Accurate spectral solutions of first- and second-order initial value problems by the ultraspherical wavelets-gauss collocation method

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1 Available at Appl. Appl. Math. ISSN: Vol. 1, Issue (December 15), pp Applications and Applied Mathematics: An International Journal (AAM) Accurate spectral solutions of first- and second-order initial value problems by the ultraspherical avelets-gauss collocation method Y. H. Youssri 1, W. M. Abd-Elhameed 1, and E. H. Doha 1 1 Department of Mathematics, Faculty of Science Cairo University, Giza-Egypt Department of Mathematics, Faculty of Science University of Jeddah, Jeddah, Saudi Arabia Corresponding Author, youssri@sci.cu.edu.eg Received: June 17, 14; Accepted: February, 15 Abstract In this paper, e present an ultraspherical avelets-gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical avelets are used to reduce the differential equations ith their initial conditions to systems of algebraic equations, hich then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a ay that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably ith the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature. Keyords: Ultraspherical polynomials, avelets, first- and second-order initial value problems, Gaussian quadrature, Bratu Equation MSC 1 No.: 65M7, 65N35, 35C1, 4C1 835

2 836 Y. H. Youssri et al. 1. Introduction In the fields of applied mathematics and physics, first- and second-order ordinary initial value problems (IVPs) are often encountered. An effective method is required to analyze the mathematical model hich provides solutions conforming to physical reality, i.e., the real orld of physics. Therefore, e should have the ability for solving nonlinear first- and second-order ordinary differential equations (ODEs). Many practical problems arising in numerous branches of science and engineering require solving high even-order differential equations and, in particular, second-order differential equations. For example, after spacial discretization, a large class of nonlinear ave equations, including Klein-Gordon and Sine-Gordon equations, are reduced to certain systems of second-order ODEs. Moreover, second-order ODEs appear in celestial mechanics, circuit theory, control theory, chemical kinetics, and biology. In the sequence of papers (Abd-Elhameed (9); Abd-Elhameed et al. (14); Abd-Elhameed et al. (13); Doha and Abd-Elhameed (); Doha et al. (13a)), the authors dealt ith the high odd- and high even-order differential equations by the Galerkin and Pertov-Galerkin methods. They constructed suitable basis functions hich satisfy the underlying boundary conditions of the given differential equation. The suggested algorithms in these articles are suitable for handling the high-order linear differential equations, hile the authors in Doha et al. (13b) used another technique based on employing operational matrix of derivatives for handling the so-called Lane-Emden equation. Among the most important second-order differential equations is the one-dimensional Bratu problem, hich has a long history. Due to its great importance in various applications, a substantial amount of research ork has been done for studying this problem (see, for example, Boyd (1986); Boyd (3); Buckmire (4)). In Boyd (1968) and Boyd (3) Boyd employed Chebyshev polynomial expansions and the Gegenbauer polynomials as base functions. Syam and Hamdan (6) presented the Laplace Adomian decomposition method for solving such problem. Also, Aksoy and Pakdemirli (1) suggested ne perturbation iteration solutions for Bratu-type equations, hile Wazaz (5) presented the Adomian Decomposition method for solving this problem. Also, Venkatesh et al. (1) used Legendre avelets for solving IVPs of Bratu-type and discussed the analysis concerned ith the convergence and error analysis. The subject of avelets has recently dran a great deal of attention from mathematical scientists in various disciplines. It is creating a common link beteen mathematicians, physicists, and electrical engineers. Wavelet theory is a relatively ne and an emerging area in mathematical research. It has been applied to a ide range of engineering disciplines; in particular, avelets are very successfully used in signal analysis for ave 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, avelets establish a connection ith fast numerical algorithms (see Constantinides (1987); Neland (1993)). Recently, avelets represent a significant approach in different fields of science and engineering. In literature, many authors have used various avelets (Beylkin et al. (1991); Chen and Hsiao

3 AAM: Intern. J., Vol. 1, Issue (December 15) 837 (1997); Razzaghi and Yousefi ()) for analyzing problems of greater computational complexity, and they have pointed out that avelets represent poerful tools to explore a ne direction in solving differential equations. For example, Legendre avelets are used to obtain an approximate solution of Lane-Emden type equations in Yousefi (6). The problem of approximating definite integrals is of central importance in many applications of mathematics. In practice, a mere approximation of an integral very often ill not be satisfactory unless it is accompanied by an estimate of the error. For most quadrature formulas of practical interest, error bounds are available in the literature hich use, for example, norms of higherorder derivatives or bounds for the integrand in the complex plane. Hoever, in many practical situations such information about the integrand is not available. In particular, automatic quadrature routines are designed such that the user only has to insert the limits of integration, a routine for computing the integrand, a tolerance for the error, and an upper bound for the number of function evaluations (see Krommer and Ueberhuber (1994); Krommer and Ueberhuber (1998); Davis and Rabinoitz (1984)). Most quadrature methods used in modern numerical softare packages like those of NAG (1998) and IMSL (1998) are based on Gaussian quadrature formulas. Furthermore, both numerical experience and theoretical results sho the superiority of Gaussian formulas over many other quadrature formulas in many function classes (see in particular Brass et al. (1997) and the literature cited therein). Collocation methods (see, for instance, Guo and Yan (9); Doha et al. (13b)) have become increasingly popular for solving ordinary and partial differential equations. In particular, they are very useful in providing highly accurate solutions to nonlinear differential equations. In this paper, e aim to introduce a collocation avelets algorithm for handling first- and secondorder IVPs based on using ultraspherical avelets collocation algorithm. This algorithm is based on making use of the ultraspherical avelets together ith the Gaussian integration formula. In this article, e discuss the convergence analysis and uniqueness of the proposed function approximations. Numerical examples and some applications are also considered. The paper is organized as follos. In Section, e enumerate some properties of ultraspherical polynomials and their shifted ones. Section 3 describes the properties of ultraspherical avelets. Sections 4 and 5 are devoted for solving first- and second-order IVPs based on using ultraspherical avelets collocation method. Uniqueness and convergence theorems for second-order IVPs are discussed in detail in Section 6. In Section 7, e give some numerical examples to demonstrate the validity and applicability of our proposed algorithm. Some concluding remarks are given in Section 8.. Some properties of ultraspherical polynomials and their shifted ones In this section e give some useful properties of ultraspherical polynomials and their shifted forms.

4 838 Y. H. Youssri et al..1. Some properties of ultraspherical polynomials The ultraspherical polynomials (a special type of Jacobi polynomials) associated ith the real parameter (λ > 1 ), are a sequence of orthogonal polynomials defined on ( 1, 1), ith respect to the eight function (x) = (1 x ) λ 1. The orthogonality relation is given by 1 π n! Γ(λ + 1 ) (1 x ) λ 1 C m (x)c n (x) dx = (λ) n (n + λ) Γ, m = n, (1), m n, 1 here (λ) n is the Pochhammer notation, i.e., (λ) n = Γ(n + λ). Γ(λ) It should be noted here that the ultraspherical polynomials C n (x) are normalized such that C n (1) = 1. This normalization is characterized by an advantage that the polynomials C n () (x) are identical ith the Chebyshev polynomials of the first kind T n (x), C ( 1 ) n (x) are the Legendre polynomials L n (x), and C n (1) (x) is equal to (1/(n + 1))U n (x), here U n (x) are the Chebyshev polynomials of the second kind. The polynomials C n (x) may be generated by using the recurrence relation (n + λ) C n+1(x) = (n + λ) x C n (x) n C n 1(x), n = 1,, 3,..., ith the initial values: C (x) = 1 and C 1 (x) = x. For more properties and relations of ultraspherical polynomials, see, for instance, Andres et al. (1999). (x x ) λ 1 C m (x) C n The shifted ultraspherical polynomials (x) = C n (x 1) are a sequence of orthogonal polynomials defined on (,1), ith respect to the eight function (x) = (x x ) λ 1, i.e., 1 C n (x) dx = They also may generated by using the recurrence relation (n + λ) C n+1(x) = (n + λ) (x 1) ith the initial values: C (x) = 1 and π 1 4λ Γ(n + λ) n! (n + λ) (Γ), m = n,, m n. C n (x) n C 1 (x) = x 1. C n 1(x), n = 1,, 3,..., All relations and properties of ultraspherical polynomials can be easily transformed to give the corresponding relations and properties of the shifted ultraspherical polynomials. (). Some properties of shifted ultraspherical polynomials The shifted ultraspherical polynomials (a special type of shifted Jacobi polynomials) associated ith the real parameter (λ > 1 ) are a sequence of orthogonal polynomials on the interval (,1),

5 AAM: Intern. J., Vol. 1, Issue (December 15) 839 ith respect to the eight function (x) = (x x ) λ 1 1 (x x ) λ 1 C m (x)c n (x) dx =, i.e., π 1 4λ Γ(n + λ) n! (n + λ) (Γ), m = n,, m n. Here, e present some properties of ultraspherical polynomials (see, for instance, Andres et al. (1999)). These polynomials are eigenfunctions of the folloing singular Sturm-Liouville equation ( (x x ) D φ k (x) λ + 1 ) (x 1) D φ k (x) + k(k + λ) φ k (x) =, and they may be generated ith the aid of the recurrence relation (k + λ)c k+1 (x) = (k + λ) (x 1) C k (x) k C (x), k = 1,, 3,..., starting from C (x) = 1 and C 1 (x) = λ (x 1), or obtained from the Rodrigues formula C n (x) = ( 4)n Γ(n + λ) Γ(n + λ) (x x ) 1 λ D n [(x x ) n+λ 1 ], here D d dx. 3. Ultraspherical avelets n! Γ Γ(n + λ) Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother avelet. If the dilation parameter a and the translation parameter b vary continuously, then the folloing family of continuous avelets are obtained: ( ) t b ψ a,b (t) = a 1/ ψ a, b, R, a. (4) a We define the ultraspherical avelets ψ nm(t) = ψ(k, n, m, λ, t) as they have five arguments: k, n can be assumed to be any positive integer, m is the order for the ultraspherical polynomial, λ is the knon ultraspherical parameter, and t is the normalized time. Explicitly, they are defined on the interval [, 1] as: ψ nm(t) = k ξ m,λ C m ( k t n + 1 ), t [ n 1 n, ],, otherise, (3) (5) here m M 1, 1 n, and ξ m,λ = λ m! (m + λ) Γ πγ(m + λ). (6) Remark 1: It is orthy noting here that ψ ( 1 nm(t) ) is identical to Legendre avelets (Razzaghi and Yousefi (); Yousefi (6)), ψ nm(t) () is identical to first kind Chebyshev avelets (Babolian and Fattahzadeh (7); Li (1)) and ψ nm(t) (1) is identical to second kind Chebyshev avelets (Maleknejad et al. (7)).

6 84 Y. H. Youssri et al. No, assume a function f(t) defined on [, 1] and suppose that it may be expanded in terms of ultraspherical avelets as f(t) = c nm ψ nm(t), (7) here n=1 m= c nm = ( f(t), ψ nm(t) ) 1 = (t t ) λ 1 f(t) ψ nm(t) dt. Assume that f(t) is approximated in terms of ultraspherical avelets as f(t) M 1 n=1 m= here C and Ψ (t) are M 1 matrices given by Ψ (t) = c nm ψ nm(t) = C T Ψ (t), (8) C = [ c 1,, c 1,1,..., c 1,M 1,..., c,m 1,..., c,1,..., c,m 1] T, (9) [ ψ 1,, ψ 1,1,..., ψ 1,M 1, ψ,,..., ψ,m 1,... ψ,,..., ψ,m 1] T. (1) 4. Solution of first-order initial value problem In this section, e present an ultraspherical Gaussian quadrature avelets method (UGQWM) for solving the folloing first-order IVP: If e define the integral operator L(.) = (11) yields, Let y k,m (x) = M 1 n=1 m= y (x) = f ( x, y(x) ), y() = a, x [, 1]. (11) y(x) = a + (.) dt, then applying L to both sides of Equation f ( t, y(t) ) dt. c nm ψ nm(x) = C T Ψ (x) be an approximation of y(x). Then e have C T Ψ (x) a + No, e collocate Equation (1) at the points x i, to obtain C T Ψ (x i ) = a + i f ( t, C T Ψ (t) ) dt. (1) f ( t, C T Ψ (t) ) dt, i = 1,, 3,..., M, (13) here x i are the distinct ( M ) roots of C (x 1). For the sake of applying Gaussian M integration formula on Equation (13), e make use of the transformation: t = x i (z + 1), hich

7 AAM: Intern. J., Vol. 1, Issue (December 15) 841 transforms the intervals [, x i ] into the interval [ 1, 1], and therefore Equation (13) may be ritten in the form C T Ψ (x i ) = a + x 1 ( i xi f 1 (z + 1), CT Ψ ( x i (z + 1))) dz, i = 1,, 3,..., M. (14) If e apply the Gaussian integration formula on the right hand side of (14), then e get C T Ψ (x i ) a + x s ( i xi ω j f (z j + 1), C T Ψ ( x i (z j + 1) )), i = 1,, 3,..., M, j= (15) here z j s are the (s + 1) zeros of the polynomial C s+1(x), and the eights ω j (see Ralston and Rabinoitz (1978)) can be computed by the formula 1 s ( ) z zj ω j = dz. (16) z i z j 1 j= j i The idea behind the approximation in Equation (15) is the exactness of the Gaussian integration formula for polynomials of degree not exceeding (s + 1). 5. Solution of second-order initial value problem In this section, e are interested in solving the folloing second-order IVP subject to homogenous initial conditions y (x) + α f(x) y (x) + β g(x, y(x) ) =, y() = y () =, x [, 1], (17) here α, β are real constants. Applying L on both sides of Equation (17) yields y (x) + α f(t) y (t) dt + β Equation (18) gives after integration by parts y (x) = α f(x) y(x) + g ( t, y(t) ) dt =. (18) [ α f (t) y(t) β g ( t, y(t) )] dt. (19) No, the application of the operator L again on both sides of Equation (19), and performing some manipulations, yield y(x) = Therefore e have C T Ψ (x) α f(t) y(t) dt + (x t) [ α f (t) y(t) β g ( t, y(t) )] dt. () (x t) [ α f (t) C T Ψ (t) β g ( t, C T Ψ (t) )] dt We no collocate Equation (1) (see Canuto et al. (1988)) as α f(t) C T Ψ (t) dt. (1)

8 84 Y. H. Youssri et al. C T Ψ (x i ) = i i (x i t) [ α f (t) C T Ψ (t) β g ( t, C T Ψ (t) )] dt α f(t) C T Ψ (t) dt, 1 i M, here x i are the distinct ( M) roots of C (x 1). As in Section 4, and if e make use M of the transformation t = x i (z + 1), then after applying Gaussian integration formula, Equation () may be ritten in the form s [ ( C T Ψ (x i ) x i ω j (1 z j ) α f x ) ( ) i 4 (z j + 1) C T Ψ xi (z j + 1) (3) j= ( ( xi β g (z j + 1)C T Ψ x ))] i (z j + 1) x s ( ( i xi ω j α f (z j + 1) C T Ψ x )) i (z j + 1), 1 i M, j= () here z j s are the (s+1) zeros of C s+1(x), and the eights ω j can be computed from the formula in Equation (16). Remark : For the second-order IVP ith nonhomogeneous initial conditions y (x) + α f(x) y (x) + β g(x, y(x) ) =, y() = a, y () = a 1, x [, 1], (4) e use the transformation u(x) = y(x) a a 1 x, hich turns Equation (4) into the homogeneous second order IVP u (x) + α f(x) u (x) + β ḡ(x, u(x) ) =, u() = u () =, x [, 1], (5) here ḡ ( x, u(x) ) = α a 1 f(x) + β g ( x, u(x) + a + a 1 x ). No Equation (5) can be treated as Equation (17). 6. Uniqueness and convergence theorems In this section, the theoretical analysis of uniqueness and convergence of the proposed algorithms is discussed in detail. The uniqueness of the solution of Equation (11) as discussed in Burden and Faires (11). No, e state and prove to theorems. In the first e prove the uniqueness of the solution of the second-order initial value problem (17), hile in the second e sho that the truncated series given in (8) converges to the exact solutions of the to initial value problems (11) and (17).

9 AAM: Intern. J., Vol. 1, Issue (December 15) 843 Theorem 1. (Uniqueness Theorem) Assume that α f (i) (x) L i, here L, L 1 are to positive constants, g(x, y) is continuous and satisfies Lipschitz condition in the variable y ith Lipschitz constant L. Then Equation (17) has a unique solution henever < L < 1, here L = x L i. Proof: Let y 1 and y be to solutions of Equation (17). Then, y 1 (x) = and α f(t) y 1 (t) dt + i= (x t) α f (t) y 1 (t) dt (x t) α f (t) β g ( t, y 1 (t) ) dt, y (x) = α f(t) y (t) dt + (x t) α f (t) y (t) dt (x t) α f (t) β g ( t, y (t) ) dt. No y 1 y + x L y 1 y, α f(t) y 1 y dt + (x t) β g(t, y 1 ) β g(t, y ) dt (x t) α f (t) y 1 y dt and hence y 1 y (1 L), but since < L < 1, one should has y 1 y =. This proves the uniqueness and hence completes the proof. Theorem. (Convergence Theorem) The truncated series expansion of Equations (11) and (17) converge to the exact ones. Proof: Let y(t) = n=1 m= y k,m (t) = y k,n (t) = M 1 n=1 m= N 1 n=1 m= c nm ψ nm(t), c nm ψ nm(t), c nm ψ nm(t),

10 844 Y. H. Youssri et al. be the exact and approximate solutions (partial sums) to Equations (11) and (17) ith N M. Then e have ( ) ( N 1 ) y(t), y k,n (t) = y(t), c nm ψ nm(t) = = = N 1 n=1 m= N 1 n=1 m= N 1 n=1 m= n=1 m= c nm ( y(t), ψ nm(t) c nm c nm c nm. We sho that y k,n (t) is a Cauchy sequence in the complete Hilbert space L (R) and hence converges. No Bessel s inequality implies that yk,n (t) y k,m (t) = n=1 m= = N 1 n=1 m=m+1 N 1 n=1 m=m+1 ) c nm ψ nm(t) c nm. cnm is convergent and this leads to y k,n (t) y k,m (t) as M, N, and hence y k,n (t) converges to say s(t). We prove that s(t) = y(t), ( ) ( ) ( ) s(t) y(t), ψ nm(t) = s(t), ψ nm(t) y(t), ψ nm(t) ( ) = lim y k,n(t), ψ nm(t) c nm N ( ) = lim y k,n (t), ψ nm(t) c nm N =, and this proves that M 1 n=1 m= 7. Illustrative examples c nm ψ nm(t) converges to y(t). In this section, e apply UGQWM to solve first- and second-ivps as ell as Bratu s equation.

11 AAM: Intern. J., Vol. 1, Issue (December 15) 845 Example 1. Consider the first-order nonlinear IVP (see, Rajabi et al. (7); Domairry and Nadim (8); Caruntu and Bota (1)): y + y + y 4 = ; x [, 1], y() = 1. (6) We solve Equation (6) using UGQWM for the case corresponds to λ = k = 1, M = 8 and s = 1. Table I illustrates the maximum absolute error of Example 1. The comparison beteen the error resulting from the numerical solution obtained by UGQWM ith the error resulting from the application of the folloing methods: Homotopy perturbation method (HPM) in Rajabi et al. (7), Homotopy analysis method (HAM) in Domairry and Nadim (8), Squared remainder minimization method (SRMM) in Caruntu and Bota (1), is presented in Table I. Since no exact solution for Equation (6) is knon, the relative error is computed as the difference (in absolute value) beteen the approximate solution and the numerical solution given by the Mathematica Wolfram softare. In addition, in Figure 1, e give a comparison beteen different approximate solutions of (6). This figure shos that the bigger values of M attains accurate solution. Table I: Comparison of HPM, HAM, SRMM, and UGQWM for Example 1 x HPM HAM SRMM UGQWM Example. Consider the first-order linear IVP (see Caruntu and Bota (1)): y = 1 (sin x y) ; x [, 1], y() =, (7) ith the exact solution y(x) =.9999 (sin x.1 cos x +.1 e 1 x ). We solve Equation (7) using UGQWM for the case corresponds to k =, M = 4 and s = 7. Integrating Equation (7), e get y(x) = 1 ( 1 cos x ) y(t) dt. (8)

12 846 Y. H. Youssri et al Exact M= M=4 M=6 yhxl x Figure 1: Different solutions of Example 1 The approximate solution for (7) is given by y,4 (x) = c 1, ψ 1, (x) + c 1,1 ψ 1,1 (x) + + c,3 ψ,3 (x), then the application of formula (15) yields s ( C T Ψ (x i ) 1 (1 cos x) + 5 x i ω j C T Ψ xi ) (z j + 1), i = 1,,... 8, (9) j= and hence a system of equations in the unknon expansion coefficients c n,m is obtained and then solved by a suitable numerical solver. Table II illustrates the maximum absolute error of Example for various values of λ, hile in Table III, e give a comparison beteen the best errors obtained by using the methods in Caruntu and Bota (1) ith UGQWM. This table shos that our method is more accurate if compared ith all methods developed in Caruntu and Bota (1). In addition, a comparison beteen the exact solution and some approximate solutions is displayed in Figure. Table II: Maximum absolute error of Example 1 λ 1.49 E Example 3. Consider the second-order nonlinear IVP, (see Al-Khaled and Anar (7)): y + e y = ; x [, 1], y() = 1, y () = 1 e, (3)

13 AAM: Intern. J., Vol. 1, Issue (December 15) 847 Table III: Different solutions of Example Method UGQWM Cheb I Cheb II Eq. pts Lobatto III A Runge-Kutta Best error yhxl.4. Exact M=4 M=3 M= x Figure : Different solutions of Example 3 ith the exact solution y(x) = ln(e + x). Equation (3) is solved using UGQWM for the case corresponds to k = 1, M = 9 and s = 8. Table IV illustrates the maximum absolute error of Example 3 for various values of λ, and in Table V e give a comparison beteen the best errors obtained by using Adomian decomposition method (ADM), quintic spline method (C -spline) in Al-Khaled and Anar (7) ith the best error obtained by UGQWM. Table IV: Maximum absolute error of Example 3 1 λ 1.49 E Example 4. Consider the second-order nonlinear Bratu problem (see Venkatesh et al. (1)): y e y = ; x [, 1], y() = y () =, (31)

14 848 Y. H. Youssri et al. Table V: Different solutions of Example 3 Method UGQWM ADM C -spline Best error ith the exact solution y(x) = log(cos x). Equation (31) is solved using UGQWM for the case corresponds to k = 1, M = 9 and s = 8. Table VI lists some approximate solutions of (31) for various values of λ if UGQWM is applied, and the corresponding maximum absolute error for each case is displayed also in Table VII. In Table VIII, e give a comparison beteen the best absolute error obtained in Venkatesh et al. (1) by using Legendre avelets method (LWM) ith that obtained ith UGQWM. Table VI: The approximate solution and the maximum absolute error of Example 4 λ Numerical solution by UGQW M E 1.16 x 8.67 x x x x 4.45 x x.3 x x x x x x 4.48 x x.31 x x 8.63 x x x x 4.4 x x.16 x Table VII: Maximum pointise error of Example 4 x Exact λ = λ = λ = Table VIII: Different solutions of Example 4 Method UGQWM LWM Best error Remark 3: It is orthy noting here that the obtained numerical results in the previous solved four examples are very accurate, although the number of retained modes in the spectral expansion is fe, and the numerical results are comparing favorably ith the knon analytical solutions. Moreover, the numerical results sho that our method is more accurate than those discussed by some other existing techniques in literature.

15 AAM: Intern. J., Vol. 1, Issue (December 15) Conclusion In this paper, a ne ultraspherical avelets algorithm for obtaining numerical spectral solutions for first- and second-order IVPs is presented and analyzed. The derivation of this algorithm is essentially based on ultraspherical avelets basis functions and the Gaussian quadrature formula. The spectral collocation method is used for handling these equations. One of the main advantages of the presented algorithms is their availability for application on both linear and non linear first- and second-order IVPs including some important equations and also a Bratu-type equation. Another advantage of the developed algorithms is highly accurate approximate solutions are achieved by using a fe number of terms of the suggested expansion. REFERENCES Abd-Elhameed, W. M. (9). Efficient spectral Legendre dual-petrov-galerkin algorithms for the direct solution of (n+1)th-order linear differential equations, J. Egypt Math. Soc., Vol. 17, pp Abd-Elhameed, W. M., Doha, E. H. and Bassuony, M. A. (14). To Legendre-dual-Petrov- Galerkin algorithms for solving the integrated forms of high odd-order boundary value problems, The Scientific World Journal, Vol. 14, Article ID 3964, 11 pages. Abd-Elhameed, W.M., Doha, E.H. and Youssri, Y.H. (13a). Efficient spectral-petrov-galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials, Quaest. Math., Vol. 36, pp Abd-Elhameed, W. M., Doha, E. H. and Youssri, Y. H. (13b). Ne avelets collocation method for solving second-order multipoint boundary value problems using Chebyshev polynomials of third and fourth kinds, Abstract and Applied Analysis, Vol. 13, Article ID 54839, 9 pages. Abd-Elhameed, W. M., Doha, E. H. and Youssri, Y. H. (13c). Ne spectral second kind Chebyshev avelets algorithm for solving linear and nonlinear second-order differential equations involving singular and Bratu type equations, Abstract and Applied Analysis, Vol. 13, Article ID , 9 pages. Aksoy, Y. and Pakdemirli, M. (1). Ne perturbation iteration solutions for Bratu-type equations, Comput. Math. Appl., Vol. 59, pp Al-Khaled, K. and Anar, M. N. (7). Numerical comparison of methods for solving secondorder ordinary initial value problems, Appl. Math. Model., Vol. 31, pp Andres, G.E., Askey, R., and Roy, R. (1999). Special Functions, Cambridge University Press, Cambridge. Babolian, E. and Fattahzadeh, F. (7). Numerical solution of differential equations by using Chebyshev avelet operational matrix of integration, Appl. Math. Comp., Vol. 188, No. 1, pp Beylkin, G., Coifman, R. and Rokhlin, V. (1991). Fast avelet transforms and numerical algorithms, Comm. Pure Appl., Vol. 44, pp Boyd, J.P. (3). Chebyshev polynomial expansions for simultaneous approximation of to branches of a function ith application to the one dimensional Bratu equation, Appl. Math.

16 85 Y. H. Youssri et al. Comput., Vol. 14, pp Boyd, J.P. (1986). An analytical and numerical study of the to-dimensional Bratu equation, J. Sci. Comput., Vol. 1, No., pp Brass, H., Fischer, J.-W. and Petras, K. (1997). The Gaussian quadrature method, Abh. Braunscheig. Wiss. Ges., Vol. 47, pp Buckmire, R. (4), Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem, Numer. Meth. Part. D. E., Vol., No. 3, pp Burden, R. L. and Faires, J. D. (11). Numerical Analysis, Ninth Edition, Youngston State University. Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988), Spectral Methods on Fluid Dynamics, Springer-Verlag. Caruntu, B. and Bota, C. (1). Approximate polynomial solutions for nonlinear heat transfer problems using the squared remainder minimization method, Int. Commun. Heat Mass, Vol. 39, pp Constantinides, A. (1987). Applied Numerical Methods ith Personal Computers, McGra-Hill, Ne York. Costabile, F. and Napoli, A. (11). A class of collocation methods for numerical integration of initial value problems, Comput. Math. Appl., Vol. 6, pp Chen, C. F. and Hsiao, C. H. (1997). Haar avelet method for solving lumped and distributedparameter systems, IEEE Proceeding Control Theory and Applications, Vol. 144, No. 1, pp Davis, P. and Rabinoitz, P. (1984). Methods of Numerical Integration, Second Edition, Academic Press, Ne York. Doha, E. H. and Abd-Elhameed, W. M. (). Efficient spectral-galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. Sci. Comput., Vol. 4, pp Doha, E.H., Abd-Elhameed, W.M. and Bassuony, M.A. (13a). Ne algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods, J. Comput. Phys., Vol. 36, pp Doha, E.H., Abd- Elhameed, W. M. and Youssri, Y. H. (13b). Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type, Ne Astron., Vol. 3-4, pp Domairry, G. and Nadim, N. (8). Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation, Int. Commun. Heat Mass, Vol. 35, pp Guo, Ben-yu and Yan, Jian-ping. (9). Legendre-Gauss collocation method for initial value problems of second order ordinary differential equations. Appl. Numer. Math., Vol. 59, pp IMSL. (1998). Fortran Numerical Libraries, Version 3. (Visual Numerics). Krommer, A.R. and Ueberhuber, C.W. (1994). Numerical Integration: on Advanced Computer Systems (Lecture Notes in Computer Science), Vol. 848, Springer, Berlin. Krommer, A.R. and Ueberhuber, C.W. (1998). Computational Integration, SIAM, Philadelphia,

17 AAM: Intern. J., Vol. 1, Issue (December 15) 851 PA. Li, Y. (1). Solving a nonlinear fractional differential equation using Chebyshev avelets, Commun. Nonlinear Sci. Numer. Simulat., Vol. 15, No. 9, pp Maleknejad, K., Sohrabi, S. and Rostami, Y. (7). Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. Math. Comp., Vol. 188, No. 1, pp NAG (1998). Fortran Library, Mark 18, NAG, Inc. Neland, D. E. (1993). An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Longman Scientific and Technical, Ne York. Rajabi, A., Ganji, D.D. and Taherian, H. (7) Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. Lett. A, Vol. 36, pp Ralston, A. and Rabinoitz, P. (1978). A First Course in Numerical Analysis, Second Edition, McGra-Hill, Ne York. Razzaghi, M. and Yousefi, S. (). Legendre avelets direct method for variational problems, Math. Comput. Simulat., Vol. 53, pp Syam, M.I. and Hamdan, A. (6). An efficient method for solving Bratu equations, Appl. Math. Comput., Vol. 176, pp Venkatesh, S. G., Ayyasamy, S. K. and Balachandar, S. R. (1). The Legendre avelet method for solving initial value problems of Bratu-type, Comput. Math. Appl., Vol. 63, pp Wazaz, A.M. (5). Adomian decomposition method for a reliable treatment of the Bratutype equations, Appl. Math. Comput., Vol. 166, pp Yousefi, S. (6). Legendre avelets method for solving differential equations of Lane-Emden type, Appl. Math. Comput., Vol. 181, pp