Convergence analysis of the Chebyshev-Legendre spectral method for a class of Fredholm fractional integro-differential equations

A. Yousefi, S. Javadi, E. Babolian, E. Moradi Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran [email protected], [email protected], [email protected], [email protected],,

Abstract

In this paper, we propose and analyze a spectral Chebyshev-Legendre approximation for fractional order integro-differential equations of Fredholm type. The fractional derivative is described in the Caputo sense. Our proposed method is illustrated by considering some examples whose exact solutions are available. We prove that the error of the approximate solution decay exponentially in 퐿2-norm. Keyword. Chebyshev-Legendre Spectral method, Caputo derivative, Fractional integro- differential equations, Convergence analysis.

1. Introduction

Many phenomena in engineering, physics, chemistry, and the other sciences may be applied by models using mathematical tools from fractional calculus. The theory of derivatives and integrals of fractional order allow us to describe physical phenomena more accurately [1-2]. Furthermore most problems cannot be solved analytically, and hence finding good approximate solution, using numerical methods will be very helpful. Recently, several numerical methods have been given to solve fractional differential equations (FDEs) and fractional integro-differential equations (FIDEs). These methods include collocation method [3-4], variational iteration method [5], Adomian decomposition method [6], Homotopy perturbation method [5,7], fractional differential transform method [8-9], the reproducing kernel method [10], and wavelet method [11- 13].

Spectral methods are an emerging area in the field of applied sciences and engineering. These methods provide a computational approach that has achieved substantial popularity over the last three decades. They have been applied successfully to numerical simulations of many problems in fractional calculus ([14-20]).

In this paper, we are concerned with numerical solutions of the following equation: 푛 1 (푖) 훼 ∑ 푎푖 퐷 푦(푡) = 푓(푡) + ∫ 푘(푡, 푠) 퐷 푦(푠) 푑푠, 푖=0 0 푚 − 1 < 훼 ≤ 푚, 푚 ∈ ℕ, 푡 ∈ [0,1], (1) subject to the initial values (푖) 푦 (0) = 푑푖, 푖 = 0,1, …, 푛 − 1, (2) where Dα is the fractional derivative in the Caputo sense, 푓(푡) and 푘(푡, 푠) are the known functions that are supposed to be sufficiently smooth and 푑푖 for any 푖 is constant. Existence and uniqueness of the solution of the Eq. (1) have been shown in [21]. The authors in [22] applied the backward and central-difference formula for approximating solution at the mesh points. The fractional derivative are global, i.e. they are defined over the whole interval 퐼 = [0,1], and therefore global method, such as spectral methods, are better suited for FDEs and FIDEs. Yousefi and et al. [20] introduced a quadrature shifted Legendre tau method based on the Gauss- Lobatto interpolation for solving Eq. (1). Inspired by the work of [23-24], we extend the approach to Eq. (1) and provide a rigorous convergence analysis for the Chebyshev-Legendre method. We show that approximate solutions are convergent in 퐿2 −norm. The structure of this paper is as follows: In section 2, some necessary definitions and mathematical tools of the fractional calculus which are required for our subsequent developments are introduced. In section 3, the Chebyshev-Legendre method of FIDEs is obtained. The rest of this section is devoted to apply the proposed method for solving Eq. (1) by using the shifted Legendre and Chebyshev polynomials. After this section, we discuss about convergence analysis and then, some numerical experiments are presented in Section 5 to show the efficiency of Chebyshev-Legendre spectral method. The conclusion is given in section 6.

2. Basic Definitions and Fractional Derivatives

For 푚 ∈ ℕ, the smallest integer that is greater than or equal to 훼, i.e. 푚 = ⌈훼⌉, the Caputo’s fractional derivative operator of order 훼 > 0, is defined as: 퐽푚−훼 퐷푚푦(푥), 푚 − 1 < 훼 ≤ 푚, 퐷훼푦(푥) = { (3) 퐷푚푦(푥), 훼 = 푚, where

1 푥 퐽푚−훼 푦(푥) = ∫ (푥 − 푡)푚−훼−1푦(푡)푑푡 , 휈 > 0, 푥 > 0. Γ(푚 − 훼) 0

For the Caputo’s derivative we have [2]: 0, 훽 ∈ {0,1,2, … } 푎푛푑 훽 < 푚, 퐷훼 푥 훽 = { Γ(훽 + 1) (4) 푥 훽−훼 훽 ∈ {0,1,2,… } 푎푛푑 훽 ≥ 푚. Γ(훽 − 훼 + 1) Recall that for α ∈ ℕ, the Caputo differential operator coincides with the usual differential operator. Similar to standard differentiation, Caputo’s fractional differentiation is a linear operator, i.e., 퐷훼 (휆 푔(푥) + 휇 ℎ(푥)) = 휆퐷훼푔(푥) + 휇퐷훼ℎ(푥), where 휆 and 휇 are constants.

The Chebyshev polynomials {푇푖(푡); 푖 = 0,1,… } are defined on the interval [−1,1] with the following recurrence formula:

푇푖+1(푡) = 2푡 푇푖(푡) − 푇푖−1(푡), 푖 = 1,2,…, with 푇0(푡) = 1 and 푇1(푡) = t. The shifted Chebyshev polynomials are defined by introducing the change of variable 푡 = 2푥 − 1. Let the shifted Chebyshev polynomials 푇푖(2푥 − 1) be denote by 푇1,푖(푥), satisfying the relation

푇1,푖+1(푥) = 2(2푥 − 1)푇1,푖(푥) − 푇1,푖−1(푥), 푖 = 1,2, … , 푥 ∈ [0,1], (5) where 푇1,0 (푥) = 1 and 푇1,1(푥) = 2푥 − 1. By these definitions we will have [32] (푖+푘−1)! 22푘 - 푇 (푥) = 푖 ∑푖 (−1)푖−푘 푥 푘 푖 = 1,2, …. (6) 1,푖 푘=0 (푖−푘)!(2푘)! 푖 - 푇1,푖(0) = (−1) , 푇1,푖(1) = 1. (7 −1 1 2 푇 (푥) 푇 (푥)(푥 − 푥 ) 2 푑푥 = 훿 ℎ , (8) - ∫0 1,푗 1,푘 푗푘 푘 where 훿푗푘 is Kronecker delta and

휋, 푘 = 0 ℎ = {휋 , (9) 푘 , 푘 ≥ 1 2

In this paper, we will consider the Gauss-type quadrature formulas. We start by defining the Chebyshev-Gauss quadrature nodes and weights, respectively: ( ) 푥 = − cos 2푗+1 휋, 푤 = 휋 , 푗 = 0,1,… , 푁. 푗 2푁+2 푗 푁+1 With the above choices, there holds 푁 1 1 ∫ 푝(푥) 푑푥 = ∑ 푝(푥푗)푤푗, ∀푝 ∈ 푃2푁+1 , (10) √1 − 푥 2 −1 푗=0 where 푃2푁+1 is a polynomial of degree less than or equal 2푁 + 1.

We now turn to the discrete Chebyshev transforms. The transforms can be performed via a matrix- vector multiplication with 풪(푁2 ) operations as usual and when we use Chebyshev polynomials, it can be carried out with 풪(푁 푙표푔2 푁) operations via fast Fourier transform (퐅퐅퐓) [26-27]. We define the Chebyshev-Lagrange polynomial by

푇푘(푥) 퐺푘(푥) = ′ , ; = 0,1, … , 푁. (푥 − 푥푘)푇푘(푥푘) 푐 Given 푢(푥) ∈ 퐶[−1,1], the Chebyshev-Lagrange interpolation operator 퐼푁푢 is defined by 푁 푐 (퐼푁푢)(푥) = ∑ 푢푘 퐺푘(푥) ∈ ℙ푁 , (11) 푘=0 where {푢푘} are determined by the forward discrete Chebyshev transform as follows ( ) 푢 = ∑푁−1 푢(푥 )cos 2푘+1 푗휋 , 0 ≤ 푘 ≤ 푁. (12) 푘 푗=0 푗 2푁 The above transform can be computed by using 푭푭푻 in 풪(푁 푙표푔2 푁) operations [26-27].

Let 퐿 푖(푡) be the standard Legendre polynomial of degree i, then we have [20]

- Three-term recurrence relation

(푖 + 1)퐿푖+1(푡) = (2푖 + 1)푡 퐿푖(푡) − 푖퐿 푖−1(푡), 푖 ≥ 1, (13) and the first two Legendre polynomials are 퐿0(푡) = 1, 퐿1(푡) = 푡.

- The Legendre polynomial 퐿 푖(푡) has the expansion 푖 [ ] ( ) 퐿 (푡) = 1 ∑ 2 (−1)푙 2푖−2푙 ! 푡푖−2푙 . (14) 푖 2푖 푙=0 2푙 푙!(푖−푙)!(푖−2푙)! - Orthogonality 1 ∫ 퐿푗(푡)퐿푘(푡) 푑푡 = ℎ푘훿푗푘 , (15) −1 such that 2 ℎ = . 푘 2푘 + 1 - Symmetry property 푖 푖 퐿푖(−푡) = (−1) 퐿푖(푡), Li(±1) = (±1) . (16)

Hence, 퐿푖(푡) is an odd (resp. even) function, if 푖 is odd (resp. even).

Now, if we define the shifted Legendre polynomial of degree 푖 by 퐿 1,푖(푥) = 퐿푖(2푥 − 1), then we can obtain the analytic form and three-term recurrence relation of the shifted Legendre polynomials of degree 푖 by the following form, respectively 푖 (푖 + 푘)! 퐿 (푥) = ∑(−1)푖+푘 푥 푘, 1,푖 (푖 − 푘)! (푘!)2 푘=0 2푖 + 1 푖 퐿 (푥) = 푥 퐿 (푥) − 퐿 (푥), 푖 ≥ 1. (17) 1,푖 푖 + 1 1,푖 푖 + 1 1,푖−1 According to Eq. (15), the orthogonality relation of shifted Legendre polynomials is

1

∫ 퐿1,푗(푡)퐿 1,푘(푡) 푑푡 = ℎ푘훿푗푘 . (18) 0

2 2 We denote 퐿휔(퐼) by the weighted 퐿 Hilbert space with the scalar product 1 2 (푢, 푣) = ∫ 푢(푥) 푣(푥) 휔(푥)푑푡, ∀푢, 푣 ∈ 퐿 휔(퐼), 0 1 1 2 2 − and the norm ‖푢‖ 2 = (푢, 푢) , where 휔(푥) = 1 in the Legendre case and 휔(푥) = (1 − 푥 ) 2 퐿휔 휔 in the Chebyshev case. We may drop the subscript 푤 when 휔 = 1. Therefore, the corresponding norm is 1 2 ‖푢‖퐿2 = (푢, 푢) .

푖 Let 퐻푚(퐼) = {푢 ∈ 퐿2 (퐼) ∶ 푑 푢 ∈ 퐿2 (퐼), 푖 = 0,1, …푚} be the weighted Sobolev space with the 휔 휔 푑푥푖 휔 norm and semi norm defined respectively 푚 2 (푘) 2 ‖푦‖퐻푚 (퐼) = ∑‖푦 ‖ 2 , 휔 퐿휔(퐼) 푘=0 and 2 푁 (푘) 2 |푦|퐻푚 :푁 (퐼) = ∑푘=min(푚:푁)‖푦 ‖ 2 . 퐿휔(퐼)

퐻푚(퐼) by its inner product is Hilbert space.

For a function 푦(푥) ∈ 퐿2[0,1], the shifted Legendre expansion is ∞

푦(푥) = ∑ 푎푗 퐿1,푗 (푥), 푗=0 where 1 1 푎푗 = ∫ 푦(푥) 퐿1,푗(푥) 푑푥, 푗 = 0,1,2, …, (19) ℎ1,푗 0 and

1 1 ℎ = ℎ = . 1,푗 2 푗 2푗 + 1

Now, we describe the Legendre-Gauss integration in the interval (0,1). We denote by 푥푁,푗,

휔푁,푗, 푗 = 0, …, 푁, respectively the nodes and weights of the standard integration on the interval

(−1,1). We suppose 푥1,푁,푗 ,휔1,푁,푗 , 푗 = 0,… , 퐿, are nodes and weights of the Legendre-Gauss integration in the interval (0,1). Then, we have

1 1 푥 = (푥 + 1), 푤 = 푤 푗 = 0, . . , 푁. 1,푁,푗 2 푁,푗 1,푁,푗 2 푁,푗

According to Eq. (29) for any 푔 ∈ ℙ2푁 +1, set of all polynomials of degree at most 2푁 + 1, we get

1 1 1 1 ∫ 푔(푥)푑푥 = ∫ 푔 ( (푥 + 1))푑푥 0 2 −1 2 1 푁 1 = ∑ 휔푁,푗 푔 ( (푥푁,푗 + 1)) 2 푗=0 2 푁 = ∑ 휔1,푁,푗 푔(푥1,푁,푗 ). (20) 푗=0

In practice, a number of first shifted Legendre polynomials are considered. We let

푇 휙(푥) = [퐿1,0(푥), 퐿1,1(푥),… , 퐿1,푁(푥)] ,

푆푁(퐼) = 푠푝푎푛{퐿1,0(푥),퐿1,1 (푥),…, 퐿 1,푁(푥)}. (21)

Theorem 2.1 [25] suppose 휙(푥) is defined in Eq.(21) and 훼 > 0; then the following relation holds:

푫훼휙(푥) ≅ 푫(훼) 휙(푥), (22) where 푫(훼) is the (푁 + 1) × (푁 + 1) operational matrix of Caputo derivative which is given by: 0 0 0 … 0 ⋮ ( ) ( ) ( ) ( ) 푆훼 푚,0 푆훼 푚,1 푆훼 푚,2 . . . 푆훼 푚,푁 퐷(훼) = (푑 ) = ⋮ , (23) 푖푗 0≤푖,푗≤푁 푆훼(푖,0) 푆훼(푖,1) 푆훼(푖,2) … 푆훼(푖,푁) ⋮ [푆훼(푁,0) 푆훼(푁, 1) 푆훼(푁,2) … 푆훼(푁,푁)] where

푖 (−1)푖+푘 (2푗 + 1) (푖 + 푘)! Γ(푘 − 푗 − 훼 + 1) 푆 (푖,푗) = ∑ . (24) 훼 퐿훼(푖 − 푘)!푘! Γ(푘 − 훼 + 1) Γ(푘 + 푗 − 훼 + 1) 푘=푚

훼 Note that because of 퐷 퐿1,푖(푡) = 0, for 푖 = 0,1,… ,푚 − 1, the first 푚 rows are zero in 푫.

3. Chebyshev-Legendre Spectral Method

The Chebyshev-Legendre spectral method was introduced in [24] to take advantage of both the Legendre and Chebyshev polynomials. The main idea is to use the Legendre-Galerkin formulation which preserves the symmetry of the underlying problem and lead to a simple sparse linear system, while the physical values are evaluated at the Chebyshev-Gauss-type points. Thus, we may replace the expensive Legendre transform by a fast Chebyshev-Legendre transform between the coefficients of Legendre expansion and Chebyshev expansion at the Chebyshev- Gauss-type points.

The main advantage of using Chebyshev polynomials is that the discrete Chebyshev transform can be performed in 푂(푁 푙표푔2 푁) operations by using 퐹퐹푇. On the other hand, the discrete Legendre transform is expensive, and therefore in our article, the Chebyshev-Legendre method based on Legendre expansion and Chebyshev-Gauss-type points is applied to reduce the cost of solving the corresponding system (For more detail see [17, 24, 28]). Then, we use the Chebyshev 푐 interpolation operator 퐼푁 , relative to the Gauss-Chebyshev points to approximate the known functions and use of Legendre polynomials expansion to approximate the unknown function together. At last, the solution procedure is essentially the same as Legendre spectral method except that Chebyshev-Legendre transform, between the values of a function at the Gauss- Chebyshev points and the coefficients of its Legendre expansion, are needed instead of the Legendre transform. There are several efficient algorithms to transform from the coefficients of Legendre expansions to Chebyshev expansions at the Chebyshev-Gauss-Lobatto points and vice versa [24, 26-28]. We use the algorithm in [24] as follow:

We let

푁 푁

푢(푥) = ∑ 훼푗 푇1,푗 = ∑ 훽푗 퐿1,푗 , 푗=0 푗 =0

휶 = (훼0,훼1, …, 훼푁 ), 휷 = (훽0, 훽1,… , 훽푁).

In this work, what we need to apply spectral method is using the transform between 휶 and 휷. By virtue of orthogonality of Chebyshev and Legendre polynomials, the relation between 휶 and 휷 can be obtained by computing (푢, 푇 ) and (푢,퐿 ). defining 1,푗 푤 1,푗

퐴 = (푎 ) , 푖푗 0≤푖,푗≤푁 퐵 = (푏 ) , 푖푗 0≤푖,푗≤푁 then, by using Eqs. (8) and (15), we can obtain

1 푎 = (푇 , 퐿 ) , 푖푗 1,푖 1,푗 푤 ℎ푖 1 푏 = (푖 + ) (퐿 , 푇 ). 푖푗 2 1,푖 1,푗

Thus, we will have

휶 = 퐴휷, 휷 = 퐵휶, 퐴퐵 = 퐵퐴 = 퐼.

According to orthogonality and parity of the Chebyshev and Legendre polynomials, we get

푎푖푗 = 푏푖푗 = 0, for 푖 > 푗 or 푖 + 푗 odd.

Therefore, we only determine the nonzero elements of both 퐴 and 퐵 by using three-term recurrence relation of the shifted Legendre and Chebyshev polynomials. Applying definition of

푎푖푗, we can obtain recurrence formula

1 푎 = (푇 , 퐿 ) 푖푗 1,푖 1,푗 푤 ℎ푖 1 2푗 + 1 푗 = (푇1,푖, (2푥 − 1) 퐿1,푗(푥) − 퐿1,푗−1(푥)) ℎ푖 푗 + 1 푗 + 1 푤 1 2푗 + 1 푗 = { ((2푥 − 1)푇 , 퐿 ) − (푇 , 퐿 ) } 1,푖 1,푗 1,푖 1,푗−1 푤 ℎ푖 푗 + 1 푤 푗 + 1 1 2푗 + 1 푗 = { (푇 + 푇 , 퐿 ) − ℎ 푎 } 1,푖+1 1,푖−1 1,푗 푤 푖 푖,푗−1 ℎ푖 2푗 + 2 푗 + 1 ℎ푖+1 2푗 + 1 ℎ푖−1 2푗 + 1 푗 = 푎푖+1,푗 + 푎푖−1,푗 − 푎푖,푗−1. ℎ푖 2푗 + 2 ℎ푖 2푗 + 2 푗 + 1

We can similarly derive entries of matrix 퐵 as follow

1 푏 = (푖 + ) 푏̃ , 푖푗 2 푖푗 where

2푖 + 2 2푖 푏̃ : = (퐿 , 푇 ) = 푏̃ + 푏̃ − 푏̃ . 푖푗 1,푖 1,푗 2푖 + 1 푖+1,푗 2푖 + 1 푖−1,푗 푖,푗−1

Thus, we can obtain each nonzero element of 퐴 and 퐵 by just a few operations. Therefore, we can extremely apply Chebyshev-Legendre spectral method.

We now describe our spectral approximations to Eq. (1). Therefore, if 푦푁 (푡) ∈ 푆푁(퐼), then by implementing Chebyshev-Legendre spectral method for Eq.(1), we can easily obtain

푛 1 (푖) 푐 푐 훼 ∑푎푖 (퐷 푦푁, 퐿1,푘) = (퐼푁푓,퐿 1,푘) + (∫ 퐼푁푘(., 푠) 퐷 푦푁 (푠) 푑푠 , 퐿1,푘). (25) 푖=0 0 푁 We have 푦푁 (푡) = ∑푗=0 푐푗퐿1,푗(푡), then according to linearity of Caputo’s fractional differentiation, Eq.(23) can be written as:

푛 푁 (푖) ∑푎푖 ∑ 푐푗 (퐷 퐿1,푗 , 퐿1,푘) 푖=0 푗=0 푁 1 푐 푐 훼 = (퐼푁푓,퐿 1,푘) + ∑푐푗 (∫ 퐼푁푘(., 푠) 퐷 퐿1,푗(푠) 푑푠 , 퐿1,푘). (26) 푗=0 0 From Eq. (22) to (24) in Theorem 2.1, we can obtain

훼 푁 퐷 퐿1,푗(푡) = ∑푙=0 푆훼(푗, 푙)퐿1,푙(푡), 푗 = 푚,푚 + 1,… , 푁. (27)

We notice that if 훼 = 푛 ∈ ℕ, then 푆훼 defined in Eq. (24) tend to integer order case and Theorem 2.1 gives the same result as integer order case.

Inserting Eq.(27) in Eq.(26), we get

푛 푁 푁

∑∑ 푎푖 푐푗 (∑푆푖(푗, 푙) (퐿1,푙, 퐿1,푘)) 푖=0 푗=푖 푙=0 푁 푁 1 푐 푐 = (퐼푁푓, 퐿1,푘) + ∑ 푐푗 ∑ 푆훼(푗,푙) (∫ 퐼푁푘(., 푠) 퐿1,푙(푡) 푑푠 , 퐿1,푘). (28) 푗=푚 푙=0 0

Then, making use of the orthogonality relation of shifted Legendre polynomials, i.e. Eq.(18), Eq. (24) reduce to 푛 푁 푆 (푗,푘) ∑∑ 푎 푐 푖 푖 푗 2푘 + 1 푖=0 푗=푖 푁 푁 1 푐 푐 = (퐼푁푓, 퐿1,푘) + ∑ 푐푗 ∑ 푆훼(푗,푙) (∫ 퐼푁푘(., 푠) 퐿1,푙(푡) 푑푠 , 퐿1,푘). (29) 푗=푚 푙=0 0 We let 푁 1 푐 ℎ푙(푥) = ∫ 퐼푁푘(푥,푠) 퐿1,푙(푡) 푑푠 ≅ ∑ 푏푙푟 퐿1,푟(푥) , 0 푟=0 푐 푓푘 = (퐼푁푓,퐿1,푘).

Thus, again by using Eqs.(18), Eq. (29) becomes the following form

푛 푁 푁 푁 푆 (푗,푘) 푏 ∑∑ 푎 푐 푖 = 푓 + ∑ ∑ 푐 푆 (푗, 푙) 푙푘 . (30) 푖 푗 2푘 + 1 푘 푗 훼 2푘 + 1 푖=0 푗=푖 푗=푚 푙=0

It is easy to verify that initial conditions convert to following equations

푁 푁 ∑푗=0 ∑푙=0 푐푗 푆푖(푗,푙) 퐿1,푙(0) = 푑푖, 푖 = 0,1, …, 푛 − 1. (31)

Combining Eqs. (30) and (31) yields

푛 푁 푁 푁 푆푖(푗,푘) 푏푙푘 ∑ ∑ 푎 푐 − ∑ ∑ 푐 푆 (푗,푙) = 푓 , 푘 = 0,1,… 푁 − 푛, 푖 푗 2푘 + 1 푗 훼 2푘 + 1 푘 푖=0 푗=푖 푙=0 푗=푚 푁 푁 ∑ ∑ 푐푗 푆푖(푗, 푙) 퐿1,푙(0) = 푑푖 , 푖 = 0,1, …, 푛 − 1. {푗=0 푙=0

푁 By solving the above system of linear equations, we can get the value of {푐 } and obtain the 푗 푗 =0 expression of 푦푁 (푥) accordingly.

4. Convergence Analysis of the Chebyshev-Legendre Spectral method

In this section, we present a general approach to the convergence analysis for NIFDEs that is proved in 퐿2 −norm. Here, there are some properties and elementary lemmas, which are important for the derivation of the main results.

Lemma 4.1 [29] For multiple integrals, the following relation holds:

푡 푡푛 푡3 푡2 푡 1 푛−1 ∫ ∫ … ∫ ∫ 푔(푡1) 푑푡1푑푡2 … 푑푡푛 = ∫ (푡 − 푠) 푔(푠) 푑푠, (32) 0 0 0 0 (푛 − 1)! 0 where 푔 is integrable function on interval (0,푡) and 푡푖 (푖 = 2,3, …, 푛) are parameters in the purpose interval. Lemma 4.2 [30] (Granwall's Lemma) Assume that 푢, 휔, 훽 ∈ 퐶 (퐼) with 훽(푡) ≥ 0. If 푢 satisfies the inequality

푡 푢(푡) ≤ 휔(푡) + ∫ 훽(푠)푢(푠) 푑푠 , 푡 ∈ 퐼, 0 then

푡 푡 푢(푡) ≤ 휔(푡) + ∫ 훽(푠)휔(푠) exp (∫ 훽(푣) 푑푣) 푑푠 , 푡 ∈ 퐼. (33) 0 푠

On the other word, if 휔 is non-decreasing on 퐼, the above inequality reduce to

푡 푢(푡) ≤ 휔(푡) exp (∫ 훽(푣) 푑푣), 푡 ∈ 퐼. (34) 푠

Lemma 4.3 [30] Suppose that 푘 is a given kernel function on 퐼 × 퐼. If 푓 ∈ 퐿푝(푎, 푏) for 1 ≤ p ≤ ∞, the integral

푥 표푟 푏 푇푓(푥) = ∫ 푘(푥, 푡) 푓(푡) 푑푡 푎 is well-defined in 퐿푝(푎, 푏) and there exists 퐶 ∗ > 0 such that

∗ ‖푇푓‖퐿푝 (푎,푏) ≤ 퐶 ‖푓‖퐿푝 (푎,푏) . (35)

2 Let 푝푁 be the interpolation projection operator from 핃 (퐼) upon ℙ푁(퐼). Then, for any function 푓 in 퐿2(퐼) satisfies

1 ∫ (푓 − 푝푁 푓)(푡) 푔(푡)푑푡 = 0, ∀푔 ∈ ℙ푁 (퐼). 0 Also, the following relations for interpolation in shifted Legendre polynomials and shifted Gauss- Legendre nodal points 푘 ≥ 1 (or for any fixed 푘 ≤ 푁) may readily be obtained as [14]

1 2푙− −푘 2 ‖푦 − 푝푁 (푦)‖퐻푙 (퐼) ≤ 퐶1푁 |푦|퐻푘 :푁(퐼) , (36) −푘 ‖푦 − 푝푁 (푦)‖퐿2 (퐼) ≤ 퐶2푁 |푦|퐻푘 :푁(퐼) . (37)

푚 where 푦 ∈ 퐻 (퐼), and 퐶1 and 퐶2 are constants independent of 푁 and 0 ≤ 푙 ≤ 푚.

Now, we shall prove the main result in this section. In the following theorem, an error estimation for an approximate solution of Eq. (1) with supplementary conditions of Eq. (2) is obtained. Let

푒푁(푥) = 푦(푥) − 푦푁 (푥), be the error function of the Chebyshev-Legendre spectral approximation to 푦(푥). From the mathematical point of view, it is possible to keep track of the effect of the boundary conditions upon the overall accuracy of the scheme. In the other hand, the boundary treatment does not destroy the spectral accuracy of the Chebyshev-Legendre method. Theorem 4.3 For sufficiently large 푁, the Chebyshev-Legendre spectral approximations 푦푁 (푥) converge to exact solution in 퐿2-norm, i.e.

‖푒푁‖퐿2 (퐼) = ‖푦 − 푦푁 ‖퐿2(퐼) → 0.

Proof. Assume that 푦푁 (푥) is obtained by using the Chebyshev-Legendre spectral method Eq. (1) together with initial conditions Eq. (2). Then, we have

푛 1 (푖) 훼 ∑푎푖 퐷 푦푁(푡) = 푝푁 (푓(푡)) + 푝푁 (∫ 푘(. , 푠) 퐷 푦푁 (푠) 푑푠), (38) 푖=0 0 such that 푝푁 is the Lagrange interpolation polynomial operator defined for Legendre polynomial. With 푛 times integration from Eq. (38), we obtain

푛 푡 푡푛 푡3 푡2 (푖) ∑푎푖 ∫ ∫ … ∫ ∫ 푦푁 (푡1) 푑푡1푑푡2 … 푑푡푛 푖=0 0 0 0 0 푡 푡푛 푡3 푡2 = ∫ ∫ … ∫ ∫ 푝푁 (푓(푡1))푑푡1푑푡2 … 푑푡푛 0 0 0 0 푡 푡푛 푡3 푡2 1 훼 +∫ ∫ … ∫ ∫ 푝푁 (∫ 푘(푡1,푠) 퐷 푦푁(푠) 푑푠)푑푡1푑푡2 … 푑푡푛 . (39) 0 0 0 0 0

By virtue of Lemma 4.1, we can convert each of the multiple integral to single integral, so we have

푛−1 푡 푎 푎 푦 (푡) + 푔(푡) + ∑ ∫ 푖 (푡 − 푠)푛−푖−1 푦 (푠) 푑푠 푛 푁 (푛 − 푖 − 1)! 푁 푖=0 0 푡 (푡 − 푠)푛−1 = ∫ 푝푁 (푓(푠)) 푑푠 0 (푛 − 1)! 푡 ( )푛−1 1 푡 − 푠 훼 +∫ 푝푁 (∫ 푘(푠, 푠1) 퐷 푦푁(푠1) 푑푠1) 푑푠, (40) 0 (푛 − 1)! 0

where 푔 is a polynomial of degree 푛 with the initial condition coefficient. Similarly, from Eq. (1), we get

푛−1 푡 푎 푎 푦(푡) + 푔(푡) + ∑ ∫ 푖 (푡 − 푠)푛−푖−1 푦(푠) 푑푠 푛 (푛 − 푖 − 1)! 푖=0 0 푡 (푡 − 푠)푛−1 = ∫ 푓(푠) 푑푠 0 (푛 − 1)! 푡 ( )푛−1 1 푡 − 푠 훼 +∫ ∫ 푘(푠,푠1) 퐷 푦(푠1) 푑푠1 푑푠. (41) 0 (푛 − 1)! 0

By subtracting Eq. (40) from Eq. (41), we obtain

푛−1 푡 푎 푎 푒 (푡) + ∑ ∫ 푖 (푡 − 푠)푛−푖−1 푒 (푠) 푑푠 푛 푁 (푛 − 푖 − 1)! 푁 푖=0 0 푡 (푡 − 푠)푛−1 푡 (푡 − 푠)푛−1 = ∫ 푒 (푓(푠)) 푑푠 + ∫ 푒 (퐾 푦(푠)) 푑푠, (42) 푝푁 푝푁 훼 0 (푛 − 1)! 0 (푛 − 1)!

such that

푒 (푓(푠)) = 푓(푠) − 푝 (푓(푠)), 푝푁 푁 1 훼 퐾훼푦(푠) = ∫ 푘(푠, 푠1) 퐷 푦(푠1) 푑푠1 , 0 푒 (퐾 푦(푠)) = 퐾 푦(푠) − 푝 (퐾 푦 (푠)) 푝푁 훼 훼 푁 훼 푁

= 퐾훼푦(푠) − 퐾훼 푦푁 (푠) + 퐾훼푦푁 (푠) − 푝푁 (퐾훼 푦푁 (푠)) = 퐾 푒 (푠) − 푒 (퐾 푦 (푠)). 0 ≤ 푠 ≤ 푡. 훼 푁 푝푁 훼 푁 From Eq. (42), we can obtain

푛−1 푡 푎푖 푛−푖−1 |푒푁(푡)| ≤ ∑ | |∫ |(푡 − 푠) 푒푁(푠)|푑푠 −푎푛 (푛 − 푖 − 1)! 푖=0 0 1 푡 + ∫ |(푡 − 푠)푛−1 푒 (푓(푠))|푑푠 푝푁 |푎푛 |(푛 − 1)! 0 1 푡 + ∫ |(푡 − 푠)푛−1 푒 (퐾 푦(푠))|푑푠 푝푁 훼 |푎푛|(푛 − 1)! 0 푡 푡 푡 ≤ 퐶 ∫ |푒 (푠)| 푑푠 + 퐶 ∫ |푒 (푓(푠))|푑푠 + 퐶 ∫ |푒 (퐾 푦(푠))| 푑푠. (43) 2 푁 3 푝푁 4 푝푁 훼 0 0 0

Applying Lemma 4.2 leads to

푡 푡 푡 |푒 (푡)| ≤ exp (∫ 퐶 푑푠) (퐶 ∫ |푒 (푓(푠))| 푑푠 + 퐶 ∫ |푒 (퐾 푦(푠))| 푑푠) 푁 2 3 푝푁 4 푝푁 훼 0 0 0 푡 푡 ≤ 퐶 ∫ |푒 (푓(푠))| 푑푠 + 퐶 ∫ |푒 (퐾 푦(푠))| 푑푠. (44) 5 푝푁 6 푝푁 훼 0 0

Equivalently, by using the 퐿2 −norm, we get

푡 푡 ‖푒 ‖ 2 ≤ 퐶 ‖∫ |푒 (푓(푠))| 푑푠‖ + 퐶 ‖∫ |푒 (퐾 푦(푠))| 푑푠‖ . (45) 푁 퐿 (퐼) 5 푝푁 6 푝푁 훼 0 퐿2 (퐼) 0 퐿2(퐼) Bu using Lemma 4.3, the above inequality reduce to

‖푒 ‖ 2( ) ≤ 퐶 ‖푒 (푓(푠))‖ + 퐶 ‖푒 (퐾 푦(푠))‖ . (46) 푁 퐿 퐼 7 푝푁 퐿2 (퐼) 8 푝푁 훼 퐿2 (퐼)

On the other hand, we have

1 1 푒 (퐾 푦(푠)) = ∫ 푘(푠, 푠 ) 퐷훼푦(푠 ) 푑푠 − 푝 (∫ 푘(푠, 푠 ) 퐷훼푦 (푠 ) 푑푠 ) 푝푁 훼 1 1 1 푁 1 푁 1 1 0 0 1 1 훼 훼 = ∫ 푘(푠, 푠1) 퐷 푦(푠1) 푑푠1 − ∫ 푘(푠, 푠1) 퐷 푦푁(푠1) 푑푠1 0 0 1 1 훼 훼 +∫ 푘(푠, 푠1) 퐷 푦푁(푠1) 푑푠1 − 푝푁 (∫ 푘(푠, 푠1) 퐷 푦푁(푠1) 푑푠1) 0 0 1 훼 = ∫ 푘(푠, 푠1) 퐷 푒푁(푠1) 푑푠1 + 퐸(푠), (47) 0

where

1 1 훼 훼 퐸(푠) = ∫ 푘(푠, 푠1) 퐷 푦푁 (푠1) 푑푠1 − 푝푁 (∫ 푘(푠, 푠1) 퐷 푦푁(푠1) 푑푠1). 0 0 Therefor

1 훼 ‖푒 (퐾 푦(푠))‖ ≤ ‖∫ 푘(. , 푠 ) 퐷 푒 (푠 ) 푑푠 ‖ + ‖퐸(푠)‖ 2 ( ) . (48) 푝푁 훼 퐿2(퐼) 1 푁 1 1 퐿 퐼 0 퐿2(퐼)

According to relation (37) and Lemma 4.3, we have

1 −푘 훼 ‖퐸(푠)‖퐿2 (퐼) ≤ 퐶2푁 ‖∫ 푘(. , 푠1) 퐷 푦푁 (푠1) 푑푠1‖ 0 퐻푘:푁 (퐼) ∗ −푘 훼 ≤ 퐶2퐶 푁 ‖퐷 푦‖퐻푘 :푁 (퐼) (49)

훼 In the other hand, because of linear operator 퐷 ∶ ℙ푁 → ℙ푁 is continuous and bounded [31], thus there exists a constant 퐶 ∗∗ ≥ 0 such that

훼 ∗∗ ‖퐷 푦푁‖퐻푘 :푁(퐼) ≤ 퐶 ‖푦푁‖퐻푘 :푁(퐼) , 푘 ∈ ℕ 푎푛푑 푘 ≤ 푁. (50)

Therefore, by combining two recent relations, we get

∗ ∗∗ −푘 ‖퐸(푠)‖퐿2 (퐼) ≤ 퐶2퐶 퐶 푁 ‖푦푁 ‖퐻푘 :푁(퐼) −푘 = 퐶9 푁 ‖푦 − 푒푁‖퐻푘 :푁 (퐼) −푘 ≤ 퐶9 푁 (‖푦‖퐻푘 :푁 (퐼) + ‖푒푁‖퐻푘 :푁(퐼) ). (51)

Now, by using relation(36), we proceed with the above inequality as

−푘 ‖퐸(푠)‖퐿2 (퐼) ≤ 퐶9 푁 (‖푦‖퐻푘 :푁(퐼) + ‖푒푁‖퐻푘 :푁(퐼) ) −푘 ≤ 퐶9 푁 (‖푦‖퐻푘 :푁(퐼) + ‖푒푁‖퐻1:푁(퐼) ) 3 −푘 −푘 2 ≤ 퐶9 푁 (‖푦‖퐻푘 :푁(퐼) + 퐶1푁 |푦|퐻푘:푁(퐼) ) 3 −2푘 −푘 2 ≤ 퐶9 푁 ‖푦‖퐻푘 :푁 (퐼) + 퐶10 푁 |푦|퐻푘 :푁 (퐼) . (52)

Similarly, from Lemma 4.3 and relation (50), we obtain

1 훼 훼 ‖∫ 푘(., 푠1) 퐷 푒푁(푠1) 푑푠1‖ ≤ 퐶11 ‖퐷 푒푁‖퐻푘 :푁(퐼) 0 퐿2(퐼)

≤ 퐶11 ‖푒푁‖퐻1:푁(퐼) 3 −푘 2 ≤ 퐶12 푁 |푦|퐻푘:푁(퐼) . (53)

At last, combining(48),(52), and (53) gives

3 −푘 −2푘 ‖푒 (퐾 푦(푠))‖ ≤ 퐶 푁 ‖푦‖ 푘 :푁 + 퐶 푁2 |푦| 푘:푁 . (54) 푝푁 훼 퐿2 (퐼) 9 퐻 (퐼) 13 퐻 (퐼)

In a similar manner with relation (37), we may write

−푘 ‖푒 (푓(푠))‖ ≤ 퐶 푁 |푓| 푘 :푁 . (55) 푝푁 퐿2 (퐼) 2 퐻 (퐼)

Finally, by substituting (54) − (55) in (46), the following relation can be obtained

3 −2푘 −푘 −푘 2 ‖푒푁‖퐿2(퐼) ≤ 퐶7 퐶2푁 |푓|퐻푘 :푁(퐼) + 퐶8 (퐶9 푁 ‖푦‖퐻푘 :푁 (퐼) + 퐶13 푁 |푦|퐻푘 :푁 (퐼) ) 3 −2푘 −푘 2 ≤ 훾1 푁 (|푓|퐻푘 :푁(퐼) + ‖푦‖퐻푘 :푁 (퐼) ) + 훾2 푁 |푦|퐻푘 :푁 (퐼) .

The above inequality proves that the approximation is convergent in 퐿2 −norm. Hence the theorem is proved.

5. Numerical example

To show efficiency of the numerical method, the following examples are considered. Example 5.1. Consider the following fractional integro-differential equation [20]

푡 1 1 푦′(푥) = 14 (1 − ) + ∫ 푥푠 퐷2푦(푠) 푑푠, 2.5 Γ(1.5) 0

with the initial condition: 푦(0) = 0, and exact solution 푦(푥) = 14푥.

We have solved this example using Chebyshev-Legendre spectral method and approximations are obtained as follows:

푛 = 0: 푦0(푥) = 0, 푛 = 1: 푦1(푥) = 14푥,

푛 = 2: 푦2(푥) = 14푥, 푛 = 3: 푦3(푥) = 14푥,

and so on. Therefore, we obtain 푦(푥) = 14푥 which is the exact solution of the problem.

Example 5.2. Our second example is the following fractional integro-differential equation [20]

1 1 푦′(푥) = 푓(푥) + ∫ 푥 2푠2 퐷4푦(푠) 푑푠, 0 with the initial condition

푦(0) = 0,

1 3 3 3 48 Γ(2.75) 2 4 where 푓(푥) = 8푥 − 푥 2 − ( − ) 푥 , and 푦(푥) = 2푥 − 푥 2 is the exact 2 6.75 Γ(4.75) 4.25 Γ(2.25) solution. The numerical results of our method can be seen from Figure 1 and Figure 2. These results indicate that the spectral accuracy is obtained for this problem, although the given function 푓(푡) is not very smooth.

Figure 1. Comparison between exact solution and approximate solution of Example 5.2 (left), the error function for some different values (right)

Figure 2. The error of numerical and exact solution of Example 5.2 versus the number of interpolation operator in 퐿2 norm

Example 5.3. Consider the following fractional integro-differential equation

9√휋 − 12 1 3 2푦′′(푥) + 푦′(푥) = ( )푥 2 + 36푥 + 8 + ∫ 푥 2√푠 퐷2푦(푠) 푑푠, √π 0 with the initial conditions

푦(0) = 0, 푦′(0) = 8.

Taking 푁 = 4, by implementing the Chebyshev-Legendre spectral method, we get the numerical solution as follow

−12 2 3 −13 4 3 푦4 = 8푥 + 1.003417 × 10 푥 + 3푥 + 1.652105 × 10 푥 ≅ 8푥 + 3푥 .

The approximate solution 푦4 for this fractional integro-differential equation tends rapidly to exact solution, i.e. 푦(푥) = 8푥 + 3푥 3.

Example 5.4. Let us consider the following fractional integro-differential equation

32 1 1 3푦(3) (푥) − 푦′′(푥) + 푦(푥) = (7 − ) 푒푥 + 3푥푒푥 + ∫ 푒푥−푠 퐷2푦(푠) 푑푠, 15√휋 0 with the initial conditions

푦(0) = 0, 푦′(0) = 1, 푦′′(0) = 2.

The exact solution of this fractional integro-differential is 푦(푥) = 푥푒푥 . We have reported the obtained numerical results for 푁 = 4 and 8 in Table 1. Also, in Figure 3, we plot the resulting errors versus the number 푁 of the steps. This figure shows the exponential rate of convergence predicted by the proposed method.

Proposed method Proposed method t Exact solution at 푁 = 4 at 푁 = 8 0 .0000000000 .0000000000 .0000000000 0.2 .2442815491 .2442805512 .2442805516 0.4 .5967277463 .5967298754 .5967298792 0.6 1.093273679 1.093271221 1.093271280 0.8 1.780432013 1.780432791 1.780432742 1 2.718281658 2.718281815 2.718281828 Table 1. The numerical results and exact solution for Example 5.4

Figure 3. Comparison between exact solution and approximate solution of Example 5.4 (left), the error function for some different values (right)

6. Conclusion

In this paper we present a Chebyshev-Legendre spectral approximation of a class of Fredholm fractional integro-differential equations. The most important contribution of this paper is that the errors of approximations decay exponentially in 퐿2 − 푛표푟푚. We prove that our proposed method is effective and has high convergence rate. The results given in the previous section are compared with exact solutions. The satisfactory results agree very well with exact solutions only for small numbers of shifted Legendre polynomials.

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