DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING DIFFERENTIAL EQUATIONS OF LANE-EMDEN TYPE

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1 Mathematical and Computational Applications, Vol, No 3, pp 35-39, 7 Association for Scientific Research DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING DIFFERENTIAL EQUATIONS OF LANE-EMDEN TYPE Vedat Suat Ertür Department of Mathematics, Ondouz Mayıs University, 5539 Kurupelit, Samsun, Turey vsertur@omuedutr Abstract- Using differential transformation method to solve the Lane-Emden equations as singular initial value problems is introduced in this study Some numerical eamples are presented to illustrate the efficiency and reliability of the method Keywords- Differential transformation method, Lane-Emden equations INTRODUCTION Singular initial value problems in the second order ordinary differential equations occur in several models of mathematical physics and astrophysics [-3] such as the theory of stellar structure, the thermal behaviour of a spherical cloud of gas, isothermal gas spheres and theory of thermoionic currents which are modelled by means of the following Lane Emden equation: α u "( ( + f (, u) = g(, <, α, () under the following initial conditions u ( ) = A, u' () = B, () where A and B are constants, f(,u) is a continuous real valued function and g( C[,] Eq () has attracted many mathematicians Wazwaz [4,5] has given a general study to construct eact and series solutions to Lane-Emden equations by employing the Adomian decomposition method Russel and Shampine [6] have investigated threepoint difference methods of second-order Moreover, three-point difference methods of second-order have been also used by Chawla and Katti [7], Chawla et al [8] and Iyengar and Jain [9] However, Jain et al [] derived three-point difference methods of fourth and sith orders to solve this problem On the other hand, El-Sayed [] used a multi-integral method to investigate the nonlinear problem (), and Legendre wavelets method [] has been implemented independently to handle the initial value problem ()-() In this paper, we etend the application of the differential transformation method [3], which is based on Taylor series epansion, to construct analytical approimate solutions of the initial value problem ()-() The concept of differential transformation was introduced first by Zhou [3], and it was applied to solve linear and nonlinear initial value problems in electric circuit analysis With this technique, it is possible to obtain highly accurate results or eact solutions for differential equations This paper is organized as follows: In Section, the differential transformation method is described In Section 3, the method is implemented to three eamples, and conclusion is given in Section 4

2 36 V S Ertür DIFFERENTIAL TRANSFORMATION METHOD The differential transformation of the th derivative of function u( is defined as follows: d u( U ( ) = (3)! d = and the differential inverse transformation of U() is defined as follows: = u ( = U ( )( ) (4) In real applications, function u( is epressed by a finite series and Eq(4) can be written as n u ( = U ( )( ) (5) = Eq (5) implies U ( )( = + ) is negligibly small In fact, n is decided by the n convergence of natural frequency in this study The following theorems that can be deduced from Eqs (3) and (4) are given below [4,5]: Theorem If u ( = y( ± z(, then U ( ) = Y ( ) ± Z( ) Theorem If u ( = ay(, then U ( ) = ay ( ), where a is a constant m m ( m+ )! Theorem 3 If u ( = ( d y( / d ), then U ( ) = Y ( + m)! Theorem 4 If u ( = y( z(, then U ( ) = Y ( ) Z( ) Theorem 5 If u = n (, then = U ( ) = δ ( n), δ ( n) = = n, n 3 NUMERICAL EXAMPLES To demonstrate the method introduced in this study, three eamples are solved here Eample We first start by considering the following Lane-Emden equation given in [] 3 u "( ( + u( = , <, (6) with initial conditions u ( ) =, u' () = (7) By multiplying both sides of Eq (6) by and then taing differential transformation of both sides of the resulting equation using Theorems -5, the following recurrence relation is obtained:

3 Differential Transformation Method for Solving Differential Equations 37 U ( + ) = ( + )( + ) 6δ ( ) + δ ( ) + δ ( 3) + δ ( 4) δ ( l ) U ( (8) By using Eqs (3) and (7), the following transformed initial conditions at = can be obtained: U ( ) =, (9) U ( ) = () Substituting Eqs (9) and () at = into Eq (8), we have U ( ) = () Following the same recursive procedure, we find U ( + ) =, = 3,4,5, K and listing the computation and result corresponding to n = 3, we have U ( 3) = () Using Eqs(9)-) and the inverse transformation rule in Eq (5), we get the following solution: 3 u ( = + (3) Note that for n> 3 one evaluates the same solution, which is the eact solution of Eq (6) with the initial conditions in Eq (7) Eample We net consider the the following Lane-Emden equation given in [] u "( ( + u( = , <, (4) with initial conditions u ( ) =, u' () = (5) By multiplying both sides of Eq (4) by and then taing differential transformation of both sides of the resulting equation using Theorems -5, we obtain the following recurrence relation U ( + ) = ( + )( + 8) δ ( 6) δ ( 5) + 44δ ( 3) 3δ ( ) δ ( l ) U ( (6) We apply the differential transformation at, therefore, the initial conditions given in Eq (5) are transformed as follows: U ( ) =, (7) U ( ) = (8) Substituting Eqs (7) and (8) at = into Eq (6), we have U ( ) = (9) Following the same recursive procedure, we find U ( + ) =, = 4,5, K and listing the computation and result corresponding to n = 4, we have U ( 3) =, () U ( 4) = ()

4 38 V S Ertür Using Eqs (7)-() and the inverse transformation rule in Eq (5), we get the following solution: 3 4 u ( = + () For n > 4, one evaluates that the solution (), which is the eact solution of Eq (4) under the initial conditions in Eq (5) Eample 3 We finally close our analysis by studying the following Lane-Emden equation 5 3 u "( ( + u( = + 3, <, (3) subject to initial conditions u ( ) =, u' () = (4) By multiplying both sides of Eq (3) by and then taing differential transformation of both sides of the resulting equation using Theorems -5, we obtain the following recurrence relation U ( + ) = δ ( 6) + 3δ ( 4) δ ( l ) U ( ( + )( + ) (5) The initial conditions in Eq (4) can be transformed at U ( ) =, (6) U ( ) = (7) Substituting Eqs (6) and (7) at = into (5), we have U ( ) = (8) Following the same procedure, U ( 3) U (5) can be solved as follows: U ( 3) =, (9) U ( 4) =, (3) U ( 5) = (3) For n > 5, by the same way, we have U ( + ) =, = 5,6,7, K By using the inverse transformation rule in Eq (5), we obtain the solution in a closed form by 5 u ( =, (3) which is the eact solution of Eq (3) subject to the initial conditions in Eq (4) 4 CONCLUSION In this study, the differential transformation method is implemented to the Lane- Emden differential equations as singular initial value problems Three equations are solved and eact solutions are obtained It is shown that differential transformation method is a very fast convergent, precise and cost efficient tool for solving the Lane- Emden equations 5 REFERENCES S Chandrasehar, Introduction to the Study of Stellar Structure, DoverPublications, New Yor, 967 HT Davis, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, NewYor, 96

5 Differential Transformation Method for Solving Differential Equations 39 3 O U Richardson, The Emission of Electricity from Hot Bodies, Longman, Green and Co, london, New Yor, 9 4 AM Wazwaz, A new algorithm for solving differential equations of Lane Emden type, Applied Mathematics and Computation 8, 87 3, 5 AM Wazwaz, A new method for solving singular initial value problems in the second-order ordinary differential equations, Applied Mathematics and Computation 8, 45 57, 6 RD Russel, LF Shampine, Numerical methods for singular boundary value problems, SIAM Journal of Numerical Analysis 4, 3-36, M MChawla, CP Katti, A finite-difference method for a class of singular boundary value problem, IMA Journal of Numerical Analysis 4, , MM Chawla, S McKee and G Shaw, Order h method for a singular two point boundary value problem, BIT 6, 38-36, SRK Iyengar, P Jain, Spline finite difference methods for singular two point boundary value problems, Numerische Mathemati 5, , 987 RK Jain, P Jain, Finite difference methods for a class of singular two point boundary value problems, International Jıournal of Computer Mathematics 7, 3-, 989 SM El-Sayed, Multi-integral methods for nonlinear boundary value problems, a fourth-order method for a singular two point boundary value problem, International Jıournal of Computer Mathematics 7, 59-65, 998 S A Yousefi, Legendre wavelets method for solving differential equations of Lane- Emden type, Applied Mathematics and Computation 8, 47-4, 6 3 J K Zhou, Differential Transformation and Its Applications for Electrical Circuits (in Chinese), Huazhong University Press, Wuhan, China, A Arioglu, Đ Özol, Solution of difference equations by using differential transform method, Applied Mathematics and Computation 74, 6-8, 6 5 V S Ertür, S Momani, Comparing numerical methods for solving fourth-order boundary value problems, Applied Mathematics and Computation (7), doi: 6/ jamc675

An Implicit Method for Numerical Solution of Second Order Singular Initial Value Problems

Send Orders for Reprints to reprints@benthamscience.net The Open Mathematics Journal, 2014, 7, 1-5 1 Open Access An Implicit Method for Numerical Solution of Second Order Singular Initial Value Problems