Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation

Home , Jonathan Homer Lane

Applied Mathematics, 2012, 3, 851-856 http://dx.doi.org/10.4236/am.2012.38126 Published Online August 2012 (http://www.SciRP.org/journal/am)

Mohammed S. Mechee1,2, Norazak Senu3 1Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, Malaysia 2Department of Mathematics, College of Mathematics and Computer Sciences, University of Kufa, Najaf, Iraq 3Department of Mathematics, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor, Malaysia Email: [email protected], [email protected]

Received December 29, 2011; revised July 12, 2012; accepted July 19, 2012

ABSTRACT Lane-Emden differential equations of order fractional has been studied. Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced.

Keywords: Fractional Calculus; Fractional Differential Equation; Lane-Emden Equation; Numerical Collection Method

1. Introduction further explored in detail by Emden [7], represents such phenomena and having significant applications, is a Lane-Emden Differential Equation has the following second-order ordinary differential equation with an arbi- form: trary index, known as the polytropic index, involved in k yt yt   fty ,=  gt  ,0<1,0 t  k  (1) one of its terms. The Lane-Emden equation describes a t variety of phenomena in physics and astrophysics, in- with the initial condition cluding aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres,and y 0=Ay , 0= B , thermionic currents [8]. where A, B are constants, f ty,  is a continuous real The solution of the Lane-Emden problem, as well as valued function and gtC0,1 (see [1]). other various linear and nonlinear singular initial value Lane-Emden differential equations are singular initial problems in quantum mechanics and astrophysics, is value problems relating to second order differential equa- numerically challenging because of the singularity be- tions (ODEs) which have been used to model several havior at the origin. The approximate solutions to the phenomena in mathematical physics and astrophysics. Lane-Emden equation were given by homotopy pertur- In this paper we generalize the definition of Lane- bation method [9], variational iteration method [10], and Emden equations up to fractional order as following: Sinc-Collocation method [11], an implicit series solution [12]. Recently, Parand et al. [13] proposed an approxi- k Dyt Dyt fty ,=  gt , mation algorithm for the solution of the nonlinear Lane- t (2) Emden type equation using Hermite functions collo- 0

Copyright © 2012 SciRes. AM 852 M. S. MECHEE, N. SENU areas of mathematical physical and engineering sciences.  ddt t  1 It generalized the ideas of integer order differentiation Dft=d= f I ft. aad1tta  d and n-fold integration. Fractional derivatives introduce  an excellent instrument for the description of general Remark 2.1. From Definition 2.1 and Definition 2.2, properties of various materials and processes. This is the we have main advantage of fractional derivatives in comparison  1 with classical integer-order models, in which such effects Dt=,>1;0 t  <<1 are in fact neglected. The advantages of fractional de- 1 rivatives become apparent in modeling mechanical and and electrical properties of real materials, as well as in the  1 description of properties of gases, liquids and rocks, and It=,>1 t  ;>0. in many other fields (see [17]). 1 The class of fractional differential equations of various types plays important roles and tools not only in ma- In this note, we consider the fractional Lane-Emden thematics but also in physics, control systems, dynamical equations of the in Equation (2). systems and engineering to create the mathematical modeling of many physical phenomena. Naturally, such 3. Analytic Solution equations required to be solved. Many studies on Consider that we are given a power series representing fractional calculus and fractional differential equations, the solution of fractional Lane-Enden differential equa- involving different operators such as Riemann-Liouville tions: operators [18], Erdlyi-Kober operators [19], Weyl-Riesz  operators [20], Caputo operators [21] and Grnwald- Let- n y ta=  nt (3) nikov operators [22], have appeared during the past three n=0 decades. The existence of positive solution and multi- hence positive solutions for nonlinear fractional differential equ-  n 1 n ation are established and studied [23]. Moreover, by Dyt = a t (4)  n n 1  using the concepts of the subordination and superor- n=0  dination of analytic functions, the existence of analytic Theorem: The analytic solution of the IVP(2) satisfied solutions for fractional differential equations in complex the following equation: domain are suggested and posed in [24,25]. a One of the most frequently used tools in the theory of 1 t1 fractional calculus is furnished by the Riemann-Liouville 2  operators (see [22]). The Riemann-Liouville fractional  nn11  atn (5) derivative could hardly pose the physical interpretation  n  n=2 nn11  of the initial conditions required for the initial value   problems involving fractional differential equations. n  ft,= atn gt Moreover, this operator possesses advantages of fast con- n=0 vergence, higher stability and higher accuracy to derive different types of numerical algorithms [26]. proof Definition 2.1. The fractional (arbitrary) order integral Substitute (3) and (4) into Equation (2), we obtain the of the function f of order  >0 is defined by desired equation. The method of power series depends to find the co-  1 t t  efficients a as a function of n and a . Ift =d f . nk n a a  3.1. Linear Lane-Emden Fractional Differential  when a =0, we write Ia ft()= ft ()* (), t where Equation (*) denoted the convolution product (see [22]), 1 1 Consider f ty,= yt in Equation (2) thus t t 2  t=,>t0 and  tt =0, 0 and    t  k 1 Dyt Dyt yt= gt  (6) as   0 where  t is the delta function. tt 2 Definition 2.2. The fractional (arbitrary) order deri- with the initial condition vative of the function f of order 0<  1 is defined by y 0=Ay , 0= B ,

Equation (5) convert to the following equation with the initial condition

1 23t y 0=Ay , 0= B , at a t 01 1   n Consider the solution of FDE is yt=  n=0 atn  nn11 Consequently,we have aktn (7)  n  113 n=2 nn11    222 at01 a t t   n  atn2 = gt n=2   3 nn11 n In case gt = 0, we obtain aa01==0 and in general 2 akn t (13) nn11 n=2 1 n  n aann= 2 2 nn11   kn  1  (8)  3  n forn = 2,3, 2 atn2 = gt n=2 Examples with Example 3.1.1.1 1 31 nn Let =,=, we pose the linear FDE 2 22 aann= 2 1  (14) 311k nnkn1    Dyt222 Dyt tyt= gt  (9) 2 t forn = 2,3, with the initial condition y 0=Ay , 0= B , 4. Numerical Collocation Method  n Collocation method for solving differential equations is Consider the solution of FDE is yt=  n=0 atn Consequently,we have one of the most powerful approximate methods for solving fractional differential equations. This method has 1  its basis upon approximate the solution of FDE by a 13t 2 22 series of complete sequence of functions, in which we at01 a t 1 mean by a complete sequence of functions, a sequence of  2 linearly independent functions which has no non zero function perpendicular to this sequence of functions In  3 general, y(t) is approximated by nn11 n akt2 (10)  n 11 n n=2  yt= a t (15) nn   ii  22 i=0 3  n where a for i=0,1,2, ,n are an arbitrary constants  at2 =() gt i  n2 to be evaluated and  t for in=0,1,2, , are n=2 i   given set of functions. Therefore, the problem in Hence Equation (6) of evaluating y(t) is approximated by (16) 11 then is reduced to the problem of evaluating the co- nn  22 efficients ai for in=0,1,2 , . aann= 2 Let tt,,t,, t is a partition to interval [0,1] and 11 (11)  012 n nnkn1    1 22 tj= h and h = and jn=0,1,2, , j n forn = 2,3, Define Example 3.1.1.2 k = DDt  2 (16) 3  Let =,=1, we get the linear FDE t 2 Hence 31 k nn Dyt22 Dyt tyt= gt  (12)  1 atii = ai  i t (17) t 2 ii=0 =0

Consider the solution of Equation (6) as following with the initial condition yy0=0, 0=0 n Hence y xABxx=  i (18)   i ii11  i=2  xxkxii=   ixi2 ii11  operating by  we obtain   n See Table 2 and Figure 2, where the exact solution is yx=1 A B  x  xi  i=2  3 yt  = t23 t and =,=1. hence 2

iiii11  ii2  xxkx= x Table 1. Absolute error of numerical solution of Example ii11    4.1. put x = x j we get nt\ 0 0.25 0.5 0.75 1 n i 5 0 1.3345e−3 0.0015 5.0673e−3 3.6339e−3 i=2ijtgtA=1 j Bx    10 0 1.3232e−5 2.6342e−5 1.5634e−6 4.1443e−5 A linear system Ax = b of n – 1 equations in n – 1 50 0 2.3416e−7 1.6611e−7 5.1126e−7 2.1233e−7 i variables is obtained and axij=   j bgtj =  j  for 100 0 4.9383e−8 3.4453e−8 5.0347e−8 6.4332e−7 ij, = 2,3, , n 1 . Hence, from Equation (6) we obtain the linear system Graph of the function f(t)=t3−t2 Ax = b which could be solved by using any numerical 0 method for solving linear system of algebraic equations. −0.1

−0.2 Numerical Examples −0.3 To implement our examples, we used Matlab R2009b on −0.4 Intel(R)core TM2Duo processor with 3.00 GHZ and 3 −0.5 GB RAM. f(t) Example 4.1.1 −0.6

−0.7 k 1 Dyt Dyt 2 yt tt −0.8  44 k  t 2 −0.9 =6 t  (19)  44  6 −1   0 0.2 0.4 0.6 0.8 1 0 ≤ t ≤ 1 (3 ) k   3  t 2  2t 2 Figure 1. Numerical and analytic graph of solution of Ex- 33  2 ample 4.1.   with the initial condition yy0=0,  0=0 Graph of the function f(t)=t2−t3 Hence 1 0.9 iiii11  ii2  xxkx=.x 0.8 ii11  0.7 See Table 1 and Figure 1, where the exact solution is 31 0.6 yt= t3 t2 and =,=.  0.5 22 f(t) Example 4.1.2 0.4

k 1 0.3 Dyt Dyt 2 yt tt 0.2 2  33 k  t 0.1 =2  (20)   33  2 0  0 0.2 0.4 0.6 0.8 1 0 ≤ t ≤ 1 2  44 k  t 2 6ttFigure 2. Numerical and analytic graph of solution of Ex- 44  6   ample 4.2.

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