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ADOMIAN’S DECOMPOSITION METHOD(ADM)

A PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENT FOR THE DEGREE OF

M.Sc (2-year) In Mathematics

By

Anand Kumar Rai

Roll No-(11MA40003)

Under the guidance of

Prof. R.K Pandey

DEPARTMENT OF MATHEMATICS

INDIAN INSTITUTE OF TECHNOLOGY

KHARAGPUR, INDIA

CERTIFICATE:-

This is to certify that the project entitled “Adomian’s Decomposition Method (ADM)” submitted by Anand Kumar Rai Roll No-(11MA40003) during the session 2013 in partial fulfillment of the degree of M.Sc in Mathematics to the Indian Institute of Technology, Kharagpur, is the bona fide record of the work carried out under my supervision and guidance.

This report fulfills all the requirements as per regulations of the institute and has not been submitted to any other institute/university for degree or diploma.

Professor

Dr. R.K Pandey

Department of mathematics

Indian Institute of Technology

Kharagpur -721302

ACKNOWLEDGEMENTS:-

It brings me immense pleasure to express my deepest sense of gratitude to my supervisor Dr. R.K Pandey for his guidance and support throughout my project work. His valuable guidance and precious advice provided the platform for the whole project work.

I would also like to express my gratitude towards all the faculty members of the Departments for their help and for introducing me to the great areas of Mathematics during my Post graduation education and fueled my interest in the subject

Date:

Anand Kumar Rai

place-

kharagpur

INTRODUCTION

Adomian's decomposition method (ADM) is useful and powerful method for solving linear and nonlinear differential equation.adomian goal is to find the solution of linear and non linear, ordinary or partial differential equation without dependence on any small parameter like as perturbation method. In this method the solution is considered as the sum of infinite series which, rapidly convergence to an accurate solution. Adomian Decomposition Method (ADM) is one of the new methods for solving Initial Value Problem in ordinary differential equations of various kinds arising not only in the field of medicine, physical and biological science but also in the area of engineering. It is important to note that a large amount of research works has been devoted to the application of (ADM) to wide class of linear and nonlinear ordinary or partial differential equations. Adomian and Rach were firstly discussed the appearance of noise terms, they concluded that, the inhomogeneity of the differential equation is the major and only reason. The appearance of these terms in the first two components of the solution series is considered by Adomian as the first condition of demonstrating a fast convergence of the solution. The studies of singular initial value problems in the second order ordinary differential equations (ODEs) have attracted the attention of many mathematicians and physicists. And those singular initial value problems in the second order differential equation can be solved by modified adomian's decomposition method, which is stated below.

Ordinary differential equation:-

Differential equations that involve only ONE independent variable are called ordinary differential equation.

e.g dy/dx=2x+3,

Partial differential equation:-

Differential equation that involve two are more than two independent variable are called partial differential equation.

Linear ordinary differential equation:-

A differential equation is called linear if there are no multiplications among the dependent

variable and their derivatives. In other words we can say that all coefficients are the functions of

independent variables.

e.g

+P +Qy = R,

Where P, Q, R are the function of 'x' only. this equation is called linear ordinary differential equation of second order.

Non linear ordinary differential equation:-

Differential equation that does not satisfied the definition of linear are nonlinear.

e.g

= ky(1-cy), here c and k is constant ,it is called non linear ode.

Linear partial differential equation:-

The partial differential equation is called linear if the unknown functions only appear in the linear forms.

e.g

the=givenc equation is linear partial differential equation

Non linear partial differential equation:-

An equation of the form F(x,y…..Day)=0, ………………..(1)

where x=(x1,x2,……xn) ЄRn,y=(y1,y2……..ym)ЄRm

F=(F1,F2…….Fk)ЄRk,and, a=( a1, a2……, an) is a multi index of non negative integer a1, a2…,

an

 

 

 

 

and Da=

a

 

a ,where Di=

,i=1,2,3,…..n,

 

equation (1) is called the non linear partial differential equation of n order

then the

 

…….

 

 

F(x,y,z,p,q)=0,where p= ,and q=

is called partial differential of first order

e.g.

 

 

 

 

(p2+q2)y=qz

ADOMIAN’S DECOMPOSITION METHOD (ADM):-

Consider the general nonlinear functional equation:- L[u(x1,x2,x3,x4,…….xn)]+N[u(x1,x2,x3,x4,…….xn)]+R[u(x1,x2,x3,x4,…….xn)]=g(x1,x2,….,xn)

…………….(1)

Where L is the invertible operator of highest-order derivative with respect to xj (which is referred to the Integrated variable), let the order of this operator is k, N is the nonlinear operator, R is the remainder linear operators and g(x1,……….., xn) is a given function. Let the inverse operator of L

is -defined1 = as ∫∫∫…..∫..

L (.) dxjdxjdxj…..dxj …………….(2)

k-times

Where k is defined as the integration level, consequently we can get the solution of equation (1) in the following form

u(x1,x2,x3……xn)=f(x1,x2,x3……xn) - L-1[ N[u(x1,x2,x3……xn)]+R[u(x1,x2,x3……xn)] ]

………….(3)

Where f(x1…xn) is arising from integrating the source function g(x1 …xn), and using the given initial

conditions, namely

f(x1,x2,x3,…..,xn)=ψ(x1,x2,x3,…..,xn)+L-1[g(x1,x2,x3,…..,xn)] …………(4)

and ψ is the solution of the following homogeneous differential equation

L[ψ(x1,x2,x3,…..,xn)]=0 …………….(5)

Using the original Adomian’s decomposition method the unknown function u(x1,…….xn) can be expressed by an infinite series of the form

u(x1,……, xn)=

 

8

 

m(x1,x2,x3,……..xn)

………………….(6)

and the

nonlinear operator N(u(x

,……..,x )) can be decomposed as

 

∑

 

u

1

 

n

= u(x1,………..,xn) ……(11)
Where lim →8

N(u(x1,……., xn))=

8 m(x1,…….., xn)

…………….(7)

 

Each term Am is called∑

Adomian’s polynomial, it depends on the history of u

,

A

j

 

j = 0, 1,2,3,4,……m-1 and it is given by,

 

 

Am(x1,….. xn)=

!

[

N(∑ λ ui(x1,x2,x3,……..xn))]λ=0 ,

m=0, . …………(8)

where λ¸ is a parameter introduced for convenience. Finally ,the approximated solution of equation (1) is

deduced by the following recurrence relation

 

u0(x1,……, xn) = f(x1,…….,xn)

……………… ..(9)

um+1(x1,……, xn)= -L-1[R(um(x1,……, xn))+ Am(x1,…….,xn)],

m=0, …………(10)

The effectiveness of the method can be dramatically improved by taking further components of the solution Series. However, for concrete problem, the (m + 1)-terms approximate Fm defined

by, ∑ u

Fm = i(x1,x2,x3,……..xn) , m=0,

The reliability of Adomian’s method gives it in a wider applicability in handling evaluation models. Now, we consider the following example to illustrate the advantages and disadvantages of the different and previous

techniques (original ADM [10]-[15], Wazwaz [8], and Wazwaz and Salah [9]).

problem:-

Consider the nonhomogeneous differential equation.

uxx+uutt = 1 -

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

with the initial conditions:-

u(0, t) =

 

and

ux(0,t) = 0

,

 

 

 

Original ADM ([10],[11]): According to equation (4) ,where g(x, t) =1 -

 

they choosed the

 

initial approximation as

 

 

 

 

 

 

 

 

 

f(x, t)= u(0, t) + xux(0, t)+∫∫g(x,t)dxdx =

 

+0+∫

∫ (1 -

 

 

)dxdx

 

 

 

 

 

 

 

= x2/2 +t2/2 –x2t2/4 –x4/24

…..…(13)

 

then

u0(x, t) = f(x, t).

 

 

Consequently, the recurrence relation (10) becomes

 

uj(x,t) = ∫∫Aj-1 (x,t)dxdx

j=1

……….(14)

and the Adomian’s polynomials are A0 = u0(u0)tt;

A1 = u1(u0)tt + u0(u1)tt; …… (15) A2 = u2(u0)tt + u1(u1)tt + u0(u2)tt;

A3 = u3(u0)tt + u2(u1)tt + u1(u2)tt + u0(u3)tt;

now, A0=u0(u)tt.

here, u0(x,t)= x2/2 +t2/2 –x2t2/4 –x4/24

 

(u0)t=t – tx2/2.

 

 

 

(u0)tt=1- x2/2.

 

 

now,

u0(u0)tt=( x2/2 +t2/2 –x2t2/4 –x4/24)( 1- x2/2)

 

 

= x2/2 +t2/2 – (x2t2/2)

 

-7x4/24 +x4t2/8 +x6/48

now,

u1=

2 2 ( x2/24 +t2/24 –2 (x2t2/2)6

-7x4/246 2+x4t2/8 +x8 6/48)dxdx

 

=(x∫

t∫)/4+x /24-x t /24 -7x

/720 +x t /240 +x /2688.

similarly

u2=t2x4/24 +7x6/720 – t2x6/90 -67x8/40320 +t2x8/1120 +173x10/1209600 – t2x10/21600

– 43x12/1064480.

the solution u1 and u2 is called approximate series solution. and unfortunately the noise term are appearing.

Problem 1:-

We consider the problem

 

y'=y2 and y(0) = 1

…………(1)

the theoretical solution given as y=( ) , 0=x<1 we apply ADM operator to equation (1) to produce Ly=y2

By finding the inverse operator and imposing initial condition we obtain

y(x)=y(0)+L-1(y2)

……………………….(2)

The ADM introduce the solution y(x) in an infinite series form as,

y(x)=

 

8

 

n(x),

 

then by equation (2) plugging the value of y(x) we get,

 

8

∑n

 

 

-1

(A ),

 

 

(x)=1+L

 

 

 

 

y

 

n

Where A

n

is the Adomian Polynomial which be derived as follows,

∑

 

y

 

 

N(∑8

 

An=

!

[

 

ui(x1,x2,x3,……..xn))]λ=0,

where N is the non linear term here it is N=y2 now by using the formulae of An we get, A0=(y0)2,

A1=2y0y1

A2=2y0y2+(y1)2

A3=2y0y3+2y1y2

A4=2y0y4+2y1y3+(y2)2

A5=2y0y5+2y1y4+2y2y3, with y(0)=1 and yn+1=L-1(An)

We can then proceed to compute y ,y2,y3,………

 

 

 

 

 

 

now, y (x)=L-1(A )= L-1(12)=

1dx=x

 

 

 

 

 

 

 

 

 

1

 

-1

0

-1

∫

 

 

 

2

 

 

 

 

 

 

 

 

y2(x)=L-1

 

(A1)= L

 

-1(2y0y1)= ∫ 2xdx2

=x

2

 

 

2

 

3

 

y (x)=L

 

 

(A ) = L

(2y y +(y ) )=

∫

(2x

+x )dx=x

 

 

3

-1

 

 

2

-1

 

 

0 2

1

 

3

+2x

3

 

4

 

y (x)=L

 

(A )= L

 

(2y0y3+2y1y2)=

 

 

 

 

)dx=x

 

 

4

-1

 

 

 

3

 

 

-1

 

 

∫ (2x2

 

∫

 

4 4 4

5

y5(x)=L

 

(A4)

= L

(2y0y4+2y1y3+(y2) )=

 

(2x +2x +x )dx=x

 

.

 

 

.

 

 

.

 

 

.

 

 

yn(x)=xn

∑8

yn(x), n=0

Consequently, y(x)=

For application purpose, only few terms of the series will be computed. Tables (1) compare the (ADM) result with the theoretical solutions. the table is given below

Table-1:

X

ADOMIAN

EXACT

ERROR

 

 

 

 

0.00

1.000000000

1.000000000

0.000000000

 

 

 

 

0.10

1.111111164

1.111111164

0.000000000

 

 

 

 

0.20

1.249999881

1.250000000

0.000000119

 

 

 

 

0.30

1.428571224

1.428571463

0.000000238

 

 

 

 

0.40

1.666666746

1.666666627

0.000000119

 

 

 

 

0.50

2.000000000

2.000000000

0.000000000

 

 

 

 

0.60

2.500000000

2.500000238

0.000000238

 

 

 

 

0.70

3.333333969

3.333333969

0.000000000

 

 

 

 

Modified Adomian decomposition method for singular initial value problems:-

Modified Adomian decomposition method:- Algorithms:

Consider the singular initial value problem in the second order ordinary differential equation in the form y'' + (2/x)y'+f(x,y) = g(x,y) ……….(1)

with the condition y(0)=A, y'(0)=B

where f(x, y) is a real function, g(x) is given function and A and B are constants Here, we propose the new differential operator, as below

L=x-1

) xy,

………………… (2)

so, the problem (1)(

can be written as,

 

 

Ly = g(x)- f(x,y)

…………(3)

The inverse operator L-1 is therefore considered a two-fold integral operator, as below

L-1 (.) = x-1

 

x(.

)

dxdx

………….(4)

∫ ∫

 

 

 

Applying L-1 of (4) to the first two terms y''+(2/x)y' of equation (1) we get,

L-1(y''+(2/x)y')=-x1

-1

 

'

''

+(2/x)y

')dxdx

 

 

∫ ∫

x(y

 

=x ∫ (xy+y-y(0))dx = y-y(0)

By operating L-1on (3) we get,

y(x)=A+L-1(g(x)) – L-1(f(x,y)) …………………………(5)

The Adomian decomposition method introduce the solution y(x) and the nonlinear function f(x, y) by infinity series

y(x)= 08yn(x),

…………..(6)

and the∑non linear term is given by

 

f(x,y)= ∑08An

…………..(7)

Where An is the adomian polynomial

to formulate this we will use the following algorithms,

A0 = F(u0);

A1 = u1F'(u0),

A2=u2F'(u0)+ ! F''(u0), ……………(8)

A3=u3F'(u0)+u1u2F''(u0) ! F'''(u0),

……

to construct the adomian polynomials,here F(u) is the nonlinear function now,By substituting equation (6) and (7) into eqn(5) we get ,

∑80 yn(x)= A+L-1g(x)-L-1∑80 An, ……………………..(9)

Through using Adomian decomposition method, the components yn(x) can be deter- mined as

y0(x)=A+L-1g(x) ……………(10)

yk+1=-L-1(Ak), k=0,

which gives,

y0(x)=A+L-1g(x) y1(x)=-L-1(A0),

y2(x)=-L-1(A1), ………………(11)

y3(x)=-L-1(A2),

……

from eqn (8) and eqn (11) we can easily determine the componants(yn(x)),and hence the series solution of y(x) in eqn (6) can be immediately obtained.

for numerical∑ purpose,the 'n' term approximates,

Fn(x)= n0 1yn(x)

can be used to approximate the exact solution.

CONCLUSION:-

The result obtained from the test problems have shown that (ADM) is the powerful and efficient technique in finding an approximate solution to both linear and non linear first order ordinary initial value problems which occur most often in physical and biological sciences. It is interesting to note that (ADM) result tends to the theoretical solution by increasing the number of terms computed.

the appearance of noise terms in the approximates solution using adomian decomposition method is avoided by using new technique.In this study we present a new technique to get the solution of inhomogeneous differential equation using adomian decomposition method without noise terms. The construction of the suitable differential equation of the inhomogeneity function convert the problem to autonomous system and conserve the homogeneity of the approximations.

As we see from the examples the homogeneity of approximation is one of the necessary conditions to avoid the noise terms through computations.

By using ADM to solve the augmented system we get either the closed form of the exact solution or the Maclurin expansion of the solution function without any noise terms. We get the closed form of the exact solution after a few steps, if the solution function u is a finite polynomial of the variable xj , but if the solution function u is a transcendental function or an infinite polynomial, then we get the Maclurin expansion of the exact solution without any noise terms.