J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012)
Differential Transform Method For Ordinary Differential Equations NARHARI PATIL1 and AVINASH KHAMBAYAT2 1
Professor & Head Department of Mathematics Shri Sant Gajanan Maharaj College of Engineering, Shegaon, Maharashtra, INDIA. 2 Department of Mathematics G. H. Raisoni Institute of Engineering and Management, Jalgaon, Maharashtra, INDIA (Received on: May 24, 2012) ABSTRACT In this paper, we study Differential Transform Method for solving Ordinary Differential Equations. The approximate solution of the equation is calculated in the form of series with easily computable componants.This Powerful method catches the exact solution. Some types of Ordinary Differential Equations are solved as numerical examples. Keywords: Series solutions, Differential transform, System of Ordinary Differential Equations.
1. INTRODUCTION Consider the system of Ordinary Differential Equations of the first order with initial conditions,
dy1 , , , … , , dx
, 1 , 2 , … , , 2 0 y 2
,
, , , … , , = y n (1.1)
Where each equation represents the first derivative of unknown function as a mapping depending on the independent variable and n unknown functions , , . . . 2. THE DIFFERENTIAL TRANSFORM METHOD The transformation of the
derivative of a function with one variable is follows
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Narhari Patil, et al., J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012)
ŕł– ! ŕł–
,
and the inverse transformation is defined by,
2.4
The particular case of equation (2.4) when 0 is referred to as the Maclaurins series of u (t), and is express as,
∞
In this section we shall give basic theorem of one dimensional transform method.
Definition1. If u (t) is analytic in domain in the domain T then let,
The following theorems that can be deduced from equations (2.1) and (2.4) are given below
φ (t , k ) =
d ku (t ) for all t ∈ T dt k
(2.1)
for , , , where k belong to the set of non negative integers, denoted as integer K domain. Therefore (2.1) can be written as,
1 1  d k u (t )  U (k ) = φ (ti , k ) =   k! k!  dt k 
(2.2)
u (t ) = ∑ U (k )(t − t 0 )
If !" # $ % !& # $' Theorem 2
1, - * / ) * + 0 , - . * Theorem 3
Where U (K) is called spectrum of u(t) at in the k –domain and the inverse differential transform of U( k ) is defined as follows k
Theorem 1
If % ( ) *
at t = t0 for all k
∞
2.5
(2.3)
k =o
In the real application, the function u(t) is expressed by a finite series and equation (2.3) can be written as,
If %
!
Theorem 4 If u t g t h t then
∑ & 5 ' 5 Theorem 5 If %
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Narhari Patil, et al., J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012)
′ , 0 0
′ @ 2
ŕ°
4 8-%2 A , 0 2
Theorem 6 If %
ŕł™ ŕ° , ŕł™ ! !
Theorem 7 If y % 7 -8 9:%8 %
ŕł– !
,
Theorem 8 If y sin 6 ! %
! sin 6 ! , Where !, are constant ŕł–
Theorem 9 If cos 6 ! %
! cos 6 ! , Where !, are constant ŕł–
3. NUMERICAL APPLICATION Here the Differential Transform Method applied on some numerical examples to obtaining exact solution of ordinary differential equation. We considered linear and non linear system of ordinary differential equation of first order. ′ ,
0 0
′ @ 6 2
8-%
6
By using above theorems and basic properties of the Differential Transform Method,
6 %
8-%
6 4 8-%2 A , 0 1
′
6 1 6 1 1 6 1 61 Also,
sin 1 6 0 , ( 0
!
sin B C
sin 2 %
ŕł– sin !
6
2 E sin ! 2
The system of (3.1) is transformed as,
1 61 1 1 6 1 F 6 2 61 2 1 1 E 8-% 2 ! 2 2 E 2 sin G ! 2 6 1
6 1 6 1
1 (61
@ ŕł– 8-% 2 ! sin A !
2
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(2.6)
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Narhari Patil, et al., J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012)
Converting it in to a system of differential equations of the first order differential equations Example 3.1 consider the following system of order two with initial condition ′′ 6 H ′′ 4H 0 , 0 0 , ′ 0 1 ′ 6 H ′ 9:8 6 29:82 , H 0 0 , H ′ 0 2 3.1 Considering the four functions,
Thus for k = 0, 1 1, 1 0, 1 2, 1 0
(3.4)
Put k = 2, 3
As 0 0 , ′ 0 1 -*J5- 8 0 0 , 0 1 and H 0, 0 2
3 0, 3
, ,
3 0
By substituting the values % -% 2.4 0
From initial conditions,
∞
(3.2)
The values of , - 1,2,3,4 , ( 0,1,2, ‌ and put (3.2) in (2.6) we get Put k = o, 1 6 1 "-H 8 1 61 1 6 1
1 "-H 8 1 2 61 1 1 6 1 F 2 (61 2 1 1 (E 8-% 2 ! 2 2 (E 2 sin G (! 2 gives 1 0 (3.3)
6 1
Put k = 1, 2 0 , 2 , 2 0 , 2 4 (3.5)
, ′ , H , H ′
0 0 , 0 1 , 0 0 , 0 2
"-H 8 1 0
1 1 F 6 2 61 2 1 1 E 8-% 2 ! 2 2 E 2 sin G ! 2
(3.6) of
3.7
Take i = 1, ∞
8-%
!
+. . .
Take i = 3, ∞
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(3.8)
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Narhari Patil, et al., J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012)
0 6 2
sin2x
Thus sin , 8-%2 , ′ 9:8 , H ′ 29:82
(3.10)
0 0, 0 1
(3.11)
Using above mentioned theorems of differential transform method we transform as 6 1 6 1 6 6 1 M 6 N ,
6 1 M 6 N
(3.12)
with initial condition 0 0, 0 1
(3.13)
Put k = 0, 1 M 0 6 0 N = 0 + 1 = 1
1 M 0 6 0 N 0 6 1 1 Put k = 1, 2 1 , 2 0
Put k = 3, 4 0, 4 Put k = 4, 5 We have
take i = 1,
Example 3.2 Consider the following system of linear differential equations, ′ 6 ith initial conditions
Put k = 2, 3 , 3
∞
(3.9)
∞
take i = 2
, 5
ŕ°ą 6.
. .
16
ŕ°Ż
. . .
Example 3.3 consider the following system of non homogenous differential equations
9:8 , 0 1
, 0 0
, 0 0
3.14
using basic theorem of differential transform method, 6 1
6 1
∞
ŕ°Ż 6 6
1 O 61 1 E cos O PP 0 1 ! 2
1 1 O P , 0 1 61 ! ଵ
ଷ 1 ௾ାଵ ଵ ଶ , ଷ 0 2 (3.15)
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Narhari Patil, et al., J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012)
Put k = 0 in above equations, we get 1 1 ,
1 1 , 1 1
Put k = 1, 2 2 0
Put k = 2 , 3 1 3 6 Put k = 3 , 4 1 4 12
, 2 0 ,
, 3 ,
, 4 0 ,
Put k = 4, 5 , 5 1 5 5! Put k = 5 , 6 ! , 6 0 , 6 0 (3.16) , !
!
By substituting these value of , , in to (3.6) we obtain , , ∞
take i = 1, 1 6 6 6 . . . = take i = 2 ,
ŕ°Ż 6 !
. . . 8-%
take i = 1, 2 6 1 6 0
ŕ°Ż 6
6
ŕ°° 6.
. . = 6 9:8
3.17
Example 3.4 Consider the following differential equation of first order,
′ 2 , ′ , ′ 6 With initial conditions ଵ 0 1 , ଶ 0 1 , ଷ 0 0
3.18
By applying basic theorems of the differential transformation on the system (3.15), (3.16) respectively, we get 2 6 1 5 5 61
1 6 1 1 5 61
ଷ 1
1 1 ଶ 1 ! 1 ௄
ŕŻ&#x;ŕ€ŕŹ´
0 1 , 0 1, 0 0 Gives 0 1 , 0 1, 0 0
(3.19)
(3.20)
take k = 0 in (3.17) we get 1 2 , 1 1 also 5 0 , k = 0 gives 1 1
Put k = 1, 5 0
2 1 , 2
,
Put k = 2, 5 0
3 , 3
,
2
3
Now for ,
(3.21) - 1 ,2, 3 . . . % 0 , 1 , 2
Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)
Narhari Patil, et al., J. Comp. & Math. Sci. Vol.3 (3), 330-337 (2012) ∞
2.
for i = 1 1 6 2 6
4 6 6. . . 2 3
3.
take i = 2
4 1 6 2 6 6 6 . . . 2 3
Take i = 3 1 6 6
6. . . 2
4.
Thus , , (3.22) 4. CONCLUSION In this study, we successfully apply DTM to find the solution of a system of differential equations of first order and any order ordinary differential equations. It is observed that DTM is an effective and reliable tool for the solution of system of ordinary differential equations. The method gives rapidly converging series solution. The accuracy of the obtaining solution can be improved by take in more terms in solution. This method is very effective to solve most of differential system.
5.
6.
7.
8.
9.
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Journal of Computer and Mathematical Sciences Vol. 3, Issue 3, 30 June, 2012 Pages (248-421)