Selçuk J. Appl. Math. Selçuk Journal of Vol. 9. No.1. pp. 69-76, 2008 Applied Mathematics
The Differential Transform Methods For Nonlinear Functions And Its Applications
Yıldıray Keskin and Galip Oturanç
Research Center of Applied Mathematics Selcuk University Campus, 42003, Konya, Turkey
e-mail: yildiraykeskin@ yaho o.com , goturanc@ selcuk.edu.tr
Received: February 25, 2008
Abstract. In this study, new differential transform methods were presented to solve some nonlinear functional. The suggested methods are convenient for computational purpose because of its simple computer coding and its providing detailed solutions. The algorithms of the new methods were applied to the different types of nonlinear functional models.
Key words: Differential transformation, differential inverse transformation, Emdem-Fowler differential equation, nonlinear differential equation.
2000 Mathematics Subject Classification. 26D15, 74G15, 35A22. 1.Introduction
Differential transform method (DT) is a numerical method for solving differen-tial equations or system of the differendifferen-tial equations [1-7]. Differendifferen-tial transform method recently had interest because of some important applications for solving engineering problems [1-7]. There is no known method in literature for solving differential equations which have nonlinear dependent variable by transform method. This paper presents new formulae for solving these problems i.e. the nonlinear Emden—Fowler and Lane-Emden [8-12] type differential equations. In the next section we begin by introducing the definitions of the differential transform methods. In section 3, we give new formulae for solving differential equations by transform methods. In section 4, we show how to apply the new formulae using for ordinary differential equations. Finally, we conclude this paper with a brief discussion in section 5.
2. Basic Definitions
In this sections, we shall state some definitions and transform table which will be needed in sequel. For detail we refer to Refs.[1-7]
The differential transform of the function () is defined as follows [4]: (1) () = 1 ! ∙ () ¸ =0
where () is the original function and () is the transformed function. Here
means the th derivative with respect to .
The differential inverse transform of () is defined as
(2) () =
∞
X
=0
() Combining (1) and (2) we obtain
(3) () = ∞ X =0 1 ! ∙ () ¸ =0
From above definitions it is easy to see that the concept of differential transform is derived from Taylor series expansion. With the aid of (1) and (2) the basic mathematical operations are readily be obtained and given in Table 1.
O r i g i n a l F u n c t i o n T r a n s f o r m e d F o r m () = ()± () () = ()± () () = · () () = · () ( w h e r e i s a c o n s t a n t ) () = () () =(+)!! · ( + ) () = ()· () () = ()⊗ () = =0 ()· ( − ) () = () = (− ) = 1 = 0 () = () () = (− )
Table 1. Basic operations of the differential transformation
3. The Differential Transformation of Nonlinear Functional Differential transform of nonlinear function () is defined as follows:
(4) () = 1 ! ∙ () ¸ =0
where () is the original nonlinear function and () is the transformed function. Differential inverse transform of () is defined as
(5) () =
∞
X
=0
() From (4) and (5) we get
() = ∞ X =0 1 ! ∙ () ¸ =0 = ∞ X =0 1 ! " Ã∞ X =0 () !# =0
which implies that the concept of differential transform is derived from Taylor series expansion, but the method does not evaluate the derivatives symboli-cally. However, relative derivatives are calculated by an iterative way which is described by the transformed equations of the original functions. From the definition of (4) and (5), it is easy to prove that the transformed functions com-plete with the basic mathematical operations shown in the Table 2. In actual applications, the function () is expressed by a finite series and (5) can be written as () = X =0 ()
Here is decided by the convergence of natural frequency in this study. The nonlinear differential transform does not require a formula for each polyno-mial. The elementary expansion operations algebraic, trigonometric or Taylor expansion, are the only operations needed. The algebraic computation lan-guages such as Maple may be used to facilitate the computational work. The nonlinear differential transform will be clarified by discussing the following suit-able forms of nonlinearity. The applications of suggested methods were pre-sented for following different cases.
Case 1: If () = 1(()) + 2(()) then () = 1() + 2() where
1(()) = ∞ X =0 1() = 1 ! " 1 Ã∞ X =0 () !# =0 and 2(()) = ∞ X =0 2()= 1 ! " 2 Ã∞ X =0 () !# =0 From definition of transform in (4), we have
() = 1 ! h (1(()) + 2(())) i =0 = 1 ! ∙ 1 µ∞ P =0 () ¶¸ =0 +!1 ∙ 2 µ∞ P =0 () ¶¸ =0 where () = 1() + 2() Case 2: If () = 1(())2(()) then () = P =0 1()2( − ) where 1(()) = ∞ X =0 1() = 1 ! " 1 Ã∞ X =0 () !# =0 and 2(()) = ∞ X =0 2()= 1 ! " 2 Ã∞ X =0 () !# =0
From definition of transform in (4), we have () = 1 ! h 1(())2(()) i =0 = 1 ! ∙ 1 µ∞ P =0 () ¶ 2 µ∞ P =0 () ¶¸ =0 where () = P =0 1()2( − ) Case 3: If () = 2() then () = 1 ! " µ∞ P =0 () ¶2# =0 We first set (6) () = ∞ X =0 ()
Substituting (6) into () = 2() gives
(7) () = ( (0) + (1) + (2)2+ (3)3+ )2
The expansion in (7) can be rearranged by grouping all terms with the sum of the subscripts of the components of is the same. This means that we can
rewrite (7) as () = 2(0) | {z } (0) + 2 (0) (1) | {z } (1) + (2 (0) (2) + 2(1)) | {z } (2) 2 +(2 (0) (3) + 2 (2) (1)) | {z } (3) 3+ (2 (0) (4) + 2 (3) (1) + 2(2)) | {z } (4) 4+ )2
This gives differential transformation of () = 2()
(0) = 2(0) (1) = 2 (0) (1) (2) = 2 (0) (2) + 2(1) (3) = 2 (0) (3) + 2 (2) (1) (4) = 2 (0) (4) + 2 (3) (1) + 2(2) .. .
Case 4: If () = sin(()), then () = 1 ! ∙ sin µ∞ P =0 () ¶¸ =0 This gives differential transformation of () = sin(())
(0) = sin( (0)) (1) = cos( (0)) (1)
(2) = cos( (0)) (2) −2!1 sin( (0))2(1)
(3) = cos( (0)) (3) − sin( (0)) (2) (1) −3!1 cos( (0))3(1)
.. .
For each case we presented for each functional and their transform forms and general solution MAPLE code in Table 2.
Nonlinear Function Transform ed Form
() = () (0) = (0) (1) = −1(0) (1) (2) =1 2(− 1)−2(0) 2(1) + (0)−1 (2) (3) =1 6(− 1)( − 2)−3(0) 3(1)+ (− 1)−2(0) (2) (1) + −1(0) (3) . . . () = cos(()) (0) = cos( (0)) (1) =− sin( (0)) (1) (2) =− sin( (0)) (2) − 1 2!cos( (0)) 2(1) (3) =− sin( (0)) (3) − cos( (0)) (2) (1)+ 1 3!sin( (0)) 3(1) . . . () = () (0) = (0) (1) = (1) (0) (2) = (2) + 1 2! 2 (1) (0) (3) = (3) + (1) (2) + 1 3! 3 (1) (0) . . . () = ln(()) (0) = ln( (0)) (1) = (1) (0) (2) = (2) (0)−1 2 2 (1) 2 (0) (3) = (3) (0)− (1) (2) 2(0) + 1 3 3 (1) 3 (0) . . . () M aple C o de restart;
N F:= N (y(x)): # N onlinear Function m := 5: # O rder
y[x]:= sum (y[b]*x^b,b= 0..m ): N F[x]:= subs(y(x)= y[x],N F): s:= expand(N F[x],x): dt:= unapply(s,x): for i from 0 to m do n[i]:= ((D @ @ i)(dt)(0)/i!):
print(N [i],n[i]);# Transform Function o d:
Table 2. The Differential Transformation of Nonlinear Functional and MAPLE codes In this section, we give two numerical examples for demonstrating the validity of the method provided in the previous section
Example 1: Consider the nonlinear differential equation
with initial condition
(9) (0) = 0
The exact solution of the system (8) is known () = ln(1 + )
Then, by using the basic properties of the transformation form Table 1 and Table 2, we can find the transformed from of equation (4.2) as
(10) ( + 1) ( + 1) = ()
where () and () are the transformations of the functions () and () = ()respectively. Using the initial condition (4.2), we have
(11) (0) = 0
Now, substituting (11) into (10), we obtain the following () values succes-sively, (1) = (0)1 = 1 (0) | {z } (0) = 1 (2) = (1)2 =1 2 µ − (1) (0) ¶ | {z } (1) = −1 2 (3) = (2)3 =13 µ 2(1) 2 (0) − (2) (0) ¶ | {z } (2) = 13 (4) = (3)4 =14 µ − 3(1) 6 (0) + (1) (2) (0) − (3) (0) ¶ | {z } (3) = −14 .. . () = (−1)+1
Finally, using the inverse transformation of { ()}∞=0, we find
(12) () = ∞ X =0 ()= ∞ X =0 (−1)+1
Hence the closed form of (12) is
() = ln(1 + )
Example 2: Consider the nonlinear Emden—Fowler type equation [11,12]
(13) ” + 5
with initial condition
(14) (0) = 0 0(0) = 0
The exact solution of the equation (13) is known () = −2 ln(1 + 2)
Then, by using the basic properties of the transformation from Table 1 and Table 2, we can find the transformed form of equation (4.2) as
( − 1) ⊗( + 2)!! ( + 2) + 5( + 1) ( + 1) = −( − 1) ⊗ () or
(15) ( + 1) ( + 1) + 5( + 1) ( + 1) = −( − 1)
where () and () are the transformations of the functions () and () = 8³()+ 2()2
´
respectively. Using the initial condition (14), we have
(16) (0) = 0 (1) = 0
Now, substituting (16) into (15), we obtain the following () values succes-sively, (2) = − (0)12 = − 1 12 µ 8 (0)+ 16 (0)2 ¶ | {z } (0) = −2 (3) = − (1)21 = − 1 21 ³ 8 (0) (1) + 8 (0)2 (1) ´ | {z } (1) = 0 (4) = − (2)32 = − 1 32 ³ 8 (0) (2) + 4 (0)2(1) + 8 (0)2 (2) + 2 (0) 2 2(1) ´ | {z } (2) = 1 (5) = 0 (6) =−2 3 (7) = 0 (8) =1 2 . . .
Finally, using the inverse transformation of { ()}8=0, we find
(17) () ∼= 8 X =0 () = −22+ 4−2 3 6+1 2 8+
Hence the closed form of (17) is
5. Concluding remarks
The main goal of this paper is suggesting the differential transformation method in order to solve nonlinear differential equations. The approximate solutions of the problems can be obtained to any desired order of the suggested method. The numerical scheme gives almost analytical solution depending on the type of the transformation method. It is also showed that the numerical algorithm of this method is easy to compute the necessary coefficients or to set a computer code in order to get as many terms of the series solution as we need.
Acknowledgements
This study was supported by the Coordinatorship of Selcuk University’s Scien-tific Research Projects (BAP).
References
1. Keskin,Y. , Kurnaz, A., Kiris, M. E. ,Oturanc G. (2007): Approximate Solutions of Generalized Pantograph Equations by the Differential Transform Method, Interna-tional Journal of Nonlinear Sciences and Numerical Simulation, 8(2), 159-164. 2. Bildik N., Konuralp A. (2006): The Use of Variational Iteration Method, Differential Transform Method and Adomian Decomposition Method for Solving Different Types of Nonlinear Partial Differential Equations,International Journal of Nonlinear Sciences and Numerical Simulation, 7(1), 65-71.
3. Abdel-Halim Hassan I.H. (2002): Different applications for the differential trans-formation in the differential equations, Applied Mathematics and Computation 129 (2-3), pp. 183-201.
4. Kurnaz A., Oturanç G. (2005): The differential transform approximation for the system of ordinary differential equations, International Journal of Computer Mathe-matics 82 (6) , pp. 709-719.
5. Ayaz F. (2004): Applications of differential transform method to differential-algebraic equations, Applied Mathematics and Computation,152,3,649-657.
6. Liu H. ,Song Y. (2007): Differential transform method applied to high index differential—algebraic equations, Applied Mathematics and Computation, 184(2),748-753.
7. Zhou JK. (1986): Differential Transformation and its Applications for Electrical Circuits, Huarjung University Press, Wuuhahn, China.
8. He Ji-Huan (2003): Variational approach to the Lane—Emden equation, Applied Mathematics and Computation, 143, 2-3, 539-541.
9. Ramos J.I. (2006): Series approach to the Lane—Emden equation and comparison with the homotopy perturbation method, Chaos, Solitons & Fractals.
10. Wazwaz A.M. (2001): A new algorithm for solving differential equations of Lane— Emden type. Appl. Math. Comput. 118, pp. 287—310.
11. Wazwaz A.M. (2005): , Adomian decomposition method for a reliable treatment of the Emden—Fowler equation,Applied Mathematics and Computation, 161(2), 543-560. 12. Ramos J. I. (2005): Linearization techniques for singular initial-value problems of ordinary differential equations, Applied Mathematics and Computation, 161(2), 525-542.
