SPECTRAL SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS VIA A NOVEL LUCAS OPERATIONAL MATRIX OF FRACTIONAL DERIVATIVES

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1 SPECTRAL SOLUTIONS FOR FRACTIONAL DIFFERENTIAL EQUATIONS VIA A NOVEL LUCAS OPERATIONAL MATRIX OF FRACTIONAL DERIVATIVES W M ABD-ELHAMEED 1,, Y H YOUSSRI 1 Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia walee 9@yahoocom Department of Mathematics, Faculty of Science, Cairo University, Giza 1613, Egypt youssri@scicuedueg Received October 14, 015 In this research article, a novel operational matrix of fractional-order differentiation of Lucas polynomials in the Caputo sense is established Based on this matrix along with the application of tau and collocation spectral methods, two efficient numerical algorithms for solving multi-term fractional differential equations are proposed and analyzed Some new formulae for Lucas polynomials are stated and proved for investigating the new algorithms The convergence and error analysis of the suggested Lucas expansion are investigated carefully Some new inequalities including the modified Bessel function of the first kind and the well-known golden ratio are stated and proved Some numerical tests are carried out for some specific and important types of problems including the Bagley-Torvik, Ricatti, Lane-Emden and oscillator equations The results obtained are compared with some existing ones in open literature and it is noticed that the two proposed algorithms are robust, accurate and easy to apply Key words: Lucas polynomials, tau and collocation spectral methods, fractional-order differential equations, error analysis PACS: 060Cb, 030Mv, 070Hm, 070Jn, 030Gp 1 INTRODUCTION Fractional calculus is a very important branch of mathematical analysis There are extensive studies concerning fractional calculus from both theoretical and practical points of view It is well-known that the fractional calculus deals with derivatives and integrals to an arbitrary order (real or complex) The applications of fractional calculus are numerous in many fields For example, several problems in mechanics (theory of viscoelasticity and viscoplasticity), (bio-)chemistry (modeling of polymers and proteins), medicine (modeling of human tissue under mechanical loads) electrical engineering (transmission of ultrasound waves), and other problems can be modeled by fractional differential equations Analytical solutions of fractionaldifferential equations are not always available, and therefore, it is an important matter to obtain numerical solutions for such equations via several techniques In this respect, a great number of researchers have considerable interests in investigating RJP Rom 61(Nos Journ Phys, 5-6), Vol , Nos 5-6, (016) P , (c) 016 Bucharest, - v13a*01670

2 796 W M Abd-Elhameed, Y H Youssri numerically various types of fractional differential equations (FDEs) For example, Taylor collocation method is followed in [1]; Adomian s decomposition method is followed in [, 3]; finite difference method is followed in [4]; variational iteration method is followed in [5], homotopy analysis method and homotopy perturbation methods are followed respectively in [6] and [7] For some relevant recent papers in the area of fractional differential equations and their applications one can be referred for example to [8 1] Spectral methods are crucial in handling various types of differential equations The most frequently used trial functions are the various orthogonal polynomials For example, Jacobi polynomials and their six special polynomials are extensively employed for obtaining numerical solutions for several kinds of differential equations (see, for example [13 19]) For comprehensive study on different versions of spectral method, one cane be referred to the books of Canuto et al [0], Hesthaven et al [1], Boyd [] and Trefethen [3] There are other methods employed for solving different problems arise in various disciplines, among these methods homotopy analysis method (see for example [4, 5]), and extended homotopy analysis method (see for example [6]) The Lucas polynomials and their generalizations are of fundamental interest The well-known sequence of Lucas numbers can be obtained from the sequence of Lucas polynomials by setting x = 1 The celebrated Lucas numbers and the golden ratio appear in a variety of applications in various disciplines such as biology, physics, computer science, statistics and graph theory Some of these applications can be found in the important book of Koshy [7] Although, several authors have considerable contributions in discussing Lucas polynomials form a theoretical point of view (see, for example [8, 9]), but the practical studies about Lucas polynomials and their related numbers are traceless in literature This gives us a strong motivation for discussing these polynomials practically for solving fractional differential equations The approach of utilizing operational matrices is a powerful tool for the treatment of various differential equations This approach is characterized by its simplicity and applicability It is widely used by many researches in a variety of papers For example, Abd-Elhameed in [30] has developed and used a novel harmonic numbers operational matrix of derivatives to solve linear and nonlinear sixth-order BVPs Moreover, Abd-Elhameed in [31] and Doha et al in [3] introduced and used tau and Galerkin operational matrices of derivatives to solve the singular Lane-Emden differential equations The employment of the operational matrices is not restricted in application to ordinary differential equations, but it can be also followed to treat FDEs There is a great number of articles in this direction For example, Saadatmandi and Dehghan in [33] established a Legendre operational matrix of fractional derivatives They used this operational matrix for solving some fractional-order differential

3 3 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 797 equations (see, [34, 35]) Also, the authors in [36, 37] used numerical spectral methods for solving important partial fractional differential problems Rostamy et al in [38] introduced a Bernstein operational matrix of fractional derivatives for handling these equations The main objective of this article is twofold: Developing a new operational matrix of fractional derivatives of Lucas polynomials based on using new relations of Lucas polynomials Presenting and implementing two algorithms based on the application of tau and collocation spectral methods for solving multi-term fractional-order differential equations The rest of the paper is as follows In the next section, some preliminaries including some elementary definitions of the fractional calculus theory are presented Moreover, in this section some relevant properties of Lucas polynomials are given and also some new relations concerning the same polynomials are stated and proved Section 3 is devoted to the construction of Lucas operational matrix of the fractional derivatives in Caputo sense In Section 4, we propose two numerical algorithms for solving multi-term fractional-order differential equations based on applying tau and collocation spectral methods Section 5 is devoted to discussing the convergence and error analysis of the suggested expansion of Lucas polynomials Some numerical experiments and comparisons are displayed in Section 6 aiming to ascertain the applicability and efficiency of the two proposed algorithms Finally, some conclusions are reported in Section 7 PRELIMINARIES AND SOME NEW FORMULAE 1 SOME DEFINITIONS AND PROPERTIES OF FRACTIONAL CALCULUS In this section, we present some notations, definitions and preliminary facts of the fractional calculus theory which will be useful throughout this article Definition 1 The Riemann-Liouville fractional integral operator I α of order α on the usual Lebesgue space L 1 [0,1] is defined as t I α 1 f(t) = Γ(α) (t τ) α 1 f(τ)dτ, α > 0 0 (1) f(t), α = 0 Definition The Riemann-Liouville fractional derivative of order α > 0 is defined by ( ) d n (D α f)(t) = (I n α f)(t), n 1 α < n, n N () dt

4 798 W M Abd-Elhameed, Y H Youssri 4 Definition 3 The fractional differential operator in Caputo sense is defined as (D α f)(t) = 1 Γ(n α) t 0 (t τ) n α 1 f (n) (τ)dτ, α > 0, t > 0, (3) where n 1 α < n,n N The operator D α satisfies the following two basic properties for n 1 α < n, (D α I α f)(t) = f(t), n 1 (I α D α f)(t) = f(t) D α t k = f (k) (0 + ) (t a) k, t > 0, k! Γ(k + 1) Γ(k + 1 α) tk α, k N,k α, (4) where α denotes the smallest integer greater than or equal to α For more properties of fractional derivatives and integrals, see for example, [39, 40] SOME RELEVANT PROPERTIES AND RELATIONS OF LUCAS POLYNOMIALS It is well-known that the Lucas polynomials can be generated with the aid of the recurrence relation L i+ (x) = xl i+1 (x) + L i (x), L 0 (x) =, L 1 (x) = x, i 0 (5) These polynomials have the following explicit form ( x + ) i ( x x ) i x + 4 L i (x) = i, and also have the following explicit power form representation: i L i (x) = i (i k + 1) k 1 k! x i k, (6) where the notation z represents the largest integer less than or equal to z, and (a) n is the Pochhammer notation ie (a) n = Γ(a+n) Γ(a) Now, we intend to state prove the following two basic theorems In the first, the inversion formula to formula (6) is given, and in the second theorem, the first derivative of Lucas polynomials is expressed in terms of the Lucas polynomials themselves Theorem 1 For all m 1, the following inversion formula holds x m = m i=0 ( 1) i δ m i (m i + 1) i i! L m i (x), (7)

5 5 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 799 where δ i is defined as δ i = { 1 i = 0, 1, i > 0 (8) Proof We proceed by induction on m It is clear that for m = 0, each of the two sides of (7) is equal to 1 Now, assume that formula (7) holds, and we have to prove that the following formula is valid x m+1 = m+1 i=0 ( 1) i δ m i+1 (m i + ) i i! L m i+1 (x) (9) Now, If we multiply both sides of the valid relation (7) by x, and make use of the recurrence relation for Lucas polynomial in (5), then we get x m+1 = m i=0 + m i=0 ( 1) i δ m i (m i + 1) i i! ( 1) i+1 δ m i (m i + 1) i i! L m i+1 (x) L m i 1 (x), which can be turned- after performing some algebraic manipulations- into the form x m+1 = m i=1 ( 1) i δ m i+1 (m i + ) i L m i+1 (x) + δ m L m+1 i! ) m δ m m ( 1) m +1 (m + 1 m + ( m )! which is equivalent to the formula x m+1 = m+1 i=0 Theorem 1 is now proved ( 1) i δ m i+1 (m i + ) i i! L m m 1, L m i+1 (x) Theorem The first derivative of the Lucas polynomials are linked with their original polynomials by the following formula: DL i (x) = i and δ i is as defined in (7) i 1 (10) (11) ( 1) m δ i m 1 L i m 1 (x), (1) m=0

6 800 W M Abd-Elhameed, Y H Youssri 6 Proof If we differentiate the power form representation of the Lucas polynomials (6) with respect to x, then the following relation is obtained DL i (x) = i i 1 ( i k 1 k ) x i k 1 (13) With the aid of the inversion formula (7), the last relation can be turned into DL i (x) = i i 1 ( ) i 1 i k 1 k k r=0 ( 1) r (i k r) r δ i k r 1 r! L i k r 1 (x) (14) If we expand the right hand side in the latter relation, and rearrange the terms, then after some lengthy algebraic computations, we get where B i,m is given by DL i (x) = i 1 m=0 j=0 B i,m L i m 1 (x), (15) m ( 1) m+j (i j 1)! B i,m = iδ i m 1 j! (i j m 1)! (m j)! In order to obtain a reduction formula for B i,m, we note that it can be written equivalently as ( ) ( i B i,m = ( 1) m m, i + m 1 (i m) δ i m 1 F 1 m 1 i ) 1 The last hypergeometric F 1 can be summed with the aid of Chu-Vandermonde identity (see Koepf [41]) to give ( ) m, i + m 1 F 1 1 i 1 (i m 1)!m! =, (16) (i 1)! and accordingly, B i,m takes the following reduced form Theorem is now proved B i,m = i( 1) m δ i m 1 In the upcoming computations, it is useful to write the power form representation of the Lucas polynomials in (6) and its inversion formula (7) in the following two alternative forms: L i (x) = i i (k+i) even (k + 1) i k ) 1 x k, i 1, (17)! ( i k

7 7 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 801 x m = m (k+m) even ( 1) m k δ k ( k+m ( m k + 1 ) m k ) L k (x), m 0 (18)! For more properties of Lucas polynomials, one can be refereed to the important book of Koshy [7] 3 CONSTRUCTION OF LUCAS OPERATIONAL MATRIX OF FRACTIONAL DERIVATIVE Assume that u(x) is a square Lebesgue integrable function on (0,1), and suppose that it can be expressed in terms of the linearly independent Lucas polynomials as: u(x) = c k L k (x) Consider the following approximation to u(x) u(x) u N (x) = N c k L k (x) = C T Φ(x), (19) where C T = [c 0,c 1,,c N ], (0) and Φ(x) = [L 0 (x),l 1 (x),,l N (x)] T (1) Now, the first derivative of the vector Φ(x) can be written in the form dφ(x) dx = M(1) Φ(x), () ( ) where M (1) = m (1) ij, is the (N + 1) (N + 1) operational matrix of derivatives whose nonzero elements can be determined form Theorem They can be written explicitly as {i( 1) i j 1 m (1) δ j, if i > j, (i + j) odd, ij = 0, otherwise 31 LUCAS OPERATIONAL MATRIX OF FRACTIONAL DERIVATIVES The principal objective of this section is to find the generalization of the operational matrix of derivatives for the fractional case

8 80 W M Abd-Elhameed, Y H Youssri 8 Corollary 1 For any positive integer r, one has d r Φ(x) dx r = M (r) Φ(x) = ( M (1)) r Φ(x) (3) Proof By induction on r and making use of (1) the corollary is obtained Now, we are going to state and prove the main theorem concerning the fractional derivatives of Lucas polynomials Theorem 3 Let Φ(x) be the Lucas polynomial vector defined in Eq (1) For any α > 0, one has D α Φ(x) = x α M (α) Φ(x), (4) where M (α) = (m α i,j ) is the (N + 1) (N + 1) lower triangular Lucas operational matrix of derivatives of order α in the Caputo sense and it is given explicitly as: η α ( α,0) η α ( α, α ) 0 0 M (α) = (5) η α (i,0) η α (i,i) 0 η α (N,0) η α (N,1) η α (N,) η α (N,N) ( ) In fact, the elements m α i,j are given explicitly by where η α (i,j) = i { m α ηα (i,j), i α, i j; i,j = 0, otherwise i k= α (i+k) even (j+k) even ( 1) k j ( δ i+k j ( ) ( i k! k j ) ( )! j+k+ k j )!Γ(k + 1 α) (6) Proof If we apply the fractional differential operator D α to Eq (17), then with the aid of relation (4), the following relation is obtained i D α i ( ) i+k 1! L i (x) = ) x k α (7)!Γ(k + 1 α) k= α (k+i) odd ( i k

9 9 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 803 The inversion formula (18) enables one after performing some manipulations to write i D α L i (x) = x α η α (i,j)l j (x), (8) j=0 where η α (i,j) is given in (6) Now, we can rewrite Eq (8) in the following vector form: D α Φ i (x) = x α [η α (i,0),η α (i,1),,η α (i,i),0,0,,0] Φ(x), α i N +1, (9) and also we can write D α Φ i (x) = x α [0,0,,0], 0 i α 1 (30) Finally, Eq (9) together with Eq (30) yield the desired result Note 1 It is worthy to note here that the fractional operational matrix of Lucas polynomials in the sense of Riemann-Liouville definition may be obtained in a similar approach discussed in this section 4 TWO MATRIX ALGORITHMS FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATION The main aim of this section is to develop two numerical algorithms for handling linear and nonlinear FDEs The algorithm, namely tau Lucas matrix method (TLMM) is applied to linear equations, while the algorithm, namely collocation Lucas matrix method (CLMM) is applied to nonlinear equations 41 LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS Consider the following linear fractional differential equation with variable coefficients q 1 D αq u(x) + λ i (x)d α i u(x) + µ(x)u(x) = f(x), x (0,1), (31) i=1 where α i < α i+1, and i < α i i + 1, i = 1,,,q 1, subject to the initial conditions u (i) (0) = a i, i = 0,1,,q 1, (3) where λ i (x),µ(x) and f(x) are known continuous functions Now, assume that u(x) can be approximated as u(x) u N (x) = C T Φ(x) (33)

10 804 W M Abd-Elhameed, Y H Youssri 10 In virtue of Theorem 3, the following approximations can be obtained D α i u(x) x α i C T M (α i) Φ(x) (34) Making use of the approximations in (33), (34), the residual of (31) is given by q 1 R(x) =C T M (αq) Φ(x) + x αq α i λ i (x)c T M (αi) Φ(x) + x αq µ(x)c T Φ(x) x αq f(x), i=1 and hence applying tau method (see for example [3]) leads to 1 Moreover the initial conditions (3) yield 0 (35) R(x)L i (x)dx = 0, i = 1,,N q (36) C T M (i) Φ(0) = a i, i = 0,1,,q 1 (37) It is clear form Eqs (36) and (37), that a linear system of algebraic equations in the unknown expansion coefficients c i of dimension (N + 1) is generated This system can be solved via the Gaussian elimination technique or any suitable technique Hence, the approximate solution (33) can be obtained Note It is worthy to note here the structure of the matrix M (α) reduces the computational cost of calculating the fractional derivative of the Lucas polynomials vector where 4 HANDLING NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS Consider the following nonlinear fractional-order differential equation: D αq u(x) = Ω ( x,u(x),d α 1 u(x),d α u(x),,d α q 1 u(x) ), x (0,1), (38) α i < α i+1, and i < α i i + 1, i = 1,,,q 1, subject to the initial conditions u (i) (0) = a i, i = 0,1,,q 1 (39) If we approximate u(x),d α i u(x) as in Section 41, then the residual R(x) of Eq (38) is given by R(x) =x αq C T M (αq) Φ(x) ( ) Ω x,c T Φ(x),x α 1 C T M (α1) Φ(x),,x α q 1 C T M (αq 1) Φ(x) (40)

11 11 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 805 The application of the spectral collocation method requires ( that) R(x) must vanish at selected collocation points We select these points to be : i N+1, i = 1,,N q, and therefore ( i R ) = 0, i = 1,,N q (41) N + 1 Eqs (41) with (37) generate a nonlinear system of equations in the unknown expansion coefficients c i of dimension (N + 1) Newton s iterative technique can be employed for solving this system, and consequently, we obtain the desired approximate solution from (33) 5 CONVERGENCE AND ERROR ANALYSIS In this section, a comprehensive study for the convergence and error analysis of the suggested expansion of Lucas polynomials is investigated For this purpose, the following Lemmas are needed Lemma 1 If f(x) is an infinitely differentiable function at the origin, then f(x) can be expanded in terms of Lucas polynomials as f(x) = j=0 ( 1) j δ k f (k+j) (0) L k (x) (4) j!(k + j)! Proof Following similar procedures to those followed in [4], the expansion in (4) can be obtained Lemma [43] If I µ (x) denotes the modified Bessel function of order µ of the first kind, then the following identity holds x k+j j!(j + k)! = I k(x) (43) j=0 Lemma 3 [44] The modified Bessel function of the first kind I µ (x) satisfies the following inequality I µ (x) xµ cosh(x), x > 0 (44) Γ(µ + 1) Lemma 4 Let σ = be the well-known golden ratio The following inequality for Lucas polynomials holds L k (x) σ k (45) In the following two theorems, we discuss in detail the convergence and error analysis of the Lucas expansion

12 806 W M Abd-Elhameed, Y H Youssri 1 Theorem 4 Let f(x) be defined on [0,1] provided with f (i) (0) A i, i 0, where A is a positive constant Also, let f(x) has the expansion f(x) = c k L k (x) The following are satisfied: 1 c k Ak cosh(a) k! The series converges absolutely Proof At first, it is clear form Lemma 1 that c k = ( 1) j δ k f (j+k) (0) j!(j + k)!, j=0 and hence the assumption that f (i) (0) A i, i 0, leads to the inequality A j+k c k j!(j + k)! (46) j=0 The last inequality after the application of Lemma is turned into c k I k (A) (47) which leads with the application of Lemma 3 to the following estimation for the expansion coefficients c k Ak cosh(a) k! This proves the first part of Theorem 4 Now, we show that the series c k L k (x) converges absolutely From (47), we have and hence Lemma 4 implies that Now, since cosh( A) (Aσ) k = cosh(a)e Aσ, therefore the series converges ab- k! solutely c k L k (x) I k (A) L k (x), c k L k (x) cosh(a) (Aσ) k k!

13 13 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 807 Theorem 5 If f(x) satisfies the hypothesis of Theorem 4, and e N (x) = then we have the following global error estimate Proof By Theorem 4, we can write which can be written as e N (x) < eµ cosh(a)µ N+1 (N + 1)! e N (x) cosh(a) k=n+1 e N (x) e µ cosh(a) ( 1 µ k (k 1)!, µ = Aσ, k=n+1 c k L k (x), ) Γ(N + 1,µ), (48) Γ(N + 1) where Γ(N + 1) and Γ(N + 1,µ) denote respectively, to the gamma and the incomplete gamma functions (see [45]) The inequality in (48) can be written in the integral form e N (x) eµ cosh(a) µ t N e t dt, N! 0 but since e t < 1, t > 0, then we have e N (x) < eµ cosh(a)µ N+1, (N + 1)! which completes the proof of the theorem 6 NUMERICAL EXAMPLES In this section, some numerical results supported by some comparisons with some other results in literature are presented aiming to ensure the efficiency, applicability and higher accuracy of the two proposed algorithms in this paper In the following experiments we evaluate the error in maximum norm namely, E = max x [0,1] u(x) u N(x) Example 1 [46] Consider the following inhomogeneous Bagley-Trovik equation: subject to the initial conditions D u(x) + D 3 u(x) + u(x) = f(x), x (0,1), (49) u(0) = 0, u (0) = α, (50) where f(x) is chosen such that the exact solution of Eq (49) is u(x) = sin(αx) We apply TLMM for different values of α and N We display in Table 1, a comparison

14 808 W M Abd-Elhameed, Y H Youssri 14 Table 1 Comparison between TLMM and CSM in [46] for Example 1 α = 1 α = 4π N TLMM CSM [46] TLMM CSM [46] between the results obtained by the application of TLMM with those obtained by the Chebyshev spectral method (CSM) which developed in [46] The displayed results in this table show that our algorithm gives a lesser error in almost all cases Example [47] Consider the following linear fractional IVP: D u(x) + D 1 u(x) + u(x) = x π x 3 +, x (0,1), (51) subject to the initial conditions u(0) = u (0) = 0 (5) If TLMM is applied for the case N =, then the residual of Eq (51) is given by R(x) = xc T M () Φ(x)+C T M ( 1 ) Φ(x)+ ( xc T Φ(x) x π x + ) x, and the operational matrices M () and M ( 1 ) are given explicitly as follows: M () = 0 0 0, M ( 1 ) = π The application of the tau method transforms Eq (51) into ( π)c + ( π)c π c 0 = π (53) Moreover, the initial conditions (5) yield c + c 0 = 0, and c 1 = 0 (54) Eqs (53) and (54) can be immediately solved to give c 1 = 1, c = 0, c 3 = 1, and consequently u(x) = x which is the exact solution Example 3 [48] Consider the following nonlinear Riccati fractional differential equation D q u(x) + u (x) = 1, x (0,1), q (0,1], (55)

15 15 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 809 Table Comparison between CLMM and [48] for Example 3 q = 07 q = 08 q = 09 x [48] CLMM [48] CLMM [48] CLMM Exact subject the initial condition u(0) = 0 (56) The exact solution of (55) in case q = 1 is u(x) = tanhx In Table, we compare our results with those obtained in [48] Figure 1 illustrates that the approximate solutions for various values of q near the value 1, have a similar behavior 10 u N (x) x Exact q=175 q=15 q=15 Fig 1 Different solutions of Example 3 Example 4 [49] Consider the following nonlinear Lane-Emden equation D q u(x) + x u (x) + u 3 (x) = + x 6, x (0,1), q (0,1], (57) subject the initial conditions u(0) = u (0) = 0 (58)

16 810 W M Abd-Elhameed, Y H Youssri 16 Table 3 Comparison between CLMM and the method in [49] for Example 4 q = 15 q = 15 q = 175 x BWM [49] CLMM BWM [49] CLMM BWM [49] CLMM Exact The exact solution of (57) in case q = is u(x) = x In Table 3, we compare our results with those obtained in [49] using Bernoulli wavelets method (BWM) Figure demonstrate that the approximate solutions for various values of q have a similar behavior 08 u N (x) Exact q=09 q=08 q= x Fig Solutions of Example 4 Example 5 [50] Consider the following linear fractional oscillator equation subject the initial conditions D q u(x) + ω u(x) = 0, x (0,1), q (1,], (59) u(0) = 0 u (0) = ω (60) The exact solution of (59) in case q = is u(x) = sin(ω x) Due to the nonavailability of the exact solution in case of q (1,) we discuss the behaviour of the solution for values of q near the value In Table 4, we list the values of the approximate solution for N = 3, ω = 1 and different values of q near the value Moreover,

17 17 Spectral solutions for FDEs via novel Lucas operational matrix of fractional derivatives 811 Table 4 Exact and approximate solution of Example 5 x q = 17 q = 18 q = 19 Exact Figure 3 illustrates that the approximate solutions for various values of q have a similar behavior 10 u N (x) x Exact q=19 q=18 q=17 Fig 3 Solutions of Example 5 7 CONCLUSIONS In this paper, a novel operational matrix of fractional derivatives of Lucas polynomials is derived This matrix is utilized along with tau and collocation spectral methods in order to reduce the multi-term fractional-order differential equations into the solution of algebraic equations which can be efficiently solved We do believe that the approach followed in this paper can be applied to other kinds of fractional differential equations The tested numerical examples ascertain the high efficiency and performance of the two proposed algorithms Acknowledgements The authors would like to thank the referee for his useful comments which have improved the paper in its present form

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