Solution of Nonlinear ordinary Differential Equations by New Sumudu Variational Iteration Method

Solution of Nonlinear ordinary Differential Equations by New Sumudu Variational Iteration Method

Babiker Y . Ali1 and Tarig M. Elzaki2

1 Mathematics Department, Faculty of Sciences, Sudan University of Science and Technology, Khartoum-Sudan

2Mathematics Department, Faculty of Sciences and Arts-Alkamil, University of Jeddah, Jeddah- Saudi Arabia.

Abstract:- In this work we propose a combined Sumudu transform (ST) and the variational iteration method (VIM) to solve nonlinear ordinary differential equations. The elegant coupling is called the Sumudu transform variational iteration method (STVIM). The strategy is outlined and then illustrated through a number of test examples.it is possible to find the exact solutions or better approximate solutions of these equations. In this method, a correction functional is constructed by a general Lagrange multiplier, which can be identified via variational theory. The solutions obtained by this method show the accuracy and efficiency of the method.

1. INTRODUCTION

   In the last two decades, many analytical approximate methods have been presented to solve nonlinear ordinary differential equations. Most of these problems generally occur commonly in many areas of engineering, physics, chemistry, and applied mathematics. Recently, many researchers have introduced various methods to obtain approximate solutions for nonlinear differential equations (NDEs), such as variational iteration method (VIM), which was developed by JiHuan He for solving linear, nonlinear initial and BVPs (He, 1997; He, 1999; He, 2000.

   Motivated and inspired by Wus thinking, and combing with the Sumudu transform (ST), we give a new modified variational iteration method (VIM),which is based on variational iteration theory and Sumudu transform (ST).The balance in this paper is as follows: the Sumudu transform (ST), variational iteration method (VIM), and the combination of Sumudu transform (ST) and variational iteration method (VIM) are presented in sections 2, 3, and 4. In section 5, numerical application of the method is illustrated by two test examples to demonstrate the efficiency of the method. Section 6 includes a conclusion that briefly summarizes the results.

2. SUMUDU TRANSFROM (ST)

   Watugula (Watugula,1993) introduced Sumudu transform as a new integral and Sumudu transform is defined over the set of functions,

   by,

   or equivalently,

   ||

   = {()\, 1, 2 > 0, |()| < , (1) × [0, )}

   0

   () = {()} = () , (1, 2)

   (1)

   () = {()} = 1 (), ( , )

   (2)

   0 1 2

   The inverse of Sumudu transform of function () is denoted by symbol 1[()] = () and is defined with Bromwich

   contour integral by

   1

   1[()] = () = lim

   +

   ()

   2

   Proposition 1 It deals with the effect of the differentiation of the function(), times on the Sumudu transform () if

   {()} = () then

   (i) (()) = 1 () 1 (0)

   (ii) (()) = 1 () 1 (0) 1 (0)

   2 2

   =0

   (iii) (()) = (() 1 (0))

   Where (0)(0) = (0) and ()(0); = 1,2,3, , 1

   are the kth-order derivatives of the function () evaluated at = 0

   Proposition 2 Let () and () be a functions with the Laplace transforms () and () respectively and Sumudu transform

   () and (), respectively. Then the Sumudu transform of the convolution of and .

   ( )() = ()( )

   0

   is given by: (( )()) = () ()

3. THE VARIATIONAL ITERATION METHOD (VIM)

   To clarify the base idea of the Variational Iteration Method (VIM). Consider the nonlinear differential equation

   + = (), (3)

   Where and are linear and nonlinear operators respectively, and () is the source inhomogeneous term. We can construct a

   correction functional according to the variational iteration method (VIM) for Eq. (3) as follows:

   +1() = () + 0 ()(() + () ()) , 0 (4) Where is a general Lagrange multiplier, which can be identified optimally via the variational theory, is the approximate solution and is a restricted variation which means = 0.

4. SUMUDU TRANSFORM VARIATIONAL ITERATION METHOD (STVIM)

   In a wide range of problems that appear in the literature, the general form of Lagrange multiplier is found to be of the form:

   = ( ).

   In this section, we will make the assumption that is expressed in this latter way. In such a case, the integration is basically the

   convolution; hence Sumudu transform (ST) is appropriate to use. Applying Sumudu transform (ST) on both sides of (4) the

   correction functional will be constructed in the following manner:

   (+1()) = (()) + (0 ()(() + () ()) ), 0 (5)

   Therefore

   (+1()) = (()) + (() (() + () ())) (6)

   (+1()) = (()) + (()) (() + () ()) (7) To find the optimal value of ( ) we first take the variation with respect to () .Thus

   (

   ()) =

   ( ()) +

   (()) ( () + () ()) (8)

   +1

   And hence upon applying the variation this simplifies to

   (+1()) = (()) + (()) (()) (9) We assume that is a linear differential operator with constant coefficients given by

   () = () + 1(1) + 2(2) + + 2 + 1 + 0, (10)

   Where are ,s constants. It is important to note that if the coefficients contain only non-constant terms , then the Sumudu

   variational approach is still valid.

   The Sumudu transform of the first term of the operator is given by

   (()) =

   ()

   =0

   1 ()(0), (11)

   so the variation with respect to is

   (()) =

   (). (12)

   The other term in the operator , namely 1(1) + 2(2) + 2 + 1 + 0, yields similar results. Hence using

   Eq. (12) , Eq. (9) reduces to

   (+1()) = (()) + (()) (1

   ) ( ()), (13)

   =0

   =0

   (+1()) = [1 + (()) (1

   )] (()). (14)

   The extremum condition of +1 requires that +1 = 0. This means that the right -hand side Eq. (14) should be set to zero.

   Hence, we have the stationary condition

   1

   (()) = 1

   =01

   (15)

   Taking the Sumudu inverse of the last equation gives the optimal value of . For this value of , we have the following

   formulation:

   (+1()) = (()) + (0 ( )(() + () ()) ), 0 (16)

5. APPLICATIONS

   In this section, we apply the Sumudu variational iteration method for solving nonlinear ordinary differential equations.

   Example1. Consider the following nonlinear differential equation

   (5)() = 2(), (17)

   (0) = 1, (0) = 1, (0) = 1, (1) = , (1) =

   The exact solution for the given differential equation is =

   For this case the Sumudu variational iteration correction functional will be constructed in the following manner:

   0

   +1() = () + ( ) ((5)() (2)()) , 0 (18)

   Next, by applying Sumudu transform, we have:

   0

   (+1()) = (()) + ( ( ) ((5)() (2)()) ) , 0 (19)

   or equivalent, by applying the convolution property, we get:

   (+1()) = (()) + () ((5)() (2)()), (20)

   (+1()) = (()) + () [

   (21)

   1

   5

   (())

   (0 )

   5

   (0)

   4

   (0)

   3

   (0)

   2

   (4)(0)

   ( (2)())].

   Applying the variation on the Eq. (21), we get

   (0 )

   (0)

   (0)

   (4)(0)

   (

   ()) =

   ( ()) +

   (()) [ 1 ( ()) (0)

   +1

   5

   5 4

   3 2

   ( (2)())]. (22)

   By simplifying Eq. (22), we get

   1

   4

   (+1()) = (()) + (())

   1

   (()) (23)

   (+1()) = (()) [1 + 4 (())] (24) The extremum condition of +1 requires that +1 = 0, then

   ( ()) [1 + 1 (())] = 0 (25)

   4

   (()) = 4 (26)

   Applying the inverse Sumudu transform, we get:

   () = 1 4 (27)

   24

   Substituting Eq. (27) into Eq. (19), we get

   1

   4 (5)

   (2)

   (+1()) = (()) 24 (0 ( )

   or

   (

   ()

   ()) ) , 0 (28)

   ( ()) = ( ()) 1 (

   +1

   1

   24

   4) (

   (5)

   ()

   (2)

   ()), (29)

   Suppose that () = (0) + (0) +

   0

   2!

   2(0) + 1

   3!

   3(0) + 1

   4!

   4(4)(0)

   Therefore

   = 1 + + 1 2 + 3 + 4

   2!

   ( ()) = (1 + + 1 2

   1

   2!

   (30)

   + 3

   +

   4) 1 (

   24

   4) (

   (1 + + 1 2

   2!

   + 3

   +

   2

   4) ),

   Applying the inverse of Sumudu transform, the resulting expression for 1 is too kong so we choose not to include it. Upon using the two conditions (1) = and (1) = , the values of and are respectively = 0.1666728160 and = 0.04166338259.

   The following Taylor series expansion of 1 was obtained:

   1 = 1.00000 + 1.000006 + 0.52 + 0.16667281603 + 0.041663382594 +

   1

   120

   5 + , (31)

   which matches highly accurately with Taylor expansion of the exact solution of the given nonlinear differential equation.

   Example2. Consider the following nonlinear differential equation

   () 2() + 2() = 1 + ( 2), (32)

   (0) = 0.

   The exact solution for the given differential equation is = 1 + 2 tanh (2 tanp (2 ))

   2

   For this case the Sumudu variational iteration correction functional will be constructed in the following manner:

   0

   +1() = () + ( ) (() 2() 1 ( 2) + (2)()) , 0 (33)

   Next, by applying Sumudu transform, we have:

   0

   (+1()) = (()) + ( ( ) (() 2() 1 ( 2) + (2)()) ) , 0

   (34)

   or equivalent, by applying the convolution property, we get:

   (+1()) = (()) + () (() 2() 1 ( 2) +

   (2)

   ()), (35)

   (+1()) = (()) + () [(

   Applying the variation on the Eq. (36), we get

   1

   2) (())

   (0 )

   1

   1

   2

   ( (2)())]. (36)

   (

   ()) =

   ( ()) +

   2

   (()) [( 1 2) ( ()) (0 ) 1 1

   ( (2)())].

   (37)

   +1

   By simplifying Eq. (37), we get

   1

   (+1()) = (()) + (()) (

   2) (()) (38)

   (+1()) = (()) [1 + (1 2) (())] (39) The extremum condition of +1 requires that +1 = 0, then

   (()) [1 + (1 2) (())] = 0 (40)

   (()) = 1

   12

   Applying the inverse Sumudu transform, we get:

   (41)

   () = 2 (42)

   Substituting Eq. (42) into Eq. (34), we get

   0

   (+1()) = (()) ( 2() (() 2() 1 ( 2) + (2)()) ) , 0

   (43)

   (+1()) = (()) (2) (() 2() 1 ( 2) + (2)()), (44) Suppose that 0() = (0) = 0, since (0) = 0 , then :

   (1()) = (0()) (2) (0() 20() 1 ( 2) + 0(2)()), (45)

   1

   2

   (1()) = 0 + 12 (1 +

   or

   ), (46)

   1 2

   (1()) = 12 + 12

   (47)

   Applying the inverse of Sumudu transform, we have

   () = 1 2

   1

   2

   1 +

   2

   24

   (1 (2 )) (48)

   Where is the Heaviside unit step function. In a similar fashion we obtain the higher iterates, for instance the second iterate is

   3

   1 4

   1

   28

   38

   32

   1

   2 1

   2() = 8 8 + 2

   + sinh

   sinh( 2)( 2) + (1 + )

   2

   + (1 (2 ))(2

   2

   1)24 + (1 (2 ) 1) 48 . (49)

   2

   The first few terms of Taylor expansion of 2, for 2, is

   1

   2() = + 2 +

   3

   1

   3

   3

   4

   which matches with the exact solution of the given nonlinear differential equation.

6. CONCLUSION

In this paper, Sumudu transform variational iteration method has been efficiently applied for solving nonlinear ordinary differential equations to give rapid convergent successive approximations without any linearization, discretization or restrictive assumptions that may change the physical behavior of the problem and absorb the positive features of the coupled techniques.

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