Comp. Appl. Math. DOI 10.1007/s40314-017-0488-z
Fifth-kind orthonormal Chebyshev polynomial solutions for fractional differential equations W. M. Abd-Elhameed1 · Y. H. Youssri1
Received: 18 December 2016 / Revised: 4 July 2017 / Accepted: 17 July 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate. Keywords Fifth-kind Chebyshev polynomials · Spectral methods · Romberg’s quadrature formulae · Connection problem · Fractional differential equations Mathematics Subject Classification 65M70 · 34A08 · 33C45 · 11B83
1 Introduction Spectral methods are crucial for obtaining solutions of ordinary, partial and fractional differential equations (FDEs). These methods have many advantages if compared with other methods such as finite element and finite difference methods. In many physical and chemical applications, solutions with many decimal places of accuracy are needed, so it is very useful to employ various spectral methods because they yield exponential convergence of the solutions. For applications of spectral methods in different disciplines, one can consult Shizgal
B 1
Y. H. Youssri
[email protected] Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt
123
W. M. Abd-Elhameed, Y. H. Youssri
(2014), Hesthaven et al. (2007), Boyd (2001), Trefethen (2000) and Canuto et al. (1988). The philosophy of the application of various spectral methods is built on writing the solution of a certain problem as a suitable combination of certain polynomials which are often orthogonal. There are three popular types of spectral methods, they are tau, collocation and Galerkin methods. The philosophy of applying Galerkin method is built on choosing suitable combinations of orthogonal polynomials satisfying the underlying initial/boundary conditions, and after that enforcing the residual to be orthogonal with the selected basis functions. This method is successfully applied to linear boundary value problems (see, for example Doha and Abd-Elhameed 2014). The tau method has the advantage that it avoids some of the problems of the Galerkin methods, since we can choose any set of orthogonal polynomials as basis functions and the boundary (initial) conditions are set as constraints (see for example Doha et al. 2013). The philosophy of applying the collocation method is to select certain collocation points such that the residual of the differential equation vanishes at these points. This method is convenient for treating nonlinear problems (see for example Dabiri and Butcher 2017; Costabile and Napoli 2015a, b; Esmaeili et al. 2011; Lamnii et al. 2011). In spectral methods, it is very useful to approximate integrals by using quadrature rules. In quadrature rules, we select (n + 1) nodes in the integration interval, and the weights wk are determined such that the definite integrals of all polynomials of degree n are exactly represented by the weighted sums of the values of the integrand at the given nodes (see, Waldvogel 2006). The Chebyshev polynomials have become increasingly crucial in analysis and its applications. In literature, there are four well-known classes of Chebyshev polynomials. The first and second kinds are ultraspherical polynomials, however the third and fourth kinds are special classes of nonsymmetric Jacobi polynomials of certain choices of their parameters. All four kinds have roles in numerical analysis. There are numerous old and recent studies concerned with the first and second kinds (see, for example Abd-Elhameed et al. 2016). Moreover, there are recent articles that investigate theoretically and practically third and fourth kinds (see, for example Abd-Elhameed et al. 2015). In the interesting Ph.D. dissertation of Masjed-Jamei (2006), the author applied the extended Sturm–Liouville problem for symmetric functions, and a basic class of symmetric orthogonal polynomials (BCSOP) is introduced. This class involves four parameters. Some elementary properties, such as a generic second-order differential equation, a generic three-term recurrence relation and some other important formulae of these polynomials are also introduced in Masjed-Jamei (2006). The main advantage of introducing this class of orthogonal polynomials is that it generalizes several important well-known classes of orthogonal polynomials and also some other new kinds of orthogonal polynomials are special polynomials of the introduced class. More definitely, the well-known four kinds of Chebyshev polynomials are particular polynomials of BCSOP. Other two new classes of orthogonal polynomials, namely Chebyshev polynomials of fifth and sixth kinds can be also deduced. There are some theoretical studies concerned with BCSOP (see, Koepf and Masjed-Jamei 2006a, b; Masjed-Jamei 2007), however, and to the best of our knowledge, no practical studies concerned with these polynomials. This motivates us to utilize these polynomials for solving numerically some types of differential equations, and in particular FDEs. Fractional calculus becomes a central branch of mathematical analysis. The importance of this branch has been increased in the recent decades. The main reason for this importance is due to the fact that many phenomena in different disciplines such as biology, physics and fluid mechanics can be modeled by certain FDEs. In the majority of these equations, it is not easy to obtain their analytical solutions, so obtaining numerical solutions for these kinds of equations becomes very necessary. A great number of authors have significant contributions
123
Fifth-kind orthonormal Chebyshev polynomial solutions for…
which deal with these types of equations. In fact, different numerical algorithms are developed for obtaining numerical solutions of different types of FDEs. In this regard, Adomian’s decomposition method is used in Daftardar-Gejji and Jafari (2007), finite difference method is used in Meerschaert and Tadjeran (2006), Taylor collocation method is used in Çenesiz et al. (2010), spline finite difference method in Moghaddam and Machado (2017), An extended predictor–corrector algorithm is used in Moghaddam et al. (2016), An improved collocation method is applied in Bhrawy and Zaky (2017), tau method is applied in Bhrawy et al. (2016b), Mokhtary et al. (2016) and Saadatmandi and Dehghan (2011), wavelets methods are employed in Abd-Elhameed and Youssri (2015), and ur Rehman and Khan (2011) and operational matrix methods are employed in Abd-Elhameed and Youssri (2016a, b), and Zaky et al. (2016). For some other numerical techniques and applications concerned with FDEs, see Abd-Elhameed and Youssri (2017), Bhrawy et al. (2015), Moghaddam and Machado (2017), Bhrawy and Zaky (2016, 2017), Salgado and Aguirre (2016), Tang and Xu (2016), and Yaghoobi et al. (2017). The main objective of the current paper is to analyze two algorithms for solving numerically some linear and nonlinear FDEs via shifted Chebyshev polynomials of the fifth kind. The linear case is treated by the application of a modified tau method, and the nonlinear case is treated by a modified tau-quadrature method. The contents of the paper are as follows. The next section concerns some preliminaries and fundamentals of the fractional calculus. Also, some properties and relations of a symmetric class of orthogonal polynomials involving the class of Chebyshev polynomials of the fifth kind are exhibited. In addition, connection formulae between Chebyshev polynomials of the first and fifth kinds are given in this section. Two numerical algorithms based on applying certain modified tau spectral methods for solving some types of linear and nonlinear FDEs are presented and analyzed in Sect. 3. A detailed study of the convergence and error analysis of the proposed Chebyshev expansion is displayed in Sect. 4. Numerical examples accompanied with some comparisons are given in Sect. 5. Finally, some remarks and discussions are presented in Sect. 6.
2 Preliminaries and some fundamentals This section aims to present some fundamentals of the fractional calculus theory. We also give an overview on a basic class of symmetric orthogonal polynomials. Moreover, we introduce the so-called the orthonormal Chebyshev polynomials of the fifth kind and develop their connections with Chebyshev polynomials of the first kind which will be very useful in developing our proposed algorithms.
2.1 A brief account on some essentials of the fractional calculus ρ
Definition 1 Let 0 It denote the Riemann–Liouville fractional integral operator of order ρ on the usual Lebesgue space L 1 [0, 1]. For all t ∈ (0, 1), it is defined as: ⎧ t ⎨ 1 (t − τ )ρ−1 f (τ ) dτ, ρ > 0, ρ (1) (0 It f )(t) = (ρ) 0 ⎩ f (t), ρ = 0. Definition 2 The right side Riemann–Liouville fractional derivative of order ρ > 0 is defined as: m d ρ m−ρ (D∗ f )(t) = (0 It f )(t), m − 1 < ρ ≤ m, m ∈ N. (2) dt
123
W. M. Abd-Elhameed, Y. H. Youssri
Definition 3 In Caputo sense, the fractional differential operator is defined as: t 1 ρ (D f )(t) = (t − τ )m−ρ−1 f (m) (τ ) dτ, ρ > 0, t > 0, (m − ρ) 0
(3)
where m − 1 < ρ ≤ m, m ∈ N. Remark 1 The reason for using the fractional derivative in the Caputo sense is that it is mathematically rigorous than the Riemann–Liouville sense (one can be referred to Changpin te al. 2011; Li and Zhao 2011). Caputo definition is advantageous in many disciplines such as applied science and engineering (Li and Deng 2007). Moreover, properties of the Caputo derivative are useful in translating the higher fractional-order differential systems into lower ones (Li and Deng 2007). Some comparisons between the two operators of Caputo and Riemann–Liouville can be found in Ishteva (2005). Now, for m − 1 ≤ γ < m, D γ satisfies the following properties: γ
(D γ 0 It f )(t) = f (t), γ
( 0 It D γ f )(t) = f (t) −
m−1 =0
f () (0+ ) (t − a) , t > 0, !
( + 1) D t = t −γ , ∈ N, ≥ γ . ( + 1 − γ ) γ
Note that the notation γ denotes the smallest integer greater than or equal to γ . For survey on fractional derivatives and integrals, one can consult Machado and Kiryakova (2017), Oldham (1974), and Podlubny (1998).
2.2 A symmetric class of orthogonal polynomials This section is concerned with presenting some properties of the polynomials of the BCSOP which was developed in Masjed-Jamei (2006). The main idea for establishing this class of polynomials is based on using an extended Sturm–Liouville problem. To be more precise, the author in Masjed-Jamei (2006) assumed that y = φi (t) is a sequence of symmetric functions which satisfies the following second-order differential equation: A1 (t) φi (t) + A2 (t) φi (t) 1 + (−1)i+1 + μi A3 (t) + A4 (t) + A5 (t) φi (t) = 0, 2
(4)
where Ai (t), 1 ≤ i ≤ 5 are independent functions and {μi } is a sequence of constants. In Masjed-Jamei (2006), it has been showed that Ai (t), i = 1, 3, 4, 5 are even functions, while A2 (t) is odd. The desired symmetric class of orthogonal polynomials can be obtained if Ai (t), 1 ≤ i ≤ 5, {μi } are chosen as follows: A1 (t) = t 2 ( p t 2 + q), A4 (t) = 0,
A2 (t) = t (m t 2 + n),
A3 (t) = t 2 ,
A5 (t) = −n, μi = −i (m + (i − 1) p) ,
where m, n, p, q are free real parameters. The above choices yield the following differential equation: t 2 ( p t 2 + q) φi (t) + t (m t 2 + n)φi (t)
n − i (m + (i − 1) p) t 2 + (1 + (−1)i+1 ) φi (t) = 0. 2
123
(5)
Fifth-kind orthonormal Chebyshev polynomial solutions for… m,n, p,q
The polynomial solution of (5) is the generalized polynomials denoted by G i have the following explicit form: ⎛ ⎞ ⎞ ⎛ 2i i 2i −k−1
1 ⎜ 2 ⎜ ⎟ i−2k ⎟ m,n, p,q (t) = Gi ⎝ ⎠t ⎠, ⎝ i, j,m,n, p,q k k=0
(t) which
j=0
where i, j,m,n, p,q =
(2 j + (−1)i+1 + 2)q + n (2 j + (−1)i+1 + 2 2i ) p + m
.
In addition, the author in Masjed-Jamei (2006) introduced the monic symmetric orthogonal m,n, p,q polynomials G¯ i (t) defined as ⎛ ⎞ 2i −1 ⎜ ⎟ m,n, p,q m,n, p,q (t) = ⎝ i, j,m,n, p,q ⎠ G i (t). G¯ i j=0
The monic polynomials G¯ i
m,n, p,q
(t) satisfy the following recurrence relation:
m,n, p,q m,n, p,q m,n, p,q (t) + Ai,m,n, p,q G¯ i−1 (t), i ≥ 0, G¯ i+1 (t) = t G¯ i
(6)
with the following initial values: m,n, p,q m,n, p,q G¯ 0 (t) = 1, G¯ 1 (t) = t,
and Ai,m,n, p,q
p q i 2 + (m − 2 p) q + (−1)i+1 p n i + 21 (m − 2 p)n 1 + (−1)i+1 = . (2 p i + m − p)(2 p i + m − 3 p) (7) m,n, p,q (t) G¯ i
can be found in Masjed-Jamei More properties of the general class of (2006). Many classes of important orthogonal polynomials are particular polynomials of G¯ i, j,m,n, p,q (t). The monic Chebyshev polynomials of the four kinds can be obtained by the following expressions: T¯i (t) = G¯ i−1,0,−1,1 (t), U¯ i (t) = G¯ i−3,0,−1,1 (t), 1+t 1−t , W¯ i (t) = 2i G¯ −3,2,−1,1 , V¯i (t) = 2i G¯ −3,2,−1,1 2i
2
2i
2
and T¯i (t), U¯ i (t), V¯i (t), W¯ i (t), are, respectively, the monic Chebyshev polynomials of the first, second, third and fourth kinds. Moreover, some other classes of polynomials can be m,n, p,q obtained as particular classes of G¯ i (t). Among these classes, the two kinds of orthogonal polynomials introduced in Masjed-Jamei (2006), namely, Chebyshev polynomials of the fifth and sixth kinds. These polynomials can be, respectively, defined as: X¯ i (t) = G¯ i−3,2,−1,1 (t), Y¯i (t) = G¯ i−5,2,−1,1 (t).
123
W. M. Abd-Elhameed, Y. H. Youssri
From now on, we restrict our study for Chebyshev polynomials of the fifth kind and their shifted ones. The orthogonality relation of X¯ i (t) is ⎧ i ⎪ ⎪ π 1 ⎨ i A,−3,2,−1,1 , if i = j, (−1) t2 2 X¯ i (t) X¯ j (t) dt = (8) √ =1 2 ⎪ 1−t −1 ⎪ ⎩0, if i = j, and Ai,m,n, p,q is given in (7). The above orthogonality relation can be written alternatively as: 1 t2 h i , if i = j, X¯ i (t) X¯ j (t) dt = √ 0, if i = j, 1 − t2 −1 and
⎧ π ⎪ if i even, ⎨ 2i+1 , 2 h i = π(i + 2) ⎪ ⎩ , if i odd. i 22i+1
(9)
(10)
In the upcoming sections, it is more convenient to normalize the monic Chebyshev polynomials of the fifth kind. For this purpose, we define 1 X i (t) = √ X¯ i (t). hi Therefore, X i (t) are orthonormal on [−1, 1] in the sense that: 1 t2 1, if i = j, X i (t) X j (t) dt = √ 0, if i = j. 1 − t2 −1
(11)
2.3 Shifted orthonormal Chebyshev polynomials of the fifth kind The shifted fifth-kind Chebyshev polynomials Ci (t) can be defined on [0, 1] by: 1 Ci (t) = √ X¯ i (2t − 1), hi
(12)
and h i is given in (10). From (11), it is not difficult to note that Ci (t), i ≥ 0, are orthonormal on [0, 1]. Explicitly, we have 1 1, if i = j, ∗ w (t) Ci (t) C j (t) dt = (13) 0, if i = j, 0 (2t − 1)2 . and w ∗ (t) = √ t − t2 Now, we are going to derive the connection formulae between the introduced polynomials Ci (t) and the shifted Chebyshev polynomials of the first kind Ti∗ (t). These formulae will be very useful in what follows.
123
Fifth-kind orthonormal Chebyshev polynomial solutions for…
Theorem 1 The polynomials Ci (t) defined in (12) are linked with Ti∗ (t) by the formula Ci (t) =
i
gi, j T j∗ (t),
(14)
j=0
where gi, j = 2
i− j 2 (−1) 2 π
and
δj =
⎧ ⎪ δ , i f i and j ar e even, ⎪ ⎨ j j , i f i and j ar e odd, ⎪ i ⎪ ⎩ 0 other wise, 1 2,
i f j = 0, 1, i f j > 0.
(15)
Proof We show (14) by induction. Let the connection formula (14) be true for (i − 1) and (i − 2) and we will show that (14) is itself valid. From (6), it is not difficult to see that the following recurrence relation is satisfied by Ci (t): h i−1 h i−2 Ci−1 (t) + Bi Ci−2 (t), (16) Ci (t) = (2t − 1) hi hi where h i is given in (10) and Bi = −
(i − 1)2 + i + (−1)i (2i − 1) . 4 i(i − 1)
Based on the above relation and some elementary properties of Ti∗ (t), Theorem 1 can be proved.
Remark 2 The connection formula in Theorem 1 can be split into two separate connection formulae. Thus we can rewrite Theorem 1 as follows. Theorem 2 The polynomials Ci (t) defined in (12) are linked with Ti∗ (t) by the following two formulae: i 2 C2i (t) = 2 (−1)i+r δr T2r∗ (t), (17) π r =0
where δr is defined in (15), and
√ i 2 2 C2i+1 (t) = √ (−1)i+r (2r + 1) T2r∗ +1 (t). π (2i + 1)(2i + 3) r =0
(18)
Now, we are able to give a trigonometric representations for X 2i (t) and X 2i+1 (t) . The following corollary displays these important representations. Corollary 1 The following trigonometric representations are valid: 2 cos((2i + 1) φ) X 2i (cos φ) = , π cos φ
(19)
123
W. M. Abd-Elhameed, Y. H. Youssri
and
2 π (2i + 1)(2i + 3)
X 2i+1 (cos φ) = ×
(2i + 3) cos φ cos((2i + 2) φ) − cos((2i + 3) φ) . cos2 φ
(20)
Proof The two trigonometric representations in (19) and (20) can follow from the two connection formula (17) and (18) together with the two trigonometric identities: (−1)i + 2
i cos((2i + 1) φ) (−1)i+r cos(2 r φ) = , cos φ r =1
and i (−1)i+r (2r + 1) cos((2r + 1) φ) r =0
=
(2i + 3) cos φ cos((2i + 2) φ) − cos((2i + 3) φ) . 2 cos2 φ
Now, we give the inversion formulae to the connection formulae (17) and (18). More precisely, we give the connection formulae that links Ti∗ (t) and Ci (t). Theorem 3 The polynomials Tr∗ (t) are linked with the polynomials Ci (t) by the following two formulae: 1 π C2r (t) ∗ T2r (t) = + C2r −2 (t) , r ≥ 0, (21) 2 2 δr and T2r∗ +1 (t)
1 = 2
π 2
2r + 3 C2r +1 (t) + 2r + 1
2r − 1 C2r −1 (t) , r ≥ 1. 2r + 1
(22)
Proof The two connection formulae (21) and (22) can be obtained directly from the two trigonometric identities: cos ((2r − 1) φ) + cos ((2r + 1) φ) = 2 cos φ cos(2 r φ),
(23)
and (2r + 1) cos φ cos(2 r φ) + (2r + 3) cos φ cos((2r + 2) φ) − cos((2r + 1) φ) − cos((2r + 3) φ) = 2(2r + 1) cos2 φ cos(2r + 1) φ.
(24)
Now, we are in a position to sate two important theorems. The first gives the analytic form of Ci (t) and the second gives its inversion formula. First, the following Lemma is needed.
123
Fifth-kind orthonormal Chebyshev polynomial solutions for…
Lemma 1 The power form representation of the polynomial T∗ (t) and its inversion formula are given as follows: T∗ (t) =
(−1)r + 22r (r + − 1)! r t , (2r )! ( − r )!
(25)
r =0
and t = 21−2 (2)!
j=0
δj T ∗ (t). ( − j)! ( + j)! j
(26)
Theorem 4 The analytic form of Ci (t) is given explicitly as Ci (t) =
i
p,i t p ,
(27)
p=0
where
p,i
⎧ i 2i ⎪ (−1) 2 + j− p j δ j (2 j + p − 1)! ⎪ ⎪ , ⎨2 2 j= p+1 (2 j − p)! 2 = √ i+1 i−1 π (2 p)! ⎪ 1 (−1) 2 + j− p (2 j + 1)2 (2 j + p)! ⎪ 2 ⎪ ⎩√ , p j= 2
(2 j − p + 1)! i(i + 2) 2 p+ 23
i even, i odd. (28)
Theorem 5 The inversion formula to the analytic formula (27) can be written explicitly as tm =
m
q,m C (t),
(29)
=0
where
q,m
√ 1 = π 2−2m− 2
⎧ 2 ( + 1)2 + m 2 + m ⎪ ⎪ ⎪ , ⎪ ⎨ (m − )!( + m + 2)! (2m)! +2 ⎪ ⎪ ⎪ +2 ⎪ ⎩ + , (m − )!(m + )! (m − − 2)!(m + + 2)!
even, odd.
Proof The proofs of Theorems 4 and 5 are based essentially on the application of Lemma 1 along with the four connection formulae (17), (18), (21) and (22).
Now, some important special values of the qth derivative of Ci (t) can be deduced from Theorem 1. The following corollary displays these results. Corollary 2 For every nonnegative integer q, the following two identities hold: D q Ci (t) t=1 = αi,q , D q Ci (t) t=0 = (−1)i+q αi,q ,
(30) (31)
where αi,q =
i
α¯ i, j,q ,
(32)
j=0
123
W. M. Abd-Elhameed, Y. H. Youssri
and
α¯ i, j,q
⎧ ⎪ if i and j are even, ⎪ √ i− j ⎪δ j , 2 2 2 (−1) ( j)q ( j − q + 1)q ⎨ j i = , if i and j are odd, ⎪ q + 21 ⎪i i +2 ⎪ ⎩0 otherwise,
(33)
where the notation ( j)q denotes the well-known Pochhammer symbol. Proof The two identities in (30) and (31) can be easily obtained with the aid of the connection formula (14) along with the following two identities: √ π(i)q (i − q + 1)q q ∗ D Ti (t) t=1 = , q + 21 and D
q
Ti∗ (t) t=0
= (−1)
i+q
√ π(i)q (i − q + 1)q . q + 21
Now, let f (t) ∈ L 2w∗ [0, 1] and assume that it can be approximated as: f (t) ≈
N
a C (t),
=0
where a =
0
1
(2t − 1)2 f (t) C (t) dt. √ t − t2
(34)
The fractional derivatives of Ck (t) are given in the following theorem. Theorem 6 The Caputo fractional derivative of order α of Ck (t) is given by ⎧ k ⎪ ⎨ t j−α , k ≥ α; j,k α D Ck (t) = j=α ⎪ ⎩ 0, otherwise, where j,k =
(35)
j! j,k , and j,k is defined in (28). ( j + 1 − α)
3 Two numerical algorithms for FDEs This section describes two algorithms for solving numerically some types of linear and nonlinear FDEs. The linear case is treated with the aid of a modified tau method, while the nonlinear case is handled with the aid of a certain tau-quadrature algorithm. The following theorem is useful in the sequel.
123
Fifth-kind orthonormal Chebyshev polynomial solutions for…
Theorem 7 Mysovskikh (2002) (Romberg’s integration formula) Suppose that z(t) is sufficiently differentiable function on [0, 1], then
1
z(t) dt = rn,m (z) + E n,m ,
0
where rn,m are given iteratively by: r0,0 (z) =
z(0) + z(1) , 2 n−1
2 1 z 2−n (2k − 1) , rn,0 (z) = rn−1,0 + 2−n 2 k=1
1 (rn,m−1 − rn−1,m ), rn,m (z) = rn,m−1 + m 4 −1 with n ≥ m ≥ 1 and
E n,m = O 4−n(m+1) . For more details about numerical quadrature formulae, one can consult Gautschi (2012).
3.1 Shifted fifth-kind Chebyshev modified tau method for solving linear FDEs This section presents a numerical technique built on the application of the shifted fifth-kind Chebyshev modified tau method (S5CMTM) for solving the following linear FDEs: D αn y(t) +
n−1
ξ (t) D α y(t) + ρ(t) y(t) = σ (t), t ∈ (0, 1),
(36)
=1
subject to the initial conditions: y ( j) (0) = a¯ j ,
j = 0, 1, . . . , n − 1, n ≥ 1,
(37)
or the boundary conditions ⎧ ⎪ ⎨ y ( j) (0) = a¯ j ,
y ( j) (1) = b¯ j ,
⎪ ⎩ y ( j) (0) = a¯ , j
y ( j) (1) = b¯ j ,
n−2 , if n even and n ≥ 2, 2 n−1 n−3 , if n odd and n ≥ 3, y ( 2 ) (0) = a, ¯ j = 0, 1, . . . , 2 j = 0, 1, . . . ,
(38) where α1 < α2 < · · · < αn and α j ∈ ( j − 1, j]; j ∈ N, and ξ (t), ρ(t) and σ (t) are continuous functions. We seek for an approximate solution of problem (36) in terms of the polynomials Ci (t) defined in (12). Consider the following approximate solution: y(t) ≈ y N (t) =
N
a C (t).
(39)
=0
123
W. M. Abd-Elhameed, Y. H. Youssri
Based on Theorem 6, the residual of Eq. (36) is given by R N (t) =
r N
ar i,r t r −αn +
r =0 i=αn
+
N r
N r n−1
ar i,r ξ (t) t r −α
=1 r =0 i=α
ar ρ(t) qi,r t i − σ (t).
r =0 i=0
Now, if we choose the following test functions φs (t) = t s ,
s = 0, 1, . . . N − n,
then the application of a modified spectral tau method yields the system r N r =0 i=αn
+
ar i,r r + s − αn + 1
r N n−1
ar i,r λr,s,
=1 r =0 i=α
+
N r
ar qi,r μi,s − σs = 0, s = 0, 1, . . . , N − n,
(40)
r =0 i=0
where
λr,s, =
1
0
μi,s =
1
ξ (t) t r +s−α dt, ρ(t) t i+s dt,
0
σs =
1
σ (t) t s dt.
0
The use of the initial conditions (37) along with relation (31) yields N
ar (−1)r +i αr,i = a¯ i , i = 0, 1, . . . , n − 1,
n ≥ 1,
(41)
r =0
where αr,i are given by (32). Moreover, the use of the boundary conditions (38) along with the two relations (30) and (31) yields ⎧ N N n−2 ⎪ ar (−1)i+r αr,i = a¯ i , ar αr,i = b¯i , i = 0, 1, . . . , , if n even and n ≥ 2, ⎪ ⎪ r =0 r =0 ⎪ 2 ⎨ N N N n−1 n−3 ar (−1)i+r αr,i = a¯ i , ar αr,i = b¯i , ar (−1)r + 2 αr, n−1 = a, ¯ i = 0, 1, . . . , , ⎪ ⎪ r =0 r =0 r =0 2 2 ⎪ ⎪ ⎩ if n odd and n ≥ 3.
(42) Finally, equations (40) and (41) or (40) and (42), generate a system of (N +1) linear algebraic equations in the (N +1) unknowns {ai }0≤i≤N . Thanks to the Gaussian elimination procedure or any other suitable procedure, the desired numerical solution in (39) can be found.
123
Fifth-kind orthonormal Chebyshev polynomial solutions for…
3.2 Shifted fifth-kind Chebyshev modified tau-quadrature method for nonlinear FDEs This section proposes a numerical method based on the application of shifted fifth-kind Chebyshev modified tau-quadrature method (S5C M T Q M) to numerically solve the following nonlinear fractional differential equation: D αn y(t) = F t, D α1 y(t), D α2 y(t), . . . , D αn−1 y(t) , t ∈ (0, 1), (43) subject to the initial conditions (37) or the boundary conditions (38), where α1 < α2 < · · · < αn , and αi ∈ (i − 1, i]; i ∈ N, and F is a nonlinear (transcendental) continuous function. We propose a numerical solution given in (39). The residual of Eq. (43) is given by R N (t) = D αn y N (t) − F t, D α1 y N (t), D α2 y N (t), . . . , D αn−1 y N (t) ⎛ N N N r r r ar i,r t r −αn − F ⎝t, ar i,r t r −α1 , ar i,r t r −α2 , . . . , = r =0 i=αn N
r
⎞
r =0 i=α1
r =0 i=α2
ar i,r t r −αn−1 ⎠ .
r =0 i=αn−1
Again, we choose the following test functions φs (t) = t s , s = 0, 1, . . . N − n, and apply a modified tau method to obtain 1 R N (t) t s dt = 0, s = 0, 1, . . . , N − n. 0
Direct application of the Romberg’s integration formula mentioned in Theorem 7 with z(t) = t s R N (t) = χs (t), and n = 4, m = 6, i.e., E n,m = O(10−17 ), yields 1 χs (t) dt ≈ r4,6 (χs ) = 0, s = 0, 1, . . . , N − n.
(44)
0
Finally from (44) and (41) or (44) and (42), we have a system of (N + 1) nonlinear algebraic equations in the (N + 1) unknowns {ai }0≤i≤N . Thanks to the Newton’s iterative procedure, the desired numerical solution can be obtained. Remark 3 Although our algorithms are applied to the FDEs (36) and (43) defined in (0, 1), they can be easily extended to solve the same FDEs but defined in (0, L). This needs some minor modifications in our proposed algorithms.
4 Convergence and error analysis of the proposed expansion In this section, we present a detailed study on the convergence and error analysis of the shifted orthonormal fifth Chebyshev kind expansion. First, we need the following lemma in which we give an upper bound to the polynomials C (t), ≥ 0.
123
W. M. Abd-Elhameed, Y. H. Youssri
Lemma 2 The polynomials C (t) are bounded on [0, 1]. Explicitly, the following inequality is satisfied for all ≥ 0 2 |C (t)| < ( + 2), ∀ t ∈ [0, 1]. (45) π Proof To show the inequality in (45), the following two cases are considered. The first case: = 2m: The connection formula in (17) along with the simple inequality: |T∗ (t)| ≤ 1, yields
|C (t)| ≤ 2 =2 =
m 2 1 π r =0
2 (m + 1) π
2 ( + 2). π
The second case: = 2m + 1: Making use of the connection formula in (18) together with the inequality: |T∗ (t)| ≤ 1, yields √ m 2 2 (2r + 1) |C (t)| ≤ √ π (2m + 1)(2m + 3) r =0 √ 2 2 = √ (m + 1)2 π (2m + 1)(2m + 3) √ 2 2 (m + 1)2 < √ π 2m + 1 √ 2 ≤ √ (2m + 2) π √ 2 = √ ( + 1) π 2 ( + 2). < π From the previous two cases, the following estimate holds for all ≥ 0 2 |C (t)| < ( + 2), ∀ t ∈ [0, 1]. π
Theorem 8 Assume that a function f (t) ∈ L 2w∗ [0, 1] with | f (3) (t)| L , and assume that it has the following expansion: f (t) =
∞ =0
123
a C (t).
(46)
Fifth-kind orthonormal Chebyshev polynomial solutions for…
The series in (46) is uniformly convergent to f (t). In addition, the following bound on the expansion coefficients in (46) holds √ 2π L |a | < , ∀ > 3. (47) 2 3 Proof We start from (34) to write a =
1 0
w1∗ f (t) C (t) dt.
With the aid of the substitution: x = 21 (1 + cos φ), the coefficients a can be written as 1 1 a = f (48) (1 + cos φ) X (cos φ) cos2 φ dφ. 2 0 Now, to show the validity of the inequality in (47), the following two cases are considered: The first case: even: With the aid of Eq. (19), Eq. (48) can be turned into π 1 1 (1 + cos φ) [cos(φ) + cos(( + 2)φ)] dφ. f a = √ 2 2π 0 The integration by parts two times on the right-hand side of the latter equation leads to the formula π 1 1 a = √ f (49) (1 + cos φ) (φ) dφ, 2 16 2π 0 where
1 1 (φ) = (cos(( − 2)φ) − cos( φ)) −1 1 (cos(φ) − cos(( + 3)φ)) − +1 1 1 (cos( φ) − cos(( + 2)φ)) − +2 +1 1 − (cos(( + 2)φ) − cos(( + 4)φ)) . +3
(50)
If we integrate the right-hand side of (49) again by parts and make use of the hypothesis of the theorem, then the following inequality is obtained: √ 2π L ( + 1) |a | < . 128 − 21 4
With the aid of the inequality: +1 128( − 21 )4 we get
<
1 , 23
√ 2π L . |a | < 2 3
(51)
123
W. M. Abd-Elhameed, Y. H. Youssri
The second case: odd: From (20) and (48), one can write π 1 2 a = (1 + cos φ) f π ( + 1) 0 2 [( + 2) cos(( + 1)φ) cos φ − cos(( + 2)φ)] dφ.
(52)
The application of integration by parts three times on the right-hand side of (52) along with the hypothesis of the theorem yields 2 3 + 11 2 + 38 + 24 2 |a | ≤ L π. π ( + 1) 16( − 2)( + 2)2 ( + 3) Now, and based on the two inequalities: 1 1 < , +1
2 3 + 11 2 + 38 + 24 1 < 3, 16( − 2)( + 2)2 ( + 3) 2
we get √ |a | <
2π L . 2 4
(53)
Finally, from (51) and (53), for every > 3, we have √ 2π L |a | < , 2 3
and this completes the proof of the theorem.
Theorem 9 If u(t) satisfies the hypothesis of Theorem 8 and u N +1 (t), u N (t) are two consecutive approximations of u(t), and if we define e¯ N (t) = u N +1 (t) − u N (t), then we have the following error estimate: e¯ N (t)2,w∗ = O(N −3 ), where e¯ N (t)2,w∗ denotes the L 2 -norm of e¯ N (t). Proof From the orthonormality of Ck (t), and direct application of Theorem 8, we have e¯ N (t)2,w∗ = u N +1 (t) − u N (t)2,w∗ =
N +1
ak Ck (t) −
k=0
N
ak Ck (t)2,w∗
k=0
= a N +1 C N +1 (t)2,w∗ = |a N +1 | √ 2π L < 2 (N + 1)3 = O(N −3 ). The theorem is now proved.
123
Fifth-kind orthonormal Chebyshev polynomial solutions for…
Theorem 10 Assume that f (t) satisfies the assumptions of Theorem 8, and let e N (t) = ∞ a C (t) be the global error. e N (t) can be estimated as: =N +1
|e N (t)| < 3L/N . Proof Direct application of Theorem 8 together with Lemma 2 leads to |e N (t)| < L
∞ +2 . 3
=N +1
Moreover, the inequality:
+2 +2 , < 3 (2 − 1)
implies that ∞
|e N (t)| < L
=N +1
Finally, noting the identity:
2N +3 2N (N +1)
<
+2 (2N + 3)L = . (2 − 1) 2N (N + 1)
3 N,
∀ N ≥ 1, the following estimation is obtained:
|e N (t)| < 3L/N .
The proof of Theorem 10 is now complete.
5 Some illustrative examples This section presents some illustrative examples supported with some comparisons aiming to demonstrate the efficiency, applicability and high accuracy of the suggested algorithms. Let E denote the error in maximum norm, that is E = max |y(t) − y N (t)|. t∈[0,1]
Example 1 (Diethelm and Ford 2002; Bhrawy and Zaky 2016; Youssri 2017) Consider the linear Bagley–Torvik IVP: 3
D 2 y(t) + D 2 y(t) + y(t) = 1 + t,
y(0) = y (0) = 1, t ∈ (0, 1),
with the exact smooth solution y(t) = 1+t. We apply S5CMTM for the case N = 2. Indeed, let 2 2 c0 + √ c1 (2t − 1) + c2 (16 t 2 − 16 t + 1) . y(t) ≈ y2 (t) = π 3 The application of the S5CMTM which is introduced in Sect. 3.1 yields the following system: √ 3 3 60 π 2 c0 + (1920 π 2 + 7168) c2 = 45 2π 2 , 4 π , √ c1 − 16 c2 = 2 3 √ π . 3 c0 − 2 3 c 1 + 3 c 2 = 3 2
123
W. M. Abd-Elhameed, Y. H. Youssri Table 1 Maximum absolute error of Example 2 N
2
4
6
8
10
12
14
E
3.55 × 10−5
6.67 × 10−8
1.99 × 10−10
1.05 × 10−12
1.35 × 10−13
9.41 × 10−13
2.73 × 10−15
Error
1.2 × 10−10 1.× 10−10 8.× 10−11 6.× 10−11 4.× 10−11 2.× 10−11
t
0 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 1 Error of Example 2 in case N = 6
The solution of the latter system is 3 π 1 3π , c1 = , c2 = 0, c0 = 2 2 4 2 and therefore y2 (t) = 1 + t, which coincides with the exact solution. Example 2 Consider the nonlinear fractional IVP 1
D 2 y(t) + e y(t) = f (t), t ∈ (0, 1),
(54)
y(0) = ln 9,
(55)
with
where f (t) = 9 + t + √
√ 2 ar csinh( t/3). π(9 + t)
The analytic solution of (54) is y(t) = ln(9 + t). We apply S5CMTQM. In Table 1, we list the maximum absolute error of problem (54)–(55) for different values of N . Moreover, the absolute error in case N = 6 is displayed in Fig. 1. Example 3 (Abd-Elhameed and Youssri 2014) Consider the Riccati fractional differential equation D α y(t) + y 2 (t) = 1, t ∈ (0, 1), α ∈ (0, 1],
(56)
governed by the initial condition y(0) = 0.
123
(57)
Fifth-kind orthonormal Chebyshev polynomial solutions for… Table 2 Comparison between S5C M T Q M and the method in Abd-Elhameed and Youssri (2014) for Example 3 t
α = 0.7
α = 0.8
Abd-Elhameed and Youssri (2014)
α = 0.9
Exact
S5CMTQM Abd-Elhameed and Youssri (2014)
S5CMTQM Abd-Elhameed and Youssri (2014)
S5CMTQM
0.1 0.209216
0.171333
0.165498
0.138091
0.129138
0.101874
0.099684
0.3 0.429549
0.393889
0.383197
0.351883
0.336448
0.295195
0.291316
0.5 0.556331
0.537862
0.530743
0.510705
0.498915
0.465588
0.462083
0.7 0.643854
0.636781
0.637215
0.628023
0.624307
0.606359
0.604405
0.9 0.707567
0.706583
0.714519
0.713562
0.717972
0.716482
0.716264
0.8 uN (t)
0.6
Exact ( =1) =0.9
0.4
=0.8 =0.7
0.2
t
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 2 Different solutions of Example 3
For α = 1, the exact solution of (56) is y(t) =
e2t − 1 . e2t + 1
We compare our results with those obtained by the method developed in Abd-Elhameed and Youssri (2014) in Table 2. Figure 2 illustrates that the approximate solutions have smaller variations for values of α near the value 1. The results of Table 2 ensure that the obtained numerical results are closer to the exact one than those resulted from the application of the method developed in Abd-Elhameed and Youssri (2014). Example 4 (Alkan et al. 2016) Consider the fractional BVP y (t) + t 2 y (t) − D 0.7 y(t) = g(t),
t ∈ (0, 1),
(58)
subject to the boundary conditions y(0) = y(1) = 0, and g(t) is chosen to be compatible with the exact solution: y(t) = t 4 (t − 1). We apply S5CMTM in case N = 5. In Table 3, our results are compared with those obtained in Alkan et al. (2016) using sinc approximation method. Figure 3 illustrates the absolute error of
123
W. M. Abd-Elhameed, Y. H. Youssri Table 3 Comparison between S5CMTM and the method in Alkan et al. (2016) for Example 4 t
0
Exact
0
Method in Alkan et al. (2016)
S5CMTM
Approximate
Approximate
E
0
0
E
0
0
0.1
−0.00009
−0.0029191
2.82 × 10−3
−0.00009
2.51 × 10−16
0.3
−0.00567
−0.0053383
3.31 × 10−4
−0.00567
3.90 × 10−16
−0.0294963
1.75 × 10−3
−0.03125
1.57 × 10−17
−0.0705373
1.49 × 10−3
−0.07203
3.88 × 10−16
−0.0704965
4.88 × 10−3
−0.06561
5.27 × 10−16
0.5 0.7 0.9 1
−0.03125 −0.07203 −0.06561 0
0
0
0
0
Error
8.× 10−16 6.× 10−16 4.× 10−16 2.× 10−16
t
0 0.0
0.2
0.4
0.6
0.8
1.0
Fig. 3 Absolute error of Example 4
Example 4. The results of Table 3 show that our numerical results with taking few terms of the proposed Chebyshev expansion are more accurate than those obtained by the method developed in Alkan et al. (2016). This of course, demonstrates the advantage of our method when compared with some other numerical methods. Example 5 Consider the fractional IVP y (4.5) (t) = F(t),
t ∈ (0, 1),
(59)
subject to the initial conditions y(0) = y (2) (0) = y (4) (0) = 0, where
t
F(t) = 0
y (1) (0) = −y (3) (0) = 1,
cos(t − y) dy. √ πy
The exact solution of (59) is y(t) = sin t. We apply S5CMTM to (59). In Table 4, we list the maximum pointwise error for different values of N .
123
Fifth-kind orthonormal Chebyshev polynomial solutions for… Table 4 Maximum absolute error of Example 5 N
7
9
11
13
15
17
19
E
9.47 × 10−6
7.15 × 10−9
2.53 × 10−11
3.68 × 10−12
2.57 × 10−14
2.22 × 10−16
2.22 × 10−16
Example 6 Consider the fractional IVP D 0.5 u(t) + u(t) = t 0.9 +
(1.9) 0.4 t , u(0) = 0, t ∈ (0, 1), (1.4)
(60)
with the exact non-smooth solution, u(t) = t 0.9 . With the aid of the transformation u = y 0.9 , Eq. (60) is turned into the nonlinear fractional initial value problem D 0.5 y 0.9 (t) + y 0.9 (t) = t 0.9 +
(1.9) 0.4 t , (1.4)
y(0) = 0, t ∈ (0, 1),
(61)
with the exact solution y(t) = t.
We start with y(t) ≈ y1 (t) = π2 a0 + π2 a1 (2t − 1), apply S5CMTQM for the case corresponding to N = 1, to get a0 = 0.6266570686577501, a1 = 0.5427009409187007, and consequently, 2 (0.6266570686577501) y1 (t) = π 2 + (0.5427009409187007)(2t − 1) = t + 5.55 × 10−17 , π which is almost the exact solution. Example 7 Consider the fractional oscillator problem D μ u(t) + u(t) = 0, u(0) = 1, u (0) = 0,
t ∈ (0, L),
(62)
where 1 < μ ≤ 2. The exact solution of Eq. (62), is given by u(t) =
∞ k=0
(−t μ )k = E μ −t μ , (k μ + 1)
where E μ (t) is the well-known Mittag-Leffler function. In case of μ = 2, this exact solution reduces to u(t) = cos t. We solve Eq. (62) for the two cases corresponding to L = 1 and L = 25. N Case 1: Assume the approximate solution u(t) ≈ u N (t) = ai Ci (t), and apply i=0 S5CMTM. We compare our proposed numerical results with the truncated series u(t) ˜ =
10 k=0
(−t μ )k , (k μ + 1)
for the case corresponding to N = 16 and different values of μ. Let E˜ represent the maximum absolute difference between the truncated series solution with the proposed numerical solution, that is ˜ E˜ = max |u N (t) − u(t)|. t∈[0,L]
123
W. M. Abd-Elhameed, Y. H. Youssri
All computations are executed by Mathematica 11. In addition, the steps required for performing the Mathematica program of Example 7 are given in Algorithm 1. The presented results in Table 5 illustrate that the method is efficient and accurate in the domain [0, 1]. Algorithm 1 Algorithm for Example 7 Given μ and N
Step 1. Step 2.
List
N r r =0 i=2
Step 3. Step 4. Step 5.
ar i,r qi,r L i+s+1 + = 0, ar r +s−μ+1 i +s+1 N
r
s = 0, 1, . . . , N − 2
r =0 i=0 Evaluate u N (0) = ω, u N (0) = 0
Join[Output 2, Output 3] Solve [Output 4]
Case 2: The spectral methods are inaccurate enough in the domain [0, L], when L >> 1. In such case, we can convert the Eq. (62) into a modified one defined in (0, 1). This will accelerate the convergence and lessen the resulting errors. For this purpose, we make use the transformation z = Lt , to get the modified equation L −μ D μ u(L z) + u(L z) = 0, u(0) = 1, u (0) = 0, z ∈ (0, 1).
(63)
In Fig. 4, we illustrate the behavior of the solution u(L z) = u(t), for different values of μ. Remark 4 It is important to report here that, the relation between the fractional derivatives over two different intervals is discussed before (see, relation (14) in Zayernouri and Karniadakis 2014). Also, the transformation used to handle Example 7 is similar to that used in Zayernouri and Karniadakis (2014).
Table 5 Maximum pointwise error of Example 7 (N = 16) μ
2
1.95
1.9
1.85
1.8
1.75
E
2.22 × 10−16
3.47 × 10−14
2.52 × 10−13
5.78 × 10−12
2.71 × 10−11
3.25 × 10−11
1.0 uN (t)
0.5
Exact ( =2) =1.9
0.0
=1.8 =1.7
−0.5 t
−1.0 0
5
10
Fig. 4 Different solutions of Example 7 (L = 25)
123
15
20
25
Fifth-kind orthonormal Chebyshev polynomial solutions for… Table 6 Maximum stepwise error of Example 8 N
2
4
6
8
10
12
14
eN
9.62 × 10−2
7.23 × 10−5
3.67 × 10−7
2.39 × 10−10
5.28 × 10−11
1.78 × 10−12
1.53 × 10−14
τ
3.31
7.24
16.74
35.24
81.27
211.83
296.34
Table 7 Maximum absolute error of Example 8 N
2
4
6
8
10
12
14
E
8.27 × 10−3
5.29 × 10−7
3.64 × 10−9
7.34 × 10−11
8.53 × 10−13
6.82 × 10−15
2.22 × 10−17
τ
2.42
6.34
10.84
25.17
53.17
86.12
128.68
Example 8 Consider the linear fractional eigenvalue problem D γ u(t) = λu(t), u(0) = 1, t ∈ (0, 1),
(64)
where 0 < γ ≤ 1. The exact solution of Eq. (64), in case γ = 1 is given by u(t) = eλ t . N We start with u(t) ≈ u N (t) = ai Ci (t). The exact solution of Eq. (64) is not available i=0 if γ < 1. We list in Table 6, the maximum stepwise error defined by: e N = maxt∈[0,1] |u N (t)− u(t)| for the case corresponding to γ = 21 and λ = 1 for different values of N . In addition, we list the CPU time τ (the time used to run the algorithm in seconds) in this table. Moreover, Table 7 lists the maximum pointwise error E and τ for the case corresponding to γ = 1 for different values of N .
6 Conclusions and discussion In this paper, a new kind of Chebyshev polynomials called Chebyshev polynomials of the fifth-kind was employed for treating some types of linear and nonlinear FDEs. The main idea behind the proposed algorithms was built on utilizing a modified spectral tau method along with the application of the Romberg’s integration formula to reduce the FDEs governed by their initial/boundary conditions into systems of linear/nonlinear algebraic equations which can be solved. The convergence and error analysis were deeply discussed through developing new connection formulae between Chebyshev polynomials of the first and fifth kinds. Finally, some numerical examples are presented to demonstrate the high performance of the proposed algorithms. The obtained results ensure that our results compare favorably with the exact ones. As far as we know, this is the first time this class of polynomials is used for handling fractional differential equations. In addition, we do believe that this kind of polynomials can be utilized for solving other types of differential equations. Acknowledgements The authors would like to thank the referees for carefully reading the paper and also for their constructive and valuable comments which have greatly improved the paper.
123
W. M. Abd-Elhameed, Y. H. Youssri
References Abd-Elhameed WM, Youssri YH (2014) New ultraspherical wavelets spectral solutions for fractional Riccati differential equations. Abstr Appl Anal 2014. doi:10.1155/2014/626275 Abd-Elhameed WM, Doha EH, Bassuony MA (2015) On the coefficients of differentiated expansions and derivatives of Chebyshev polynomials of the third and fourth kinds. Acta Math Sci 35(2):326–338 Abd-Elhameed WM, Doha EH, Youssri YH, Bassuony MA (2016) New Tchebyshev–Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations. Numer Methods Partial Differ Equ 32(6):1553–1571 Abd-Elhameed WM, Youssri YH (2015) New spectral solutions of multi-term fractional order initial value problems with error analysis. Cmes Comput Model Eng 105:375–398 Abd-Elhameed WM, Youssri YH (2016) A novel operational matrix of Caputo fractional derivatives of Fibonacci polynomials: spectral solutions of fractional differential equations. Entropy 18(10):345 Abd-Elhameed WM, Youssri YH (2016) Spectral solutions for fractional differential equations via a novel Lucas operational matrix of fractional derivatives. Rom J Phys 61(5–6):795–813 Abd-Elhameed WM, Youssri YH (2017) Generalized Lucas polynomial sequence approach for fractional differential equations. Nonlinear Dyn 89(2):1341–1355 Alkan S, Yildirim K, Secer A (2016) An efficient algorithm for solving fractional differential equations with boundary conditions. Open Phys 14(1):6–14 Bhrawy AH, Zaky MA, Machado JAT (2016) Numerical solution of the two-sided space-time fractional telegraph equation via Chebyshev tau approximation. J Optim Theory Appl 174(1):321–341 Bhrawy AH, Taha TM, Machado JAT (2015) A review of operational matrices and spectral techniques for fractional calculus. Nonlinear Dyn 81(3):1023–1052 Bhrawy AH, Zaky MA, Machado JAT (2016) Efficient Legendre spectral tau algorithm for solving the twosided space–time caputo fractional advection–dispersion equation. J Vib Control 22(8):2053–2068 Bhrawy AH, Zaky MA (2016) Shifted fractional-order Jacobi orthogonal functions: application to a system of fractional differential equations. Appl Math Model 40(2):832–845 Bhrawy AH, Zaky MA (2017) An improved collocation method for multi-dimensional space–time variableorder fractional Schrödinger equations. Appl Numer Math 111:197–218 Bhrawy AH, Zaky MA (2017) Highly accurate numerical schemes for multi-dimensional space variable-order fractional Schrödinger equations. Comput Math Appl 73(6):1100–1117 Boyd JP (2001) Chebyshev and fourier spectral methods. Courier Corporation, North Chelmsford Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics. Springer, New York Çenesiz Y, Keskin Y, Kurnaz A (2010) The solution of the Bagley–Torvik equation with the generalized Taylor collocation method. J Franklin Inst 347(2):452–466 Changpin Li, Deliang Qian, YangQuan Chen (2011) On Riemann–Liouville and caputo derivatives. Discrete Dyn Nat Soc 2011:15. Article ID 562494 Costabile F, Napoli A (2015) A method for high-order multipoint boundary value problems with Birkhoff-type conditions. Int J Comput Math 92(1):192–200 Costabile F, Napoli A (2015) Collocation for high order differential equations with two-points Hermite boundary conditions. Appl Numer Math 87:157–167 Dabiri A, Butcher EA (2017) Efficient modified Chebyshev differentiation matrices for fractional differential equations. Commun Nonlinear Sci Numer Simul 50:284–310 Daftardar-Gejji V, Jafari H (2007) Solving a multi-order fractional differential equation using Adomian decomposition. Appl Math Comput 189(1):541–548 Diethelm K, Ford J (2002) Numerical solution of the Bagley–Torvik equation. BIT 42(3):490–507 Doha EH, Abd-Elhameed WM, Youssri YH (2013) Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane–Emden type. New Astron 23:113–117 Doha EH, Abd-Elhameed WM (2014) On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds. Bull Malays Math Sci Soc 37(2):383–398 Esmaeili S, Shamsi M, Luchko Y (2011) Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput Math Appl 62(3):918–929 Gautschi W (2012) Numerical differentiation and integration. In: Numerical analysis. Springer, pp 159–251 Hesthaven J, Gottlieb S, Gottlieb D (2007) Spectral methods for time-dependent problems, vol 21. Cambridge University Press, Cambridge Ishteva M (2005) Properties and applications of the Caputo fractional operator. Ph.D. thesis, Dept. of Math., Universität Karlsruhe (TH), Sofia, Bulgaria
123
Fifth-kind orthonormal Chebyshev polynomial solutions for… Koepf W, Masjed-Jamei M (2006a) A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it. Integral Trans Spec Funct 17(8):559–576 Koepf W, Masjed-Jamei M (2006b) A generic formula for the values at the boundary points of monic classical orthogonal polynomials. J Comput Appl Math 191(1):98–105 Lamnii A, Mraoui H, Sbibih D, Tijini A, Zidna A (2011) Sextic spline collocation methods for nonlinear fifth-order boundary value problems. Int J Comput Math 88(10):2072–2088 Li C, Deng W (2007) Remarks on fractional derivatives. Appl Math Comput 187(2):777–784 Li CP, Zhao ZG (2011) Introduction to fractional integrability and differentiability. Eur Phys J Special Topics 193(1):5–26 Machado JAT, Kiryakova V (2017) The chronicles of fractional calculus. Fract Calc Appl Anal 20(2):307–336 Masjed-Jamei M (2006) Some new classes of orthogonal polynomials and special functions: a symmetric generalization of Sturm-Liouville problems and its consequences. Ph.D. thesis Masjed-Jamei M (2007) A basic class of symmetric orthogonal polynomials using the extended Sturm– Liouville theorem for symmetric functions. J Math Anal Appl 325(2):753–775 Meerschaert M, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations. Appl Numer Math 56(1):80–90 Moghaddam BP, Yaghoobi SH, Machado JAT (2016) An extended predictor-corrector algorithm for variableorder fractional delay differential equations. J Comput Nonlinear Dyn 11(6):061001 Moghaddam BP, Machado JAT (2017) A stable three-level explicit spline finite difference scheme for a class of nonlinear time variable order fractional partial differential equations. Comput Math Appl 73(6):1262– 1269 Moghaddam BP, Machado JAT (2017) Extended algorithms for approximating variable order fractional derivatives with applications. J Sci Comput. doi:10.1007/s10915-016-0343-1 Mokhtary P, Ghoreishi F, Srivastava HM (2016) The Müntz–Legendre Tau method for fractional differential equations. Appl Math Model 40(2):671–684 Mysovskikh IP (2002) Romberg method. In: Hazewinkel M (ed) Encyclopaedia of mathematics. Springer, New York, pp 1–4020 Oldham KB (1974) The fractional calculus. Elsevier, New York Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic, Cambridge Saadatmandi A, Dehghan M (2011) A tau approach for solution of the space fractional diffusion equation. Comput Math Appl 62(3):1135–1142 Salgado GHO, Aguirre LA (2016) A hybrid algorithm for Caputo fractional differential equations. Commun Nonlinear Sci Numer Simul 33:133–140 Shizgal B (2014) Spectral methods in chemistry and physics. Springer, New York Tang X, Xu H (2016) Fractional pseudospectral integration matrices for solving fractional differential, integral, and integro-differential equations. Commun Nonlinear Sci Numer Simul 30(1):248–267 Trefethen LN (2000) Spectral methods in MATLAB, vol 10. SIAM ur Rehman M, Khan RA (2011) The Legendre wavelet method for solving fractional differential equations. Commun Nonlinear Sci Numer Simul 16(11):4163–4173 Waldvogel J (2006) Fast construction of the Fejér and Clenshaw–Curtis quadrature rules. BIT 46(1):195–202 Yaghoobi SH, Moghaddam BP, Karim I (2017) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dyn 87(2):815–826 Youssri YH (2017) A new operational matrix of caputo fractional derivatives of Fermat polynomials: an application for solving the Bagley- Torvik equation. Adv Differ Equ 2017(1):73 Zaky MA, Ezz-Eldien SS, Doha EH, Machado JAT, Bhrawy AH (2016) An efficient operational matrix technique for multidimensional variable-order time fractional diffusion equations. J Comput Nonlinear Dyn 11(6):061002 Zayernouri M, Karniadakis GE (2014) Exponentially accurate spectral and spectral element methods for fractional ODEs. J Comp Phys 257:460–480
123
