 Journal of Modern Physics, 2011, 2, 559-563 doi:10.4236/jmp.2011.26065 Published Online June 2011 (http://www.SciRP.org/journal/jmp) Copyright © 2011 SciRes. JMP Solution of Modified Equations of Emden-Type by Differ ential Transform Method Supriya Mukherjee1, Banamali Roy2, Pratik Kumar Chatterjee3 1Department of Mathematics Swami Vivekananda Institute of Science & Techno logy, South Gobindapur, P.S-Sonarpur, Kolkata, West Bengal, India 2Department of Mat hem at i cs Ba ngabasi Evenin g C oll e ge, 19 Rajkumar Chakraborty Sarani, Kolkata, West Bengal, India 3Department of Computer Science and Engineering Swami Vivekananda Institute of Science & Technology, South Gobindapur, P.S-Sonarpur, Kolkat, West Bengal, India E-mail: supriya_ju@yahoo.com, banamaliroy@yahoo.co.in, pratik_kumar_chatterjee@yahoo.in Received February 2, 2011; revised March 24, 2011; accepted April 6, 2011 Abstract In this paper the Modified Equations of Emden type (MEE0), 3xxxx  is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al.  and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method. Keywords: Modified Equations of Emden Type, Differential Transforms Method, Runge-Kutta 4 (RK4) Method 1. Introduction The modified equation of Emden type (MEE), also called the modified Painleve-Ince equation, 30xxxx  (1) where over dot denotes differentiation with respect to time and  and  are arbitrary parameters, have re-ceived attention from both mathematicians and physicists for more than a century [2-6]. The above differential equation appears in a number of mathematical problems such as univalued functions defined by second order dif-ferential equations  and the Riccati equation . Phy-sicists have found this equation in the study of equilibrium configurations of a spherical gas cloud acting under the mutual attraction of its molecules and subject to the laws of thermodynamics [9-12], in spherically symmetric ex-pansion or collapse of a relativistic gravitating mass  and in the modeling of the fusion of pellets . The invariance and the integrability of this equation have been a subject of study for the past two decades by a number of authors [15-26]. This equation have been found to possess an explicit general solution for the following parametric choices, 0, (2a) 0, (2b) 29 (2c) 2 (2d) However, the general solution of Equation (1) for arbi-trary values of  and  was explored for the first time by Chandrasekhar et al. . They have constructed the time-independent Hamiltonians from the time-inde- pendent integrals of Equation (1) and by the suitable use of canonical transformations, have converted these Ham-iltonians to their standard forms. The general solutions are then obtained by integrating these new Hamiltonians. We present here a humble effort to arrive at the same by the Differential Transform Method [DTM]. The concept of differential transform was first intro-duced by Zhou  in solving linear and nonlinear initial value problems in electrical circuit analysis. The tradi-tional Taylor series method takes a long time for compu- 560 S. MUKHERJEE ET AL. tation of higher order derivatives. Instead, DTM is an iterative procedure for obtaining analytic Taylor series solution of differential equations and is much easier. In our previous work we have seen that the DTM provides the solution of the Duffing-Van der Pol oscillator equa-tion in a rapidly convergent series  and that, it is in good agreement with the solution obtained by Chandra-sekar et al. . 2. The Modified Emden-Type Equations As already mentioned, the modified equation of Emden type cannot be integrated straightforwardly for arbitrary values of  and . The solution of MEE for the par-ticular choice of parameters given by (2a) and (2b) can be obtained by simple integration and for the choice (2c), the equation is linearizable to a free particle equation. In the fourth case the general solution can be expressed in terms of the Weierstrass elliptic function [2-6,15-26,30]. It has also been noted that the MEE possess the Painleve prop- erty for certain values of 284r [19, 20,22]. In  the authors have identified the first integrals of Equation (1) separately for each of the three ranges 1) 28, 2) 28, and 3) 28. The Hamilto-nians are obtained from these integrals and are given by 22222142 2224log 8411 8221log sec8244rrpxprrHpxprrxxpxp (3) For the case 28 the Hamiltonian 24log 4Hpxp reduces to the standard form 221log2Hp U32, (4) under the canonical transformation 4,PxU 28Up . (5) The general solution thus obtained by integrating the new Hamiltonian (4) and by using the canonical equa- tions and UP2PU is given by,   21181exp 22xterf zEerf zi (6) where, 02expπtit Ez2, 212log2EU U, 0 is an arbitrary constant of integration and erf is the error function . tIn our present work we have solved the modified equ-ation of Emden type by the Differential transform method and we have compared the results with Equation (6) . We have also compared the results with those obtained by Runge-Kutta 4 Method. 3. The Differential Transform Method Differential transform of a function is defined as follows )(xf 0d1!dkkxfxFk kx. (7) In (7), fx is the original function and Fk is the transformed function. The Taylor series expansion of the function fx about a point is given as 0x 00d!dkkkkxfxxfx kx. Replacing 0d1!dkkxfxkx by Fk, we have 0kkfx xFx (8) which may be defined as the inverse differential trans-form. From (7) and (8) it is easy to obtain the following ma-thematical operations: 1) If fxgxhx, then  FkGkHk 2) If fxcgx, then  FkcGk, where c is a constant. 3) If then ddnngxfxx, then !!knFkGkkn. 4) If fxgxhx 0kl, then FkGlHkl. 5) If fx xn, then  ,Fkkn where  is Kronecker delta.  0d,xfxgtt then  1GkFk k,6) If where 1k. Copyright © 2011 SciRes. JMP S. MUKHERJEE ET AL. 5617) If  fxuxvxwxkks, then  00smFkUsVmWks m. where Fk, Gk, Hk, Uk, Vk, Wk are then differe the tial transform offunctionsfx, gx, hx , ux, vx, wx respectively. 4. Solu oMedeEquations Using Differential Transform tionf the odifi Emdn-Type Method The equation of the modified Emden type is given as 2ddxx320ddxxtt (9) The initial conditions are (0) 0x and (0)x1 (where prime denotes difftime). rm (DT) thave erentiation with respect to Applying Differential Transfoo (9), we 23dd 0xxDTx x 2ddttor 232dd 0ddxxDTDTxDT xtt     , l(10) The inverse differential transform of T(k) is defined as i.e.    00021 2110.klkkssmkkTk TlklTkTsTmTks m   0kkxtTk.t (11) Using the initial conditions (0) 0x; d(0) 1xdt we have, and For e abon, we have (0) 0Tk = 0 in th(1) 1T. ove equati 200 2T T20 1000TTTTT . (12) For k = 1, we have      10632021 10mTT11003.6TTTTTmTmTTTT(13) For k = 2, we have ,0,     230220012 4332040.lssmTlTlTlTsTmTs mT   (14) For k = 3, 4, 5, 6, 7, we have 2530 20T, (15) 60T, (16) 317 272520 105T, (17) 80T, (18) 421395 379291020600 30240T31440and , (19) 10 0T. (20) For the case 28, from Equahave, tions (11) to (19) we 20T, (21) 36T, (22) 40T, (23) 221320 60T530 , (24) 60T, (25) 3317 223172520 10552920T, (26) 80T, (27) 4341395 602591020600 9072000921607351000000T, (28) 10 0T. From Equation (11) we have (29)  0llxttTl 237231t354913660 52920735 1000000xtt ttt (30) Copyright © 2011 SciRes. JMP S. MUKHERJEE ET AL. 562 5. Comparison of Results The solution plot of the Modified Equations otype using DTM is given in Figure 1 for the parametric choice f Emden 28 for different values of . The graph-ical represetion of the solutionDTM in this paper is in good agreemta nd thereby illustrate the reliability and the performance of the differential trans-form method. Figure 2 gives us a comparison of the so-lution for MEE obtained by DTM with the sotained by classical Runge-Kutta 4 Method. Table 1 gives the estimate of absolute error between the DTM-solu-lear from Figures 1, 2 ained by DTM is a bet-nta (30) obtained by the ent with those ob-ined by Chandrasekar et al.  alution ob-tions with RK4 solutions. It is cnd Tab le 1, that the solution obtater approximation to the exact solution (as obtained in ) than the classical RK4 method. Therefore, the DTM is a very efficient and accurate method that can be used to provide analytical solution for nonlinear differential equ- Figure 1. Plot of solution (30) of Modified Equations of Emden type for the case α2 = 8β taking α = 4 and α = 5. Figure 2. Plot of solution of Modified Equations of Emden type for the case α2 = 8β taking α = 4 using DTM [Solid line] and RK4 method [Dotted line]. Table 1. Comparison of the DTM- solutions with RK4 solu-tions and calculation of Absolute error. Time RK4 solutionDTM solution x(t) Absolute error 0 0 0 0 0.1 0.09934 0.42503 0.32569 0.2 0.1948 0.67042 0.47562 0.3 0.28299 0.80679 0.5238 0.4 0.36135 0.87702 0.51567 0.5 0.42833 0.90712 0.47879 0.6 0.48334 0.91287 0.42953 0.7 0.52667 0.90384 0.37717 0.8 0.55915 0.88584 0.32669 0.9 0.58201 0.8624 0.28039 1.4 0.59858 0.71514 0.11656 1 0.59662 0.83567 0.23905 1.1 0.60436 0.80693 0.20257 1.2 0.60652 0.77697 0.17045 1.3 0.60427 0.74628 0.14201 1.5 0.5903 0.68373 0.09343 1.6 0.0.56852 5801 0.65215 0.62048 0.07205 0.05196 1.7 1.8 0.55602 0.58874 0.03272 1.9 0.54293 0.55697 0.01404 2 0.52952 0.52517 0.00435 atio 6. clus The main co this pa to cons ap-proate ansolution Modifieions of Een tyave achis goal ing DThe aof DTfact thaides its withlytical mation. con that posed dal transf is afectivs of solious lineon-lineuationsalso indat a small numbeuments icient to an accurate solupracticplicatios is that, oly a limber of terms are requio be ummed, and he ay be cd owever the method gives more satisfactory r small times which are evident from Figure 1 r results by adding more terms on Equation (30) for longer time in-ns. 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