ULTRASPHERICAL WAVELETS METHOD FOR SOLVING LANE-EMDEN TYPE EQUATIONS

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1 ULTRASPHERICAL WAVELETS METHOD FOR SOLVING LANE-EMDEN TYPE EQUATIONS Y. H. YOUSSRI, W. M. ABD-ELHAMEED,, E. H. DOHA Department of Mathematics, Faculty of Science, Cairo University, Giza 63, Egypt Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia walee Received April 5, 05 In this paper, a new shifted ultraspherical wavelets operational matrix of derivatives is introduced. The two wavelets operational matrices, namely Legendre and first kind Chebyshev operational matrices can be deduced as two special cases. Two numerical algorithms based on employing the shifted ultraspherical wavelets operational matrix of derivatives for solving linear and nonlinear differential equations of Lane-Emden type are developed. The main idea for obtaining the presented algorithm is essentially based on reducing the linear or nonlinear equations with their initial conditions to systems of linear or nonlinear algebraic equations, which can be efficiently solved. Some numerical examples are given to demonstrate the validity and the applicability of the algorithms. Key words: Lane-Emden equations, ultraspherical polynomials, wavelets, operational matrix, spectral methods, collocation methods. PACS: 0.60.Cb, 0.30.Mv, 0.70.Hm, 0.70.Jn, 0.30.Gp.. INTRODUCTION Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied to a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for wave form representation and segmentations, time frequency analysis and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms, (see [, ]). The application of Legendre wavelets for solving differential and integral equations is thoroughly considered by many authors (see, [3 8]). Also, Chebyshev wavelets are used for solving some fractional and integral equations (see, [9 3]). In recent years, the studies of singular initial value problems (IVPs) for second order ordinary differential equations (ODEs) have attracted the attention of many mathematicians and physicists. One of the equations describing this type is the Lane- Emden type equation which is formulated as y + α x y + f(x,y) = g(x), 0 < x, α 0, () RJP Rom. 60(Nos. Journ. Phys., 9-0), Vol , Nos. 9-0, (05) P , (c) 05 Bucharest, - v..3a*05..0

2 Ultraspherical wavelets method for solving Lane-Emden type equations 99 subject to the initial conditions y(0) = A, y (0) = B, () where A and B are constants, f(x,y) is a continuous real-valued function, and g(x) C[0,]. Eq. () was named after the astrophysicists Jonathan H. Lane and Robert Emden (870), as it was first studied by them. Eq. () 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. Isothermal gas sphere equation is modeled by y + x y + e y = 0, 0 < x, (3) subject to the initial conditions y(0) = 0, y (0) = 0. (4) Singular IVPs of Lane-Emden type have been investigated by a large number of authors. The solution of the Lane-Emden problem, as well as other various linear and nonlinear singular initial value problems in quantum mechanics and astrophysics, is numerically challenging because of the singularity behavior at the origin. The approximate solutions to the Lane-Emden equation were given by many numerical techniques such as homotopy perturbation method [4 6], variational iteration method [7], sinc-collocation method [8], an implicit series solution [9] and a Jacobi-Gauss collocation method [0]. Parand et al. [] obtained another approximate solution based on rational Legendre pseudospectral approach. Recently, some other approximate solutions of Lane-Emden type equations are obtained using perturbation techniques [, 3], optimal homotopy method [4], generalized Jacobi- Galerkin method [5], and third and fourth kinds Chebyshev operational matrices [6]. Spectral methods play prominent roles in solving various kinds of differential equations. It is known that there are three most widely used spectral methods, they are the tau, collocation, and Galerkin methods. Collocation methods have become increasingly popular for solving differential equations, in particular, they are very useful in providing highly accurate solutions to nonlinear differential equations (see, for example [7 34]). One approach for solving differential equations is based on converting the underlying differential equations into integral equations through integration, approximating various signals involved in the equation by truncated orthogonal series and using the operational matrix of integration, to eliminate the integral operations. Spe-

3 300 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha 3 cial attentions have been given to applications of block pulse functions [35], Legendre polynomials [36], Chebyshev polynomials [37], Haar wavelets [38], Legendre wavelets [6, 39], and Chebyshev wavelets [9]. Another approach is based on using operational matrix of derivatives in order to reduce the underlying problem into solving a system of algebraic equations (see, [3]). The ultraspherical polynomials have been used in various applications. For instance, Doha and Abd-Elhameed [40, 4] and Doha et al. [4] have constructed efficient spectral-galerkin algorithms using compact combinations of ultraspherical polynomials for solving second-, high even- and high odd-order elliptic equations. Moreover, Doha and Abd-Elhameed [43] have obtained accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method and they have pointed out that the expansion based on Chebyshev polynomials is not always better than ultraspherical series. The expansion for the solution enables one to get the sought-for approximation for any possible value of the ultraspherical parameter λ >. The main aim of this paper is to develop and implement two algorithms for solving differential equations of Lane-Emden type based on employing shifted ultraspherical wavelets operational matrix of derivatives. The paper is organized as follows. In Sec., we give some relevant properties of ultraspherical polynomials and their shifted forms. Moreover, in this section, ultraspherical wavelets are constructed. In Sec. 3, we establish the operational matrix of differentiation of the ultraspherical wavelets basis. Section 4 is concerned with presenting and implementing a numerical algorithm for solving linear and nonlinear Lane-Emden equations. In Sec. 5, we give a comprehensive study for the error analysis of the suggested wavelets expansion. Section 6 is concerned with considering some numerical examples aiming to illustrate the efficiency and applicability of the developed algorithm. Conclusions are given in Sec. 7.. SOME PROPERTIES OF ULTRASPHERICAL POLYNOMIALS AND THEIR SHIFTED FORMS.. ULTRASPHERICAL POLYNOMIALS The ultraspherical polynomials (a special type of Jacobi polynomials) associated with the real parameter (λ > ) are a sequence of orthogonal polynomials on the interval (-,), with respect to the weight function w(x) = ( x ) λ, i.e. { ( x ) λ C m (x)c n 0, m n, (x) dx = h n, m = n, (5)

4 4 Ultraspherical wavelets method for solving Lane-Emden type equations 30 where π n! Γ(λ + h n = ) n (n + λ) Γ, Γ(n + λ) n =. Γ It is convenient to weigh the ultraspherical polynomials so that C This is not the usual standardization, but has the desirable properties that C n (0) n () =. (x) are identical with the Chebyshev polynomials of the first kind T n (x), C ( ) n (x) are the Legendre polynomials L n (x), and C n () Chebyshev polynomials of the second kind. (x) is equal to Un(x) n+, where U n(x) are the The derivatives of ultraspherical polynomials are given in the following theorem. Theorem. For all q, the qth derivative of the ultraspherical polynomial C k (x) is given explicitly by D q C k (x) = q k! (q )! Γ(k + λ) k q m=0 (k+m q) even ( k m + q ) ( k + m + q + λ ) (m + λ)γ(m + λ)! Γ ( ) k q m ( k + m q + λ + ) C m (x), m!! Γ k q. (6) (For a proof of Theorem, see, Doha [44]). As a direct consequence of Theorem, the first derivative of C n (x) can be easily obtained in the following corollary. Corollary. For all n, one has DC n (x) = n! Γ(n + λ) n k=0 (k+n) odd (λ + k)γ(k + λ) k! C k (x). (7).. SHIFTED ULTRASPHERICAL POLYNOMIALS The shifted ultraspherical polynomials are defined on [0,] by ). All results for ultraspherical polynomials can be easily transformed to give the corresponding results for their shifted forms. The orthogonality relation for C n (x) with respect to the weight function w(x) = (x x ) λ is given by C n (x) = C n (x 0, m n, (x x ) λ C n (x) C m (x)dx = 4 0 λ π n! Γ(λ + ) n (n + λ) Γ, m = n.

5 30 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha 5 Also, the first derivative of C n (x) is given in the following corollary. Corollary. For all n, one has D C n (x) = 4n! Γ(n + λ) n k=0 (k+n) odd (λ + k)γ(k + λ) k! C k (x). (8) 3. SHIFTED ULTRASPHERICAL WAVELETS OPERATIONAL MATRIX OF DERIVATIVES Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets: ( ) t b ψ a,b (t) = a / ψ, a,b R, a 0. (9) a Ultraspherical wavelets ψ nm(t) = ψ(k,n,m,λ,t) have five arguments: k,n can assume any positive integer, m is the order for the ultraspherical polynomial, λ is the known ultraspherical parameter and t is the normalized time. They are defined on the interval [0,] by k+ ( ) C nm(t) = m k+ t (n + ), t [ n, n+ ], Am k k 0, otherwise, ψ where m = 0,,...,M, n = 0,,..., k, and (0) A m = m (m + λ)γ π m!γ(λ + ). () A function f(t) defined on [0,] may be expanded in terms of ultraspherical wavelets as f(t) = c nm ψ nm(t), () where ( ) c nm = f(t),ψ nm(t) n=0 m=0 = w 0 (t t ) λ f(t)ψ nm(t) dt. (3) Assume that f(t) can be approximated in terms of ultraspherical wavelets as: f(t) k M n=0 m=0 c nm ψ nm(t) = C T Ψ (t),

6 6 Ultraspherical wavelets method for solving Lane-Emden type equations 303 where C and Ψ (t) are k (M + ) matrices given by C = [ c 0,0,c 0,,...,c 0,M,c,0...,c,M,...,c k,0,...,c k,m] T, (4) [ ] Ψ (t) = ψ T 0,0,ψ 0,,...,ψ 0,M,ψ,0,...,ψ,M,...ψ k,0,...,ψ k,m. (5) The ultraspherical wavelets operational matrix of the first derivative is stated and proved in the following theorem. Theorem. Let Ψ (t) be the ultraspherical wavelets vector defined in (5). The derivative of the vector Ψ (t) can be expressed by d dt Ψ (t) = DΨ (t), (6) where D is k (M + ) k (M + ) operational matrix of derivative defined as follows F F... 0 D = , (7) F where F is the (M + ) square matrix whose (r,s)th element is defined as follows: k+, r, r > s and (r + s) odd, F r,s = Ar A s 0, otherwise, where A r is as given in (). Proof. If we make use of the shifted ultraspherical polynomials, the rth element of vector Ψ (t) in (5) can be written as Ψ r (t) = ψ nm(t) = k+ C m Am ( ) k t n χ [ n k, n+ ], k i =,,..., k (M + ), (8) where r = n(m + ) + (m + ); m = 0,,...,M; n = 0,,...,( k ), and {, t [ n χ [ n k, n+ k ] =, n+ ]; k k 0, otherwise. If we differentiate (8) with respect to t, then we get dψ r (t) = ψ nm (t) = k+ d [ ( )] C m k t n χ dt Am dt [ n k, n+ k ]. (9)

7 304 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha 7 It is to be noted here that the r.h.s. of (9) is zero outside the interval [ n, n+ ], and k k hence its ultraspherical wavelets expansion only have those elements in Ψ (t) that are nonzero in the interval [ n, n+ ], i.e. Ψ k k i (t), i = n(m + ) +,n(m + ) +,...,n(m + ) + M +, this enables one to expand dψ r in terms of shifted dt ultraspherical wavelets Ψ i (t);i = n(m + ) +,n(m + ) +,...,n(m + ) + M +, in the form dψ r (t) dt = (n+)(m+) i=n(m+)+ a i Ψ i (t), and accordingly (9) implies that the operational matrix D is a block matrix defined as in (7). Moreover, we have and this implies that d dt Ψ r d C 0 (t) = 0, dt (t) (t) = 0 for r =,(M + ) +,(M + ) +,...,( k )(M + ) +. Consequently, the first row of matrix F defined in (7) is zero. Now and with the aid of Corollary, the first derivative of Ψ r (t) may be expressed in the form dψ r (t) = k+ 3 dt Am m j=0 j+m odd (λ + j) C j ( k t n) χ [ n k, n+ Expanding this equation in terms of ultraspherical wavelets basis, we get dψ r (t) dt m = k+ j=0 j+m odd k ]. (0) Ψ n(m+)+j (t). () Ar A j So if we choose F r,s as k+, r=,...,m+, s=,..., r- and (r+s) odd; F r,s = Ar A s 0, otherwise, then Eq. (6) is obtained, and the proof of the theorem is completed. Remark. It is worthy to note here that the ultraspherical wavelets operational matrix given in (7) is considered as a generalization of the Legendre wavelets operational matrix given in [3].

8 8 Ultraspherical wavelets method for solving Lane-Emden type equations 305 Corollary 3. The operational matrix for the nth derivative can be obtained from d n Ψ(t) dt n = D n Ψ(t), n =,,..., () where D n is the nth power of matrix D. 4. LANE-EMDEN DIFFERENTIAL EQUATIONS In this section, we give two numerical algorithms for solving linear and nonlinear differential equations of Lane-Emden type based on employing the shifted ultraspherical wavelets operational matrix of derivatives that was introduced in Sec LINEAR DIFFERENTIAL EQUATIONS Consider the linear Lane-Emden differential equation y (x) + α x y (x) + f(x)y(x) = g(x), y(0) = A, y (0) = B, If we set f (x) = xf(x), then Eq. (3) is turned into 0 < x, α 0. g (x) = xg(x), xy (x) + αy (x) + f (x)y(x) = g (x), y(0) = A, y (0) = B, 0 < x, α 0. If we approximate y(x),x,f (x) and g (x) by the ultraspherical wavelets, then one can write k M y(x) c nm ψ nm (x) = C T Ψ(x), k x n=0 m=0 n=0 m=0 k f (x) M q nm ψ nm (x) = Q T Ψ(x), M n=0 m=0 k g (x) n=0 m=0 f nm ψ nm (x) = F T Ψ(t), M g nm ψ nm (x) = G T Ψ(x), (3) (4) (5)

9 306 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha 9 where C T,Q T,F T and G T are defined similarly as in (4). Relations (7) and () enable one to approximate y (x) and y (x) as y (x) C T DΨ (x), y (x) C T D Ψ (x). (6) With the aid of Eqs. (5) and (6), the residual of Eq. (4) can be written explicitly in the form R(x) =Q T Ψ (x)(ψ (x)) T (D ) T C + αc T DΨ (x) + F T (Ψ (x)) T Ψ (x)c G T Ψ (x). Applying tau method (see [45]), the following ( k (M + ) ) linear equations are generated 0 (7) Ψ j (x)r(x)dx = 0, j =,,..., k (M + ). (8) Moreover, the initial conditions of Eq. (3) yield C T Ψ (0) = A, C T DΨ (0) = B. (9) Thus Eqs. (8)-(9) generate k (M + ) set of linear equations that can be solved for the unknown components of the vector C, and hence the approximate spectral wavelets solution of y(x) can be obtained. 4.. NONLINEAR LANE-EMDEN DIFFERENTIAL EQUATIONS Consider the nonlinear equation y (x) = ϕ ( x, y(x), y (x) ) = α x y (x) f(x,y) + g(x), y(0) = A, y (0) = B, 0 < x, α 0, where f(x,y) is a nonlinear function in the two variables x,y. If we set Θ ( x,y(x),y (x) ) = xϕ ( x,y(x),y (x) ) = αy (x) xf(x,y) + xg(x), then Eq. (30) is turned into (30) xy (x) = Θ ( x,y(x),y (x) ), y(0) = A, y (0) = B, 0 < x, α 0. (3) If we approximate y(x) as in (5) and make use of (7) and (), then after following the same procedure of Sec. 4., we get ( ) xc T D Ψ (x) Θ x,c T Ψ (x),c T DΨ (x). (3) Also, the initial conditions yield C T Ψ (0) = A, C T DΨ (0) = B. (33)

10 0 Ultraspherical wavelets method for solving Lane-Emden type equations 307 To find an approximate solution of y(x), we compute (3) at the first ( k (M +) ) roots of C k (M+) (x). These equations with the two Eqs. (33) generate k (M + ) nonlinear equations in the expansion coefficients, c nm, that can be solved with the aid of the well-known Newton s iterative method. 5. ERROR ANALYSIS In this section, we give a comprehensive study for the error analysis of the suggested ultraspherical wavelets expansion. In this respect, we will state and prove two important theorems, in the first one we show that the ultraspherical wavelets expansion of a function f(x) with a bounded second derivative, converges uniformly to f(x), and in the second one we give an upper bound for the error (in L ω -norm, ω(t) = (t t ) λ ) of the truncated ultraspherical wavelets expansion. The following lemma is needed. Lemma. (see, [46], p. 74) Let f(x) be a continuous, positive, decreasing function for x n. If f(k) = a k, provided that a n is convergent, and R n = a k, then R n n f(x)dx. k=n+ Theorem 3. A function f(x) L ω [0,], 0 < λ <, w(x) = (x x ) λ can be expanded as an infinite series of ultraspherical wavelets, which converges uniformly to f(x), given that f (x) L. Explicitly, the expansion coefficients in (3) satisfy the inequality c nm < 4L( + λ) (m + + λ), n 0, m >. (34) (m ) 4 (n + ) 5 Proof. For the proof of Theorem 3, see, [47]. Note. It is worthy to note here that for ) large values of n and m the expansions coefficients c nm is of O (n 5 m. Note. It is worthy to note here that the result obtained in Theorem 3 may be improved but with some extra conditions on the function f(x). To be concise, if f is differentiable p-times for ) some p > and f (p) (x) L, we can show that c nm is of O (n p m. Theorem 4. If f,λ satisfy the hypothesis of Theorem 3, and if we consider the ultraspherical wavelets expansion f k,m (t) = c nm ψ nm(t), then the k M following n=0 m=0

11 308 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha error estimate (in L ω -norm) is obtained f f k,m ω < ( + λ) L(M + λ), M > 3. (35) 4 k (M 3) 7 Proof. From Eq. (), and making use of the orthonormality property of {ψ nm(t)}, we get f f k,m ω = In virtue of Theorem 3, one can write f f k,m ω < 6L ( + λ) 4 and the application of Lemma leads to f f k,m ω < 6L ( + λ) 4 and hence k n= k m=m n= k M m=m c nm. (m + + λ) 4 (m ) 8 (n + ) 5, (x + + λ) 4 (x ) 8 (y + ) 5 dxdy = (λ + )4 L 4 3 k ( 5λ 5λ + 35λM + (M )M + 9 ) 05(M 3) 7 < ( + λ)4 L (M + λ) 4 k (M 3) 7, f f k,m ω < ( + λ) L(M + λ), 4 k (M 3) 7 which completes the proof of the theorem. Note 3. It is worthy to note here that for large values ) of k and M the expansions coefficients e k,m = f f k,m is of O (M 5 4 k. 6. ILLUSTRATIVE EXAMPLES In this section, we illustrate the two proposed algorithms for solving initial and boundary value linear and nonlinear differential equations of Lane-Emden type, by presenting four examples. 6.. LINEAR LANE-EMDEN EQUATIONS Example. Consider the Lane-Emden equation given in [48] by y + x y + y = 6 + x + x + x 3 ; 0 < x, y(0) = 0, y (0) = 0, (36)

12 Ultraspherical wavelets method for solving Lane-Emden type equations 309 with the exact solution y = x +x 3. We solve Eq. (36) using the algorithm described in Sec. 4. for the case corresponding to k = 0,M = 3. After performing some manipulations, the components of the vector C are given by c 0,0 = A0 + 6λ 6( + λ), c 0, = A 3 + 4λ 6( + λ), and consequently c 0, = A 5 + 0λ 6( + λ), c A3 + λ 0,3 = 6( + λ), y(x) = C T Ψ (x) = x + x 3, which is the exact solution. Example. Consider the Lane-Emden equation given in [49] by y + x y + y m = 0; 0 < x, y(0) =, y (0) = 0. (37) We solve Eq. (37) using the algorithm described in Sec. 4. for the two cases correspond to m = 0,. Case (i) (m = 0): In this case the exact solution of (37) is y = x 6. If we make use of (7) and (), then the two operational matrices D and D are given respectively by and D = D = F, F 3, 0, F, F 3, 0 0 where F, = Γ( + λ) ( + λ) Γ( + λ) ( + λ)( + λ)( + λ) π Γ( + λ), F 3, = π Γ( + λ), moreover the vector Ψ (x) can be evaluated to give Ψ (x) = A (x ) ( + λ)(x ), A3, + λ,

13 30 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha 3 where A m is as given in (). Now, the residual of Eq. (37) is given by R(x) = xc T D Ψ (x) + C T DΨ (x) + x. (38) If we apply the tau method, then we get A c 0, + + λ A3 c 0, + = 0, (39) + λ 8 moreover, the use of the two initial conditions yields c 0,0 A c 0, + A 3 c 0, =, (40) and A c 0, 4 + 4λ A3 c 0, = 0. (4) + λ The solution of the linear system of Eqs. (39)-(4) gives c 0,0 = and consequently λ 48( + λ), c 0, = A, c 0, = ( + λ) 48( + λ) A 3, y(x) = x 6, (4) which is the exact solution. Case (ii) (m = ): In this case the exact solution of (37) is y = sinx x. We solve Eq. (37) using the algorithm described in Sec. 4.. The maximum absolute error E is given in Table for various values of k,m, and λ. The numerical and exact solutions are depicted in Fig., in case of k = 0,M = 3 for some values of λ. Also, aiming to illustrate the super convergence of our algorithm, Eq. (37) is solved by recasting it as a system of first-order differential equations and then applying the following backward differentiation formula (BDF) (see, [50]) y n+ = 4 3 y n+ 3 y n + 3 hf(t n+,y n+ ), y(t 0 ) = y 0. We argue as follows, let y = u, then Eq. (37) is equivalent to the following system of first-order differential equations y = u, y(0) =, xu = u xy, u(0) = 0. Finally, Table displays a comparison between our proposed wavelets solution for the case k = 0,M = 0 and λ =, with the solution obtained by BDF method. The results in this Table show that our method is more accurate if compared with the BDF method.

14 4 Ultraspherical wavelets method for solving Lane-Emden type equations 3 Table Maximum absolute error E for Example case (ii) k M λ E λ E λ E λ E Table A comparison of the wavelets solution with the BDF solution for Example case (ii) x Wavelets solution BDF solution Wavelets error BDF error Exact λ=0 λ=0.5 λ= y(x) x Fig. Numerical and exact solutions of Example case (ii) for k = 0,M = NONLINEAR LANE-EMDEN EQUATIONS Example 3. Consider the Lane-Emden equation given in [49] by y + x y + y 5 = 0; 0 < x, y(0) =, y (0) = 0. (43) ( ) with the exact solution y = + x 3. We solve Eq. (43) using the algorithm described in Sec. 4. for the case corresponds to k = 0, M = 3. In Table 3, the

15 3 Y. H. Youssri, W. M. Abd-Elhameed, E. H. Doha 5 components of the vector C and the maximum absolute error E are illustrated for various values of λ. The numerical and exact solutions are depicted in Fig., in case of k = 0,M = 3 for some values of λ. Table 3 Components of C and the maximum absolute error E for Example 3 λ c 00 c 0 c 0 c 03 E Exact λ=0.5 λ=.5 λ= y(x) x Fig. Numerical and exact solutions of Example 3 for k = 0,M = 3. Example 4. Consider the Lane-Emden equation given in [48, 49] by y + x y + e y = 0; 0 < x, y(0) = 0, y (0) = 0. (44) We solve (44) using the algorithm described in Sec. 4. for the case corresponding to k = 0, M = 3. In Table 4, we give a comparison between the present solution for the three cases corresponding to λ =, 3,, with the two solutions obtained by Pandey et al. [49] and Wazwaz [48]. Remark. The results in Table 4 show that the two numerical solutions for various values of λ are closer to the Runge-Kutta solution obtained from Mathematica than the solutions obtained by Pandey [49] and Wazwaz [48].

16 6 Ultraspherical wavelets method for solving Lane-Emden type equations 33 Table 4 A comparison of the present solution with [49] and [48] for Example 4 x Pandey Wazwaz λ = λ = 3 λ = Mathematica CONCLUDING REMARKS In this paper, numerical solutions of Lane-Emden linear and nonlinear equations are given with the aid of ultraspherical wavelets algorithm. The derivation of this algorithm is essentially based on constructing the shifted ultraspherical wavelets operational matrix of differentiation. One advantage of the developed algorithms is that high accurate approximate solutions are achieved using a few number of terms in ultraspherical expansion. Another advantage is that for every value of the ultraspherical parameter, a numerical solution is obtained. The obtained numerical results are favorably compared with the analytical ones. REFERENCES. A. Constantinides, W.P. Schowalter, J.J. Carberry, J.R. Fair, Applied numerical methods with personal computers (McGraw-Hill, Inc., 987).. D.E. Newland, An introduction to random vibrations, spectral and wavelet analysis (Courier Corporation, 0). 3. F. Mohammadi, M.M. Hosseini, J. Frankl. Inst.-Eng. Appl. Math. 348 (8), (0). 4. F. Mohammadi, M. Hosseini, J. Adv. Res. () (00). 5. F. Mohammadi, M. Hosseini, S.T. Mohyud-Din, Int. J. Syst. Sci. 4 (4), (0). 6. M. Razzaghi, S. Yousefi, Int. J. Syst. Sci. 3 (4), (00). 7. M. Razzaghi, S. Yousefi, Math. Comput. Model. 34 (-), (00). 8. M. Razzaghi, S. Yousefi, Math. Meth. Appl. Sci. 5 (7), (00). 9. E. Babolian, F. Fattahzadeh, Appl. Math. Comput. 88 (), (007). 0. M. Dehghan, A. Saadatmandi, Int. J. Comput. Math. 85 (), 3 30 (008).. A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59 (3), (00).. L. Zhu, Y. Wang, Q. Fan, in The 0 International Conference on Scientific Computing, Las Vegas, USA, pp. 630, L. Zhu, Q. Fan, Commun. Nonlinear Sci. Numer. Simul. 7 (6), (0). 4. M. Chowdhury, I. Hashim, Phys. Lett. A 365 (5-6), (007). 5. M. Chowdhury, I. Hashim, Nonlinear Anal-Real 0, 04 5 (009). 6. A. Yildirim, T. Özis, Phys. Lett. A 369 (-), (007). 7. A. Yildirim, T. Özis, Nonlinear Anal.-Theory Methods Appl. 70 (6), (009). 8. K. Parand, A. Pirkhedri, New Astron. 5 (6), (00). 9. E. Momoniat, C. Harley, Math. Comput. Model. 53 (-), (0).

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