RATIONAL CHEBYSHEV COLLOCATION METHOD FOR SOLVING NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS OF LANE-EMDEN TYPE

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1 INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 6, Number 1, Pages c 2010 Institute for Scientific Computing and Information RATIONAL CHEBYSHEV COLLOCATION METHOD FOR SOLVING NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS OF LANE-EMDEN TYPE KOUROSH PARAND AND MEHDI SHAHINI (Communicated by H. Azari) Abstract. Lane-Emden equation is a nonlinear singular equation that plays an important role in the astrophysics. In this paper, we have applied the collocation method based on rational Chebyshev functions to solve Lane-Emden type equations. The method reduces solving the nonlinear ordinary differential equation to solving a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows that our solutions are highly accurate and the method is convergent. Key Words. Lane-Emden equation, rational Chebyshev, collocation method, nonlinear ODE, semi-infinite interval. 1. Introduction Many science and engineering problems arise in unbounded domains. Different spectral methods have been proposed for solving problems on unbounded domains. A direct approach is to use spectral method associated with orthogonal systems in unbounded domains, such as the Hermite spectral method and the Laguerre method [1, 2, 3, 4, 5, 6, 7]. The second approach is to reformulate original problems in unbounded domains to singular problems in bounded domains by variable transformations, and then using suitable Jacobi polynomials to approximate the resulting singular problems [8, 9, 10]. Another approach is replacing infinite domain with [ L, L] and semi-infinite interval with [0, L] by choosing L, sufficiently large. This method is named as domain truncation [11]. Another effective direct approach for solving such problems is based on rational approximations. Christov [12] and Boyd [13, 14] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of rational functions. Boyd [14] defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping to the Chebyshev polynomials. Guo et al. [15] introduced a new set of rational Legendre functions which are mutually orthogonal in L 2 (0, + ). They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line. Boyd et al. [16] applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine. Received by the editors June 17, 2008 and, in revised form, April 7, Mathematics Subject Classification. 34B16, 34B40, 65M70. 72

2 RATIONAL CHEBYSHEV COLLOCATION TO SOLVE LANE-EMDEN EQUATIONS 73 Parand and Razzaghi [17, 18, 19] applied spectral method to solve nonlinear ordinary differential equations on semi-infinite intervals. Their approach was based on rational Tau method. They obtained the operational matrices of derivative and product of rational Chebyshev and Legendre functions and then they applied these matrices together with the Tau method to reduce the solution of these problems to the solution of a system of algebraic equations. In this paper a collocation method by rational Chebyshev functions is applied to solve Lane-Emden type equations on semi-infinite domains. Many problems in mathematical physics and astrophysics which occur on semiinfinite interval, are related to the diffusion of heat perpendicular to the parallel planes and can be modeled by the heat equation (1) x k d dx or equivalently ( x k dy dx ) + f(x)g(y) = h(x), x > 0, k > 0, (2) y + k x y + f(x)g(y) = h(x), x > 0, k > 0, where y is the temperature. For the steady-state case, and for k = 2, h(x) = 0, this equation is the generalized Emden-Fowler equation [20, 21, 22] given by (3) y + 2 x y + f(x)g(y) = 0, x > 0, subject to the conditions (4) y(0) = a, y (0) = b, where f(x) and g(y) are given functions of x and y, respectively. When f(x) = 1, (3) reduces to the Lane-Emden equation which, with specified g(y), was used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas sphere and theory of thermionic currents. The case f(x) = 1 and g(y) = y m, is the standard Lane-Emden equation that corresponds to the polytropic models. This equation is one of the basic equations in the theory of stellar structure and has been the focus of many studies. This equation describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. It also describes the variation of density as a function of the radial distance for a polytrope. It was first studied by the astrophysicists Jonathan Homer Lane and Robert Emden, which considered the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics. Let us consider a spherical cloud of gas and denote its hydrostatic pressure at a distance r from the center by P. Let M(r) be the mass of the sphere at radius r, Φ is the gravitational potential of the gas and g is the acceleration of the gravity. From these we have the usual equations (5) g = GM(r) r 2 = dφ dr, where G is the gravitational constant. Three conditions are assumed for the determination of Φ and P, namely, (6) dp = gρdr = ρdφ,

3 74 K. PARAND AND M. SHAHINI where ρ is the density of the gas; (7) 2 Φ = 4πGρ, in which 2 = d 2 /dr 2 + 2/r d/dr; (8) P = Kρ γ, where γ and K are empirical constants. From (6) and (8), together with the assumption that Φ = 0, when ρ = 0, it is readily found that ρ = LΦ m, where m = (γ 1) 1 and L = (m + 1)K m. By substituting this value of ρ in (7), we obtain (9) 2 Φ = a 2 Φ m, where a 2 = 4LG. If we now let Φ = Φ 0 y, where Φ 0 is the value of Φ at the center of the sphere and let x (10) r =, aφ 1 2 (m 1) 0 then we will have the Lane-Emden equation with g(y) = y m. The equation has the form (11) y + 2 x y + y m = 0, x > 0, where y and x are dimensionless variables, and m is an index related to the ratio of specific heats of the gas comprising the star. Physically interesting values of m lie in the interval [0, 5]. The conditions associated with (11) are (12) y(0) = 1, y (0) = 0, Exact solutions for Lane-Emden equation, i.e. (11) are known only for the values m = 0, 1, and 5, for other values of m the Lane-Emden equation is to be integrated numerically. The first zero of y gives the radius of the star, so y must be computed up to this zero. The gross properties of the star such as, mass, central pressure, binding energy, etc. can be computed through their relations to y. Inserting f(x) = 1 and g(y) = (y 2 C) 3 2 into (3) gives the white-dwarf equation (13) y + 2 x y + (y 2 C) 3 2 = 0, x > 0, which was introduced by [20] in his study of the gravitational potential of the degenerate white dwarf stars. The boundary conditions of this equation are the same as the Lane-Emden boundary conditions in (12). It is interesting to point out that setting C = 0 reduces the white-dwarf equation to Lane-Emden equation of index m = 3. Different approaches were used to solve Lane-Emden type equations. Bender et al. [23] proposed a perturbative technique for solving nonlinear differential equation such as Lane-Emden, consist of replacing nonlinear terms in the Lagrangian such as y n by y 1+δ and then treating δ as a small parameter. Shawagfeh [24] applied a nonperturbative approximate analytical solution for the Lane-Emden equation using the Adomian decomposition method. His solution was in the form of a power series. He used Padé approximation method to accelerate the convergence of the power series. Mandelzweig and Tabakin [25] used quasilinearization approach to solve Lane- Emden equation. This method approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones,

4 RATIONAL CHEBYSHEV COLLOCATION TO SOLVE LANE-EMDEN EQUATIONS 75 and unlike the perturbation theories is not based on the existence of some kind of small parameters. Wazwaz [26] employed the Adomian decomposition method with an alternate framework designed to overcome the difficulty of the singular point. It was applied to the differential equations of Lane-Emden type. Further he used [27] the modified decomposition method for solving analytic treatment of nonlinear differential equations such as Lane-Emden equation. The modified method accelerates the rapid convergence of the series solution, dramatically reduces the size of work, and provides the solution by using few iterations only without any need to the so-called Adomian polynomials. Liao [28] provided an analytic algorithm for Lane-Emden type equations. This algorithm logically contains the well-known Adomian decomposition method. Different from all other analytic techniques, this algorithm itself provides us with a convenient way to adjust convergence regions even without Padé technique. By the semi-inverse method, He [29] obtained a variational principle for the Lane- Emden equation, which gives much numerical convenience when applying finite element methods or Ritz method. Parand and Razzaghi [19] presented a numerical technique to solve higher ordinary differential equations such as Lane-Emden. Their approach was based on a rational Legendre Tau method. In their work the operational matrices of the derivative and product of rational Legendre functions together with the Tau method were utilized to reduce the solution of these physical problems to the solution of systems of algebraic equations. Ramos [30, 31, 32] solved Lane-Emden equation by different methods. He presented linearization methods for singular initial-value problems in second-order ordinary differential equations such as Lane-Emden. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus they yield piecewise analytical solutions and globally smooth solutions [30]. Later, he developed piecewise-adaptive decomposition methods for the solution of nonlinear ordinary differential equations. Piecewise-decomposition methods provide series solutions in intervals which are subject to continuity conditions at the end points of each interval, and their adaption is based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size [31]. In [32], series solutions of the Lane-Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of the dependent variables. Yousefi [33] used integral operator and converted Lane-Emden equations to integral equations and then applied Legendre wavelet approximations. In his work properties of Legendre wavelet together with the Gaussian integration method are utilized to reduce the integral equations to the solution of algebraic equations. By his method the equation was formulated on [0, 1]. Chowdhury and Hashim [34] obtained analytical solutions of the generalized Emden-Fowler type equations in the second order ordinary differential equations by homotopy-perturbation method (HPM). This method is a coupling of the perturbation method and the homotopy method. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. In

5 76 K. PARAND AND M. SHAHINI this work, HPM yields solutions in convergent series forms with easily computable terms. Aslanov [35] constructed a recurrence relation for the components of the approximate solution and investigated the convergence conditions for the Emden-Fowler type of equations. He improved the previous results on the convergence radius of the series solution. Dehghan and Shakeri [36] investigated Lane-Emden equation using the variational iteration method, and showed the efficiency and applicability of this procedure for solving this equation. Their technique did not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Bataineh et al. [37] obtained analytical solutions of singular initial value problems (IVPs) of the Emden-Fowler type by the homotopy analysis method (HAM). Their solutions contained an auxiliary parameter which provided a convenient way of controlling the convergence region of the series solutions. It was shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. As one more step in this direction, we use rational Chebyshev collocation method to solve singular nonlinear differential equations, i.e. Lane-Emden and white-dwarf equations. The main point of our analysis lies in the fact that there is no reconstruction of the problem on the finite domain. We show that our results have good agreement with exact results, which demonstrate the viability of this approximate method. In this sense, this method has the potential to provide a wider applicability. On the other hand, the comparison of the results obtained by this method and others shows that this method provides more accurate and numerically stable solutions than those obtained by other methods. The organization of the paper is as follows: In Section 2 we explain the formulation of rational Chebyshev functions required for our subsequent development. In Section 3, after a short introduction to essentials of Lane-Emden equation, we summarize the application of rational Chebyshev functions method for solving Lane-Emden and white-dwarf equations. Then, a comparison is made with the existing methods in the literature. Section 4 is devoted to conclusions. 2. Rational Chebyshev interpolation In this section, at first, we introduced rational Chebyshev functions and expressed some basic properties of them. More, we presented approximating a function using Gauss-Radau integration and rational Chebyshev-Gauss-Radau points Rational Chebyshev functions. The well-known Chebyshev polynomials are orthogonal in the interval [-1,1] with respect to the weight function ρ(y) = 1/ 1 y 2 and can be determined with the help of the following recurrence formula: (14) T 0 (y) = 1, T 1 (y) = y, T n+1 (y) = 2yT n (y) T n 1 (y), n 1. The new basis functions, denoted by R n (x), are defined by [14, 11, 41] (15) R n (x) = T n (y) = cos(nt),

6 RATIONAL CHEBYSHEV COLLOCATION TO SOLVE LANE-EMDEN EQUATIONS 77 where L is a constant parameter and (16) (17) y = x L, y [ 1, 1], x + L ( ) x t = 2 cot 1, t [0, π]. L The constant parameter L sets the length scale of the mapping. boyd [38] offered guidelines for optimizing the map parameter L. R n (x) is the nth eigenfunction of the singular Sturm-Liouville problem x ( (18) (x + L) (x + L) xr L n(x)) + n 2 R n (x) = 0, x I = [0, ), n = 0, 1, 2,..., and by (14) satisfies in the following recurrence relation: (19) R n+1 (x) = 2 R 0 (x) = 1, R 1 (x) = x L x + L, ( ) x L R n (x) R n 1 (x), n 1. x + L 2.2. Function approximation. Let w(x) = L/( x(x + L)) denotes a nonnegative, integrable, real-valued function over the interval I = [0, ). We define (20) L 2 w(i) = {v : I R v is measurable and v w < }, where ( (21) v w = 0 ) 1 v(x) 2 2 w(x)dx, is the norm induced by the inner product of the space L 2 w(i), (22) < u, v > w = 0 u(x)v(x)w(x)dx. Thus {R n (x)} n 0 denotes a system which are mutually orthogonal under (22), i.e., (23) < R n, R m > w = c mπ 2 δ nm, with (24) c m = { 2 m = 0, 1 m 1. where δ nm is the Kronecker delta function. This system is complete in L 2 w(i). For any function u L 2 w(i) the following expansion holds (25) u(x) = with + k=0 a k R k (x), (26) a k = < u, R k > w R k 2. w The a k s are the discrete expansion coefficients associated with the family {R k (x)}.

7 78 K. PARAND AND M. SHAHINI 2.3. Rational Chebyshev interpolation approximation. Canuto et al. [39] and Gottlieb et al. [40] introduced Gauss-Radau integration. Further Guo et al. [41] introduced rational Chebyshev-Gauss-Radau points. Let (27) R N = span { R 0, R 1,..., R N }, and (28) y j = cos ( ) 2j + 1 π j = 0, 1,..., N, 2N + 1 are the N + 1 Chebyshev-Gauss-Radau points; thus we define (29) x j = L 1 + y j 1 y j j = 0, 1,..., N, which is named as rational Chebyshev-Gauss-Radau nodes. In fact, these points are zeros of the function R N+1 (x) + R N (x). Using Gauss-Radau integration we have: (30) (31) 0 u(x)w(x)dx = = 1 u(l 1 + y 1 1 y )ρ(y)dy N u(x j )w j u R 2N, j=0 where w j = 2π/(2N + 1), 0 j N 1 and w N = π/(2n + 1) are the corresponding weights with the N + 1 rational Chebyshev-Gauss-Radau nodes. In a collocation method the fundamental representation of a smooth function u on a semi-infinite interval is in terms of its values at discrete Gauss-Radau type points. Derivatives of the function are approximated by analytic derivatives of the interpolating function. The interpolating function is denoted by I N u. It is an element of R N and is defined as N (32) I N u(x) = a k R k (x), k=0 such that I N u(x j ) = u(x j ), 0 j N. I N u is the orthogonal projection of u upon R N with respect to the discrete inner product and discrete norm as: (33) < u, v > w,n = N u(x j )v(x j )w j, j=0 (34) u w,n =< u, u > 1/2 w,n. The Gauss-Radau integration formula implies that (35) < I N u, v > w,n =< u, v > w,n, u.v R 2N. 3. Illustrative examples In this section we apply the collocation method to find solutions for Lane-Emden and white-dwarf equations which satisfy the boundary conditions in (12). For a collocation approximation, by (32)-(35), we approximate y as the truncated series I N y. The solution I N y is represented by its values at the grid points x j introduced in (29). The grid values of I N y are related to the discrete expansion coefficients by (26). Thus, our goal is to find the coefficients a k, 0 k N.

8 RATIONAL CHEBYSHEV COLLOCATION TO SOLVE LANE-EMDEN EQUATIONS 79 Table 1. Comparison the first zero of y for m = 2, present method, perturbation method [23] and exact numerical values [42]. N L Present method Padé Exact value Table 2. Comparison the first zero of y for m = 3, present method, perturbation method [23] and exact numerical values [42]. N L Present method Padé Exact value Table 3. Comparison the first zero of y for m = 4, present method, perturbation method [23] and exact numerical values [42]. N L Present method Padé Exact value Solving Lane-Emden equation. The Lane-Emden equation is introduced in (11) with boundary conditions in (12). As mentioned before, it has exact solutions for m = 0, 1 and 5. Here, we solved it for m = 2, 3 and 4 by collocation method. By applying the above discussion for Lane-Emden equation and using rational Chebyshev-Gauss-Radau points in (29), we have (36) such that (37) (38) ( d 2 I N y dx di N y x dx + I Ny m) = 0, j = 0,..., N 1, x=x j I N y(x 0 ) = 1, di N y(x) dx = 0. x=x0 By omitting the first equation in (36) and with two boundary conditions in (37) and (38), we have N + 1 equations that generate a set of N + 1 nonlinear equations that can be solved by Newton method for unknown coefficients a k s. The first zero of y gives the radius of the star, so y must be computed up to this zero. The approximations of the first zero of y obtained by this method and perturbative technique [23] with exact numerical value [42] for m = 2 are listed in Table 1. Values of the first zero of y, for m = 3 and m = 4 are shown in Table 2 and Table 3, respectively. The comparison of the results with exact values, our solution is more accurate. Horedt [42] has given exact numerical values of y for some optional x. In Table 4 these values are compared with our results for m = 4 and N = 8. It shows that our results are highly accurate. The resulting graph of Lane-Emden equation for N = 7 is shown in Figure 1.

9 80 K. PARAND AND M. SHAHINI Table 4. Comparison of y(x), between present method and exact values given by Horedt [42], for m = 4. x Present method Exact value Figure 1. Lane-Emden graph obtained by present method (N = 7) for m = 2 (dotted line), m = 3 (solid line) and m = 4 (dashed line) y(x) x 3.2. Solving white-dwarf equation. Solving white-dwarf equation (13) is the same as solving the Lane-Emden equation. So we should find function I N y that satisfies ( d 2 I N y (39) dx ) di N y x dx + (I Ny 2 C) 3 2 = 0, j = 0,..., N 1, x=x j and boundary conditions (40) (41) I N y(x 0 ) = 1, di N y(x) dx = 0. x=x0 This function can be determined by its coefficients a k s. These coefficients can be found by solving the set of N + 1 nonlinear equations in (39), for j = 1,..., N 1, (40) and (41). In this paper, this equation is solved for C = 0.2, 0.4, 0.6, 0.8 and 1.0. The graph of white-dwarf equation for N = 6 and L = 0.4 is shown in Figure 2.

10 RATIONAL CHEBYSHEV COLLOCATION TO SOLVE LANE-EMDEN EQUATIONS 81 Figure 2. White-dwarf graph obtained by present method (N = 6, L = 0.4) for C = 0.2 (dashed line), C = 0.4 (dashed-dotted line), C = 0.6 (dotted line), C = 0.8 (spaced-dotted line) and C = 1.0 (solid line) y(x) x 4. Conclusion In the above discussion, we applied the collocation method to solve initial value problems, i.e. Lane-emden and white-dwarf. Lane-Emden equation occurs in the theory of stellar structure and describes the temperature variation of a spherical gas cloud. The white-dwarf equation appears in the gravitational potential of the degenerate white dwarf stars. The difficulty in this type of equations, due to the existence of singular point at x = 0, is overcomed here. In the Lane-Emden equation, the first zero of y is an important point of the function, so we have computed y to this zero. In this paper, this equation is solved for m = 2, 3 and 4, which does not have exact solutions. White-dwarf equation is solved for C = 0.2, 0.4, 0.6, 0.8 and 1.0. The validity of the method is based on the assumption that it converges by increasing the number of collocation points. Here, we tried to apply a method that give more accurate answers without reformulating the equation to bounded domains. Our goal has been achieved by approximations with a high degree of accuracy. References [1] Funaro, D., Computational aspects of pseudospectral Laguerre approximations, Appl. Numer. Math., 6 (1990) [2] Funaro, D. and Kavian, O., Computational aspects of pseudospectral Laguerre approximations, Math. Comp. 57 (1990) [3] Guo, B.Y., Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comp., 68 (1999)

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