DOI : 10.17577/IJERTV2IS80285
- Open Access
- Total Downloads : 409
- Authors : Manjak N. H, A. K Mishra, Rakiya M. K .A
- Paper ID : IJERTV2IS80285
- Volume & Issue : Volume 02, Issue 08 (August 2013)
- Published (First Online): 19-08-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Application of Adomian Decomposition Method for Solving a Class of Diffusion Equation
1 Manjak N. H , 2 A. K Mishra And 3 Rakiya M. K .A
1Mathematical Sciences Programme, Abubakar Tafawa Balewa University, Bauchi, Nigeria 2Department of Mathematics, Gombe State University, Gombe, Nigeria 3Department of Computer Science, Federal Polytechnic Bauchi, Nigeria
Abstract
In this paper we adopt the Adomian Decomposition Method (ADM) for approximating solutions of linear and nonlinear diffusion (parabolic) equations. The solution is calculated in the form of a series with easily computable components. The series solutions of the differential equations considered by the method in this work converge to the exact solutions which illustrate how effective the ADM is in solving such problems.
Keywords: Nonlinear parabolic equations; Diffusion
linearization, perturbation, massive computation and any transformation of the governing differential equations. Mustafa Inc [8].
-
Analysis of the Method
In this section, we demonstrate the main algorithm of ADM for linear and nonlinear parabolic equations with initial conditions, we consider the equation
u 2u
equation; Adomian polynomial; Adomian
u x2
N (u)
g(x, t).
decomposition method.
-
Introduction
(x,t)[a,b](0,T )
With the following initial condition
(2.1)
Recently a great deal of interest has been focused on the applications of Adomians decomposition method
u(x,0)
f (x),
(2.2)
to solve a wide variety of stochastic and deterministic problems. Adomians gold is to find a method to unify linear and nonlinear, ordinary and partial differential equations for solving initial and boundary value problems. Afrouzi and Khademloo [6].
Where, Nu is the nonlinear term. We are looking for a solution satisfying equation (2.1) under condition (2.2). The decomposition method consists of approximating the solution of (2.1) as an infinite series.
Various physical systems can be described by linear and nonlinear parabolic equations. These linear and nonlinear models, as well as their solutions are of fundamental importance for applied sciences.
Our objective in this paper is the desire to obtain analytic solutions to such linear and nonlinear parabolic equations.
u(x, t) un (x, t)
n
Decomposing N (the nonlinear operator) as
N (u) An (u0 , u1 ,u2, …,un )
n
(2.3)
(2.4)
The decomposition method yields rapidly convergent series solutions by using a few iterations for both linear and nonlinear deterministic and stochastic equations. The advantage of this method is that it provides a direct scheme for solving the problem without the need for
Where the Ans are the Adomian polynomials, Adomian [1], and are calculated owing to the basic formula, Adomian et al [4], Adomian [2-3] and Manjak[7].
t
t
The inverse operator L1 exists and it can conveniently
1 d n
be taken as the definite integral with respect to t, i.e
An An (u0 ,u1,u2 …un ) n! d n N u
n 1, 2, 3…
1
1
t
0
0
… (2.5)
Lt (, )
(, )dt
0
which is a one- fold definite
From which
A0 Nu0
integral, since Lt is a first- order operator.
1
Operating with Lt on both side of (2.6) yields.
d
1 1
1 1
A1 u1 du
N (u0 )
0
Lt Ltu Lt
g(x,t) Lt
Nu Lt
Lxx u
(2.7)
So that
d
u 2 d 2
u(x,t) u(x,0) L1 g(x,t) L1 Nu L1 L
u (2.8)
A u N (u
) 1 N (u )
t t t xx
2 2 du
0 2! du 2 0
0
0
Substituting initial condition (2.2) into Eq. (2.8) to have
d d 2
u(x,t) f (x) L1g(x,t) L1Nu L1L u
(2.9)
A3 u3 du N (u0 ) u1u2 du2 N (u0 )
-
t t xx
0 0
u3 d 3
Substituting (2.3) and (2.4) into (2.9) we have
: 1 N (u0 )
3! du3
1 1 1
0 u(x,t) f (x) Lt
g(x, t) Lt
( An ) Lt
n0
Lxx (un )
no
:
The An `s can be written in the following convenient way
… (2.10)
From (2.10) the Adomian decomposition scheme is defined by the recurrent relation
n
n
An
1
c( , n) f (u
), n 1,
u0 (x,t)
f (x) L1 g(x,t)
t
t
0
0
And,
-
(x,t) L1 A
-
L1L u
for n= 0, 1, 2,
Applying the decomposition method, Mustafa [8], it is convenient to re-write Eq. (2.1) in the standard operator
…
n1
t n t xx n
form as
From which
u (x,t) L1 A
-
L1 L u
Ltu Lxx u Nu g(x,t)
(2.6)
-
t 0
t xx 0
u (x,t) L1 A L1 L u
Where,
L ; L
t t xx
2
x2 ; N
is the nonlinear
-
t 1
:
-
xx 1
operator and
g(x,t) is the forcing term.
We can determine the components
un as is necessary
The decision as to which operator to solve in a multidimensional problem is made on the basis of the best known conditions and possibly also on the basis of the operator of the lowest order to minimize integration. Bellman and Adomian [9].
to enhance the desired accuracy for the approximation. So, the n-terms approximation can be used to approximate the solution.
That is
n1
u .
u(x,t) u(x,0) L1 (1 L
u )
n i
i0
t 4 xx
n
n0
(3.5)
-
-
-
-
Some Analytic Solutions
From which we obtain the recurrent relation
To give a clear view of our study and to illustrate the
-
u(x,0) sin x, and,u L1 (1 L
u ), n 0,1,2,…
above discussed technique, we will consider both linear and nonlinear diffusion equations in this section.
Problem 3.1
o n1 t
From which u L1 (1 L u )
4 xx n
Consider the following one dimensional linear heat (heat) equation given by
1 t 4 xx 0
u L1 (1 L u )
k(x)ut
uxx
(3.1)
2 t 4 xx 1
Subject to the initial boundary conditions
u(x,0) g(x),0 x 1
(3.2)
:
1 1
t 2 sin x
2 t
2 t
u(0,t) p (t),u(1,t) p (t)
u1 Lt
[ Lxx (sin x)]4 4
We let
k(x) 4 and
g(x) sin x .
u L1
[1 L (4 xx
t 2 sin x
4
)]
t 2 4 sin x 32
For if
k(x) k 2 , a constant, then the exact solution
u L1[1 L
t 2 4 sin x
(
)]
t 3 6 sin x
of (3.1) is given by
t 2
3 t 4 xx 32
:
384
u(x,t) e k 2 sin x
Bellman and Adomian [5].
:
Now, using Adomian decomposition method, re-write equation (3.1) in the general operator form as
And so in; in this manner the rest of the components of the decomposition series (2.3) can be obtained. The
4Ltu Lxx u
(3.3)
2
solution for the heat (diffusion) equation (3.1) in a seris form is given by
Where
Lt t ; Lxx
x2
u(x,t) un (x,t) u0 u1 u2 u3 …
n0
t
t
Operating with
L1 on equation (3.3) to have
Therefore, the solution obtained from the method will be
L
L
t t
t t
t
t
1 L u L1
(1 L u)
4 xx
(3.4)
u(x,t) sin x(1
t 2
4
t 2 4
32
t 3 6
384
…)
Evaluating the L.H.S of (3.4) and substitute back into (3.4) to yield
t 2
t 4 2
t 2 3
Evaluate the L.H.S of (3.9) and substitute back into the equation to obtain
4
4
4
sin x(1
…)
u(x,t) u(x,0) L1 A
-
L1 L u
(3.10)
1! 2! 3!
t n
n0
t xx n n0
t 2
u(x, t) sin x e 42
0
0
From (3.10) the solution by the decomposition method consists of the following scheme:
Yielding the same result as the exact solution of the
u u(x,0) ex
differential equation, where
u L1 A L1 L u , n 0 , From which
k(x) k 2 4 .
n1 t n t xx n
u L 1 A L 1 L u ,
-
t 0 t xx 0
u L1 A L1 L u ,
-
t 1 t xx 1
Problem 3.2
u L1 A L1 L u , . . .
-
t 2 t xx 2
x
x
Consider the following nonlinear reaction-diffusion equation
Whence,
ut uxx
u 2 (u )2
(3.6)
Nu
f (u,ux
) u 2 u 2
x
x
subject to the initial condition
Nu0
f (u0
,u0x
) u 2 (u
0
0
0
0
x
)2 e2 x e2 x
u(x,0) ex Mustafa [8].
u L1[u 2 (u
)2 ] L1 L [u ]
And
1 t o 0x
t xx 0
L1[e2 x e2 x ] L1 L [ex ]
Re-writing (3.6) in the general form, we have t
-
xx
Ltu Lxx
Where,
Nu
Lt
, L
1 t
1 t
t xx
2
x2
(3.7)
and
L1[ex ]
t
t
1
1
u te x /1!
Ie
ex dt te x
x
x
Nu u 2 (u )2
-
L1
2 t
2 t
An c(i, n) f u , n 1,2,3,…
An c(i, n) f u , n 1,2,3,…
n
A L1
Lxx u1
In this case, the operator
1
1
t
Lt t
with
And
i
0
i1
inverse L t
(.)dt . … (3.8)
0
i.e.
n
Applying the inverse operator
L 1 in (3.8), we have
A1 c(i, n) f u0 c(1,1) f u0 2u1[u0 u0x ]
i1
t
t
L1 L u L1 Nu L1 L u
(3.9)
t t t t xx
f (u ,u
) u 2 u 2 .
in a similar manner, other terms of the series can be
x
x
x
x
for
0 0x
0 0x
generated.
x
x
f (u0 ,u0
) 2u0 2u0
2(u0 u0 )
The solution u(x,t)of the reaction- diffusion equation
And
A1 2(u0
-
u0x
) 2(te x )[ex ex ] 0
(3.6) in series form is then given by
Therefore,
1
1 x
x t 2 x
t 2 x
u(x,t)
n
un (x,t) u0
-
-
u1
-
u2
-
u3
…
u2 Lt
Lxx u1 Lt
Lxx [te
] te
dt e e 0
2 3
2 3
2 2!
u L1 A
-
L1L u
x
x t x t x
x
-
t t
x t
3 t 2
n
t xx 2
e te
e e
2
2
3
3
2 3!
…
e [1 t
2! 3!
…
e e .
From
A c(i, n) f iu
n
n
0
0
i1
, n 1,2,3,…
This gives the exact solution by
2
2
A2
i1
c(i,2) f iu ,
i,e
u(x,t) ext .
0
0
A2 c(1,2) f u 0
c(2,2) f u 0
In view of the forgoing, some important conclusions can be made here.
f (u
) f (u ,u
) u 2 u 2 .
But
Therefore
0 0 0x
0 0x
-
-
Conclusion
f (u0 )
f (u0
,u0x ) 2(u0 u0x )
and
Linear and non parabolic equations play an important role in applied sciences. The basic goal of this paper
f (u0 )
We obtain
f (u0 ,u0 ) 2(11) 0
x
x
x
x
A2 u2 2[u0 u0 ]
has been to employ ADM for studying this model. The goal has been achieved by deriving the exact solutions for linear and nonlinear cases by using few terms of the series only. The decomposition introduces a significant development and improvement in the field of solution methods and this makes the scheme powerful and gives a wider applicability. The method avoids the
difficulties and massive computational work that
1
1
t x
t x
2
L
L
t [ e ]
2
t 2 x
L
L
1
1
t [ e
2
2(ex
ex )]
usually arise from other classical methods. It gives more realistic series solution that converges very rapidly in physical problems.
t
t
2
2
L1[t ex
2
] 0
-
References
t
t
t
t
2 3
x
x
e dt ex
2 3!
t x
t x
3
i.e u3 3! e
-
Adomian G. (1989). Nonlinear Stochastic System. The Theory And Application to Physics, Kluwer Academic publ.
-
Adomian G. (1991). A Review of the Decomposition Method and Some Recent Results for Nonlinear Equations. Comp. Math. Appl. 21(950): 101-127.
-
Adomian G. (1991). An Analytic Solution of Stochastic Navier-Stokes Problem. Foundation Physics. 21(7): 831-843.
-
Adomian G. (1984). Solving Frontier Problems on Physics. The Decomposition Method. Kluwer Academic Publisher.
-
Adomian G. (1994). Solution of Physical Problems by Decomposition. Comp. Math. Appl. 27 (9-10): 145-154.
-
Afrouzi G. A. And Khademloo S. (2006). On Adomian Decomposition Method for Solving Reaction Diffusion Equation. International Journal of Nonlinear Science 2(1): 11-15.
-
Mustafa Inc (2004). On Numerical Solutions of Partial Differencial Equations by the Decomposition Method. Kragujevac J. Math. 26:153-164.
-
Richard Bellman and George Adomian (1984). Partial Differential Equations. New Methods for their Treatment and Solutions. Mathematics and its Applications. D. Reidal Publishing Company. Dordrecht/Boston/Lancaster.