÞ ¼
2 expðxÞ
b1 ðxÞ ¼ 0;
f 2 ðxÞ ¼
2
þ x;
2 expðxÞ
b2 ðxÞ ¼ 1;
and
2 expðxÞ
x.
4 expðxÞ
Then, Eq. (4.7) can be written as
aðxÞ ¼
X 0 ðxÞ þ X ðxÞ ¼ X 2 ðxÞ;
with the initial condition
X ð0Þ ¼ 2;
which has an explicit solution
2
; x < ln 2.
X ðxÞ ¼
2 expðxÞ
P
We solve this problem by ADM. Writing X ðxÞ ¼ 1
n¼0 X n ðxÞ, we express the recurrent scheme of ADM as
Z x
Z x
X n dx þ
An dx; n P 0.
X 0 ¼ 2; X nþ1 ¼ 0
0
This in turn gives
X 0 ¼ 2;
X 1 ¼ 2x;
X 2 ¼ 3x2 ;
X3 ¼
13 3
x;
3
X4 ¼
15 4
x.
4
2
Consequently, the series solution is X(x) = 2 + 2x + 3x2 + and in a closed form by X ðxÞ ¼ 2expðxÞ
.
2
Now, substituting X ðxÞ ¼ 2expðxÞ
into the differential equation (4.6) we obtain
dv
þ xv ¼ xv2 ;
dx
vð0Þ ¼ 2;
which has the exact solution
v ðxÞ ¼
2
2 .
2 expðx2 Þ
Using (ADM), we get
3
13
v2 ¼ x 4 ; v3 ¼ x 6 ;
4
24
3
13
25
vðxÞ ¼ 2 þ x2 þ x4 þ x6 þ x8 þ 4
24
64
v0 ¼ 2;
v1 ¼ x2 ;
v4 ¼
25 8
x;
64
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L. Bougoffa, S. Bougouffa / Applied Mathematics and Computation 177 (2006) 553–560
It can be easily observed that this last solution is equivalent to the exact solution. Hence, it follows from (4.4)
that
uðxÞ ¼
4
.
2
ð2 expðxÞÞ 2 expðx2 Þ
5. Conclusion
The Adomian decomposition method was used successfully to solve a class of coupled systems of two linear
and nonlinear differential equations for second- and first-orders, respectively. Some examples with closed form
solutions are studied carefully in order to illustrate the possible practical use of this method, and the results
obtained are just the same as those given from applying the ADM.
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