Post on 26-Jan-2021
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
javadi@khu.ac.ir, asadyosefi@gmail.com, babolian@khu.ac.ir, eslam.moradi@gmail.com,,
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 ๐ฟ2-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:
โ๐๐ ๐ท(๐)๐ฆ(๐ก)
๐
๐=0
= ๐(๐ก) +โซ ๐(๐ก, ๐ ) ๐ท๐ผ๐ฆ(๐ ) ๐๐ 1
0
,
๐โ1 < ๐ผ โค ๐,๐ โ โ, ๐ก โ [0,1], (1)
subject to the initial values
mailto:javadi@khu.ac.irmailto:asadyosefi@gmail.commailto:babolian@khu.ac.irmailto:eslam.moradi@gmail.com
๐ฆ(๐)(0) = ๐๐ , ๐ = 0,1, โฆ , ๐ โ 1, (2)
where Dฮฑ is the fractional derivative in the Caputo sense, ๐(๐ก) and ๐(๐ก, ๐ ) are the known functions
that are supposed to be sufficiently smooth and ๐๐ for any ๐ 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 ๐ผ = [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 ๐ฟ2 โnorm.
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 ๐ โ โ, the smallest integer that is greater than or equal to ๐ผ, i.e. ๐ = โ๐ผโ, the Caputoโs
fractional derivative operator of order ๐ผ > 0, is defined as:
๐ท๐ผ๐ฆ(๐ฅ) = {๐ฝ๐โ๐ผ๐ท๐๐ฆ(๐ฅ), ๐โ 1 < ๐ผ โค ๐,
๐ท๐๐ฆ(๐ฅ), ๐ผ = ๐, (3)
where
๐ฝ๐โ๐ผ๐ฆ(๐ฅ) =1
ฮ(๐ โ๐ผ) โซ (๐ฅ โ ๐ก)๐โ๐ผโ1๐ฆ(๐ก)๐๐ก
๐ฅ
0
, ๐ > 0, ๐ฅ > 0.
For the Caputoโs derivative we have [2]:
๐ท๐ผ ๐ฅ๐ฝ = {
0, ๐ฝ โ {0,1,2, โฆ } ๐๐๐ ๐ฝ < ๐,
ฮ(๐ฝ + 1)
ฮ(๐ฝ โ ๐ผ + 1) ๐ฅ๐ฝโ๐ผ ๐ฝ โ {0,1,2,โฆ } ๐๐๐ ๐ฝ โฅ ๐.
(4)
Recall that for ฮฑ โ โ, the Caputo differential operator coincides with the usual differential
operator. Similar to standard differentiation, Caputoโs fractional differentiation is a linear
operator, i.e.,
๐ท๐ผ(๐ ๐(๐ฅ) + ๐ โ(๐ฅ)) = ๐๐ท๐ผ๐(๐ฅ) + ๐๐ท๐ผโ(๐ฅ),
where ๐ and ๐ are constants.
The Chebyshev polynomials {๐๐(๐ก); ๐ = 0,1,โฆ } are defined on the interval [โ1,1] with the
following recurrence formula:
๐๐+1(๐ก) = 2๐ก ๐๐(๐ก) โ ๐๐โ1(๐ก), ๐ = 1,2,โฆ,
with ๐0(๐ก) = 1 and ๐1(๐ก) = t. The shifted Chebyshev polynomials are defined by introducing the
change of variable ๐ก = 2๐ฅ โ 1. Let the shifted Chebyshev polynomials ๐๐(2๐ฅ โ 1) be denote by
๐1,๐(๐ฅ), satisfying the relation
๐1,๐+1(๐ฅ) = 2(2๐ฅ โ 1)๐1,๐(๐ฅ) โ ๐1,๐โ1(๐ฅ), ๐ = 1,2, โฆ , ๐ฅ โ [0,1], (5)
where ๐1,0(๐ฅ) = 1 and ๐1,1(๐ฅ) = 2๐ฅ โ 1.
By these definitions we will have [32]
- ๐1,๐(๐ฅ) = ๐ โ (โ1)๐โ๐ (๐+๐โ1)! 2
2๐
(๐โ๐)!(2๐)! ๐ฅ๐ ๐ = 1,2, โฆ๐๐=0 . (6)
- ๐1,๐(0) = (โ1)๐ , ๐1,๐(1) = 1. (7
- โซ ๐1,๐(๐ฅ) ๐1,๐(๐ฅ)(๐ฅ โ ๐ฅ2)
โ1
2 ๐๐ฅ = ๐ฟ๐๐โ๐,1
0 (8)
where ๐ฟ๐๐ is Kronecker delta and
โ๐ = {๐, ๐ = 0 ๐
2, ๐ โฅ 1
, (9)
In this paper, we will consider the Gauss-type quadrature formulas. We start by defining the
Chebyshev-Gauss quadrature nodes and weights, respectively:
๐ฅ๐ = โ cos(2๐+1)๐
2๐+2, ๐ค๐ =
๐
๐+1, ๐ = 0,1,โฆ , ๐.
With the above choices, there holds
โซ ๐(๐ฅ)1
โ1โ ๐ฅ2 ๐๐ฅ = โ๐(๐ฅ๐)๐ค๐, โ๐ โ ๐2๐+1
๐
๐=0
1
โ1
, (10)
where ๐2๐+1 is a polynomial of degree less than or equal 2๐ + 1.
We now turn to the discrete Chebyshev transforms. The transforms can be performed via a matrix-
vector multiplication with ๐ช(๐2) operations as usual and when we use Chebyshev polynomials,
it can be carried out with ๐ช(๐ ๐๐๐2 ๐) operations via fast Fourier transform (๐ ๐ ๐) [26-27].
We define the Chebyshev-Lagrange polynomial by
๐บ๐(๐ฅ) =๐๐(๐ฅ)
(๐ฅ โ ๐ฅ๐)๐๐โฒ(๐ฅ๐)
, ; = 0,1, โฆ , ๐.
Given ๐ข(๐ฅ) โ ๐ถ[โ1,1], the Chebyshev-Lagrange interpolation operator ๐ผ๐๐๐ข is defined by
(๐ผ๐๐๐ข)(๐ฅ) =โ ๐ข๐ ๐บ๐(๐ฅ) โ โ๐ ,
๐
๐=0
(11)
where {๐ข๐} are determined by the forward discrete Chebyshev transform as follows
๐ข๐ = โ ๐ข(๐ฅ๐)cos(2๐+1)๐๐
2๐, 0 โค ๐ โค ๐. (12)๐โ1๐=0
The above transform can be computed by using ๐ญ๐ญ๐ป in ๐ช(๐ ๐๐๐2๐) operations [26-27].
Let ๐ฟ ๐(๐ก) be the standard Legendre polynomial of degree i, then we have [20]
- Three-term recurrence relation
(๐ + 1)๐ฟ๐+1(๐ก) = (2๐ + 1)๐ก ๐ฟ๐(๐ก) โ ๐๐ฟ ๐โ1(๐ก), ๐ โฅ 1, (13)
and the first two Legendre polynomials are
๐ฟ0(๐ก) = 1, ๐ฟ1(๐ก) = ๐ก.
- The Legendre polynomial ๐ฟ ๐(๐ก) has the expansion
๐ฟ๐(๐ก) =1
2๐ โ (โ1)๐
(2๐โ2๐) !
2๐ ๐!(๐โ๐)!(๐โ2๐)! ๐ก๐โ2๐ . (14)
[๐
2]
๐=0
- Orthogonality
โซ ๐ฟ๐(๐ก)๐ฟ๐(๐ก) ๐๐ก = โ๐๐ฟ๐๐ , (15)1
โ1
such that
โ๐ =2
2๐ + 1.
- Symmetry property
๐ฟ๐(โ๐ก) = (โ1)๐ ๐ฟ๐(๐ก), Li(ยฑ1) = (ยฑ1)
๐ . (16)
Hence, ๐ฟ๐(๐ก) is an odd (resp. even) function, if ๐ is odd (resp. even).
Now, if we define the shifted Legendre polynomial of degree ๐ by ๐ฟ1,๐(๐ฅ) = ๐ฟ๐(2๐ฅโ 1), then we
can obtain the analytic form and three-term recurrence relation of the shifted Legendre
polynomials of degree ๐ by the following form, respectively
๐ฟ1,๐(๐ฅ) = โ(โ1)๐+๐
(๐ + ๐)!
(๐ โ ๐)! (๐!)2 ๐ฅ๐ ,
๐
๐=0
๐ฟ1,๐(๐ฅ) =2๐ + 1
๐ + 1๐ฅ ๐ฟ1,๐(๐ฅ) โ
๐
๐ + 1๐ฟ1,๐โ1(๐ฅ), ๐ โฅ 1. (17)
According to Eq. (15), the orthogonality relation of shifted Legendre polynomials is
โซ ๐ฟ1,๐(๐ก)๐ฟ1,๐(๐ก) ๐๐ก = โ๐๐ฟ๐๐ . (18)1
0
We denote ๐ฟ๐2 (๐ผ) by the weighted ๐ฟ2 Hilbert space with the scalar product
(๐ข, ๐ฃ) = โซ ๐ข(๐ฅ) ๐ฃ(๐ฅ) ๐(๐ฅ)๐๐ก1
0
, โ๐ข, ๐ฃ โ ๐ฟ๐2 (๐ผ),
and the norm โ๐ขโ๐ฟ๐2 = (๐ข, ๐ข)๐
1
2 , where ๐(๐ฅ) = 1 in the Legendre case and ๐(๐ฅ) = (1 โ ๐ฅ2)โ1
2
in the Chebyshev case. We may drop the subscript ๐ค when ๐ = 1. Therefore, the corresponding
norm is
โ๐ขโ๐ฟ2 = (๐ข, ๐ข)12 .
Let ๐ป๐๐(๐ผ) = {๐ข โ ๐ฟ๐
2 (๐ผ) โถ ๐๐๐ข
๐๐ฅ๐โ ๐ฟ๐
2 (๐ผ), ๐ = 0,1, โฆ๐} be the weighted Sobolev space with the
norm and semi norm defined respectively
โ๐ฆโ๐ป๐๐ (๐ผ)2 = โโ๐ฆ(๐)โ
๐ฟ๐2 (๐ผ)
2,
๐
๐=0
and
|๐ฆ|๐ป๐ :๐ (๐ผ)2 = โ โ๐ฆ(๐)โ
๐ฟ๐2 (๐ผ)
2.๐๐=min(๐:๐)
๐ป๐(๐ผ) by its inner product is Hilbert space.
For a function ๐ฆ(๐ฅ) โ ๐ฟ2[0,1], the shifted Legendre expansion is
๐ฆ(๐ฅ) =โ๐๐ ๐ฟ1,๐(๐ฅ),
โ
๐=0
where
๐๐ =1
โ1,๐ โซ ๐ฆ(๐ฅ) ๐ฟ1,๐(๐ฅ) ๐๐ฅ, ๐ = 0,1,2, โฆ, (19)
1
0
and
โ1,๐ =1
2โ๐ =
1
2๐ + 1.
Now, we describe the Legendre-Gauss integration in the interval (0,1). We denote by ๐ฅ๐,๐,
๐๐,๐, ๐ = 0, โฆ , ๐ , respectively the nodes and weights of the standard integration on the interval
(โ1,1). We suppose ๐ฅ1,๐,๐ ,๐1,๐,๐ , ๐ = 0,โฆ , ๐ฟ, are nodes and weights of the Legendre-Gauss
integration in the interval (0,1). Then, we have
๐ฅ1,๐,๐ =1
2 (๐ฅ๐,๐ +1), ๐ค1,๐,๐ =
1
2๐ค๐,๐ ๐ = 0, . . , ๐.
According to Eq. (29) for any ๐ โ โ2๐+1, set of all polynomials of degree at most 2๐+ 1, we
get
โซ ๐(๐ฅ)๐๐ฅ =1
2
1
0
โซ ๐(1
2(๐ฅ + 1))๐๐ฅ
1
โ1
=1
2โ ๐๐,๐ ๐ (
1
2(๐ฅ๐,๐ + 1))
๐
๐=0
=โ ๐1,๐,๐ ๐(๐ฅ1,๐,๐).๐
๐=0
(20)
In practice, a number of first shifted Legendre polynomials are considered. We let
๐(๐ฅ) = [๐ฟ1,0(๐ฅ), ๐ฟ1,1(๐ฅ),โฆ , ๐ฟ1,๐(๐ฅ)]๐,
๐๐(๐ผ) = ๐ ๐๐๐{๐ฟ1,0(๐ฅ),๐ฟ1,1(๐ฅ),โฆ , ๐ฟ1,๐(๐ฅ)}. (21)
Theorem 2.1 [25] suppose ๐(๐ฅ) is defined in Eq.(21) and ๐ผ > 0; then the following relation holds:
๐ซ๐ผ๐(๐ฅ) โ ๐ซ(๐ผ)๐(๐ฅ), (22)
where ๐ซ(๐ผ) is the (๐ + 1) ร (๐+ 1) operational matrix of Caputo derivative which is given by:
๐ท(๐ผ) = (๐๐๐)0โค๐,๐โค๐ =
[ 0 0 0 โฆ 0
โฎ๐๐ผ(๐,0) ๐๐ผ(๐,1) ๐๐ผ(๐,2) . . . ๐๐ผ(๐,๐)
โฎ๐๐ผ(๐, 0) ๐๐ผ(๐, 1) ๐๐ผ(๐, 2) โฆ ๐๐ผ(๐,๐)
โฎ๐๐ผ(๐,0) ๐๐ผ(๐,1) ๐๐ผ(๐,2) โฆ ๐๐ผ(๐,๐)]
, (23)
where
๐๐ผ(๐,๐) = โ(โ1)๐+๐ (2๐ + 1) (๐ + ๐)! ฮ(๐โ ๐โ ๐ผ+ 1)
๐ฟ๐ผ(๐ โ ๐)!๐! ฮ(๐ โ ๐ผ+ 1) ฮ(๐ + ๐ โ ๐ผ+ 1). (24)
๐
๐=๐
Note that because of ๐ท๐ผ๐ฟ1,๐(๐ก) = 0, for ๐ = 0,1,โฆ ,๐ โ 1, the first ๐ rows are zero in ๐ซ.
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 ๐(๐๐๐๐2 ๐) operations by using ๐น๐น๐. 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
interpolation operator ๐ผ๐๐ , 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
๐ข(๐ฅ) =โ๐ผ๐ ๐1,๐
๐
๐=0
=โ๐ฝ๐ ๐ฟ1,๐
๐
๐=0
,
๐ถ = (๐ผ0,๐ผ1, โฆ , ๐ผ๐),
๐ท = (๐ฝ0, ๐ฝ1,โฆ , ๐ฝ๐).
In this work, what we need to apply spectral method is using the transform between ๐ถ and ๐ท. By
virtue of orthogonality of Chebyshev and Legendre polynomials, the relation between ๐ถ and ๐ท
can be obtained by computing (๐ข, ๐1,๐)๐ค and (๐ข,๐ฟ1,๐). defining
๐ด = (๐๐๐)0โค๐,๐โค๐,
๐ต = (๐๐๐)0โค๐,๐โค๐,
then, by using Eqs. (8) and (15), we can obtain
๐๐๐ =1
โ๐ (๐1,๐ , ๐ฟ1,๐)๐ค
,
๐๐๐ = (๐ +1
2)(๐ฟ1,๐ , ๐1,๐).
Thus, we will have
๐ถ = ๐ด๐ท,
๐ท = ๐ต๐ถ,
๐ด๐ต = ๐ต๐ด = ๐ผ.
According to orthogonality and parity of the Chebyshev and Legendre polynomials, we get
๐๐๐ = ๐๐๐ = 0, for ๐ > ๐ or ๐ + ๐ odd.
Therefore, we only determine the nonzero elements of both ๐ด and ๐ต by using three-term
recurrence relation of the shifted Legendre and Chebyshev polynomials. Applying definition of
๐๐๐, we can obtain recurrence formula
๐๐๐ =1
โ๐ (๐1,๐ , ๐ฟ1,๐)๐ค
=1
โ๐ (๐1,๐ ,
2๐ + 1
๐ + 1 (2๐ฅ โ 1) ๐ฟ1,๐(๐ฅ) โ
๐
๐ + 1๐ฟ1,๐โ1(๐ฅ))
๐ค
=1
โ๐ {2๐ + 1
๐ + 1((2๐ฅ โ 1)๐1,๐ , ๐ฟ1,๐)
๐คโ
๐
๐ + 1(๐1,๐ , ๐ฟ1,๐โ1)๐ค
}
=1
โ๐ {2๐ + 1
2๐ + 2 (๐1,๐+1 + ๐1,๐โ1, ๐ฟ1,๐)๐ค
โ๐
๐ + 1 โ๐๐๐ ,๐โ1}
=โ๐+1โ๐
2๐ + 1
2๐ + 2 ๐๐+1,๐ +
โ๐โ1โ๐
2๐ + 1
2๐ + 2 ๐๐โ1,๐ โ
๐
๐ + 1 ๐๐,๐โ1.
We can similarly derive entries of matrix ๐ต as follow
๐๐๐ = (๐ +1
2) ๏ฟฝฬ๏ฟฝ๐๐,
where
๏ฟฝฬ๏ฟฝ๐๐: = (๐ฟ1,๐ , ๐1,๐) =2๐ + 2
2๐ + 1 ๏ฟฝฬ๏ฟฝ๐+1,๐ +
2๐
2๐ + 1 ๏ฟฝฬ๏ฟฝ๐โ1,๐ โ ๏ฟฝฬ๏ฟฝ๐,๐โ1.
Thus, we can obtain each nonzero element of ๐ด and ๐ต 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 ๐ฆ๐(๐ก) โ ๐๐(๐ผ), then by
implementing Chebyshev-Legendre spectral method for Eq.(1), we can easily obtain
โ๐๐
๐
๐=0
(๐ท(๐)๐ฆ๐, ๐ฟ1,๐) = (๐ผ๐๐ ๐,๐ฟ1,๐)+ (โซ ๐ผ๐
๐ ๐(. , ๐ ) ๐ท๐ผ๐ฆ๐(๐ ) ๐๐ 1
0
, ๐ฟ1,๐). (25)
We have ๐ฆ๐(๐ก) = โ ๐๐๐ฟ1,๐(๐ก),๐๐=0 then according to linearity of Caputoโs fractional
differentiation, Eq.(23) can be written as:
โ๐๐
๐
๐=0
โ๐๐
๐
๐=0
(๐ท(๐)๐ฟ1,๐ , ๐ฟ1,๐)
= (๐ผ๐๐ ๐,๐ฟ1,๐)+โ๐๐
๐
๐=0
(โซ ๐ผ๐๐ ๐(. , ๐ ) ๐ท๐ผ๐ฟ1,๐(๐ ) ๐๐
1
0
, ๐ฟ1,๐). (26)
From Eq. (22) to (24) in Theorem 2.1, we can obtain
๐ท๐ผ๐ฟ1,๐(๐ก) = โ ๐๐ผ(๐, ๐)๐ฟ1,๐(๐ก), ๐ = ๐,๐ +1,โฆ ,๐. (27)๐๐=0
We notice that if ๐ผ = ๐ โ โ, then ๐๐ผ 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
โโ๐๐ ๐๐
๐
๐=๐
(โ๐๐(๐, ๐)
๐
๐=0
(๐ฟ1,๐, ๐ฟ1,๐))
๐
๐=0
= (๐ผ๐๐ ๐, ๐ฟ1,๐)+ โ ๐๐
๐
๐=๐
โ ๐๐ผ(๐, ๐)
๐
๐=0
(โซ ๐ผ๐๐ ๐(. , ๐ ) ๐ฟ1,๐(๐ก) ๐๐
1
0
, ๐ฟ1,๐). (28)
Then, making use of the orthogonality relation of shifted Legendre polynomials, i.e. Eq.(18),
Eq. (24) reduce to
โโ๐๐ ๐๐
๐
๐=๐
๐๐(๐,๐)
2๐ + 1
๐
๐=0
= (๐ผ๐๐ ๐, ๐ฟ1,๐)+ โ ๐๐
๐
๐=๐
โ ๐๐ผ(๐, ๐)
๐
๐=0
(โซ ๐ผ๐๐ ๐(. , ๐ ) ๐ฟ1,๐(๐ก) ๐๐
1
0
, ๐ฟ1,๐). (29)
We let
โ๐(๐ฅ) = โซ ๐ผ๐๐ ๐(๐ฅ,๐ ) ๐ฟ1,๐(๐ก) ๐๐
1
0
โ โ๐๐๐ ๐ฟ1,๐(๐ฅ)
๐
๐=0
,
๐๐ = (๐ผ๐๐ ๐,๐ฟ1,๐).
Thus, again by using Eqs.(18), Eq. (29) becomes the following form
โโ๐๐ ๐๐
๐
๐=๐
๐๐(๐,๐)
2๐ + 1
๐
๐=0
= ๐๐ + โโ๐๐ ๐๐ผ(๐, ๐)
๐
๐=0
๐
๐=๐
๐๐๐2๐ + 1
. (30)
It is easy to verify that initial conditions convert to following equations
โ โ ๐๐ ๐๐(๐, ๐) ๐ฟ1,๐(0)๐๐=0
๐๐=0 = ๐๐ , ๐ = 0,1, โฆ , ๐ โ 1. (31)
Combining Eqs. (30) and (31) yields
{
โโ๐๐ ๐๐
๐
๐=๐
๐๐(๐,๐)
2๐ + 1
๐
๐=0
โโโ๐๐ ๐๐ผ(๐, ๐)
๐
๐=๐
๐
๐=0
๐๐๐2๐ + 1
= ๐๐ , ๐ = 0,1,โฆ๐ โ ๐,
โโ๐๐ ๐๐(๐, ๐) ๐ฟ1,๐(0)
๐
๐=0
๐
๐=0
= ๐๐ , ๐ = 0,1, โฆ , ๐ โ 1.
By solving the above system of linear equations, we can get the value of {๐๐}๐=0๐
and obtain the
expression of ๐ฆ๐(๐ฅ) 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 ๐ฟ2 โnorm. 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:
โซ โซ โฆโซ โซ ๐(๐ก1)๐ก2
0
๐๐ก1๐๐ก2 โฆ๐๐ก๐
๐ก3
0
๐ก๐
0
๐ก
0
=1
(๐ โ 1)! โซ (๐ก โ ๐ )๐โ1๐(๐ )
๐ก
0
๐๐ , (32)
where ๐ is integrable function on interval (0,๐ก) and ๐ก๐ (๐ = 2,3, โฆ , ๐) are parameters in the
purpose interval.
Lemma 4.2 [30] (Granwall's Lemma) Assume that ๐ข, ๐, ๐ฝ โ ๐ถ (๐ผ) with ๐ฝ(๐ก) โฅ 0. If ๐ข satisfies the
inequality
๐ข(๐ก) โค ๐(๐ก) + โซ ๐ฝ(๐ )๐ข(๐ ) ๐๐ ๐ก
0
, ๐ก โ ๐ผ,
then
๐ข(๐ก) โค ๐(๐ก) + โซ ๐ฝ(๐ )๐(๐ ) exp(โซ ๐ฝ(๐ฃ) ๐๐ฃ๐ก
๐
) ๐๐ ๐ก
0
, ๐ก โ ๐ผ. (33)
On the other word, if ๐ is non-decreasing on ๐ผ, the above inequality reduce to
๐ข(๐ก) โค ๐(๐ก) exp(โซ ๐ฝ(๐ฃ) ๐๐ฃ๐ก
๐
), ๐ก โ ๐ผ. (34)
Lemma 4.3 [30] Suppose that ๐ is a given kernel function on ๐ผ ร ๐ผ. If ๐ โ ๐ฟ๐(๐, ๐) for
1 โค p โค โ, the integral
๐๐(๐ฅ) = โซ ๐(๐ฅ, ๐ก) ๐(๐ก)๐ฅ ๐๐ ๐
๐
๐๐ก
is well-defined in ๐ฟ๐(๐, ๐) and there exists ๐ถโ > 0 such that
โ๐๐โ๐ฟ๐ (๐,๐) โค ๐ถโโ๐โ๐ฟ๐ (๐,๐) . (35)
Let ๐๐ be the interpolation projection operator from ๐2(๐ผ) upon โ๐(๐ผ). Then, for any function ๐
in ๐ฟ2(๐ผ) satisfies
โซ (๐ โ ๐๐๐)(๐ก) ๐(๐ก)๐๐ก = 0,1
0
โ๐ โ โ๐(๐ผ).
Also, the following relations for interpolation in shifted Legendre polynomials and shifted Gauss-
Legendre nodal points ๐ โฅ 1 (or for any fixed ๐ โค ๐) may readily be obtained as [14]
โ๐ฆ โ ๐๐(๐ฆ)โ๐ป๐ (๐ผ) โค ๐ถ1๐2๐โ
12โ๐|๐ฆ|๐ป๐:๐(๐ผ) , (36)
โ๐ฆ โ ๐๐(๐ฆ)โ๐ฟ2 (๐ผ) โค ๐ถ2๐โ๐|๐ฆ|๐ป๐:๐(๐ผ) . (37)
where ๐ฆ โ ๐ป๐(๐ผ), and ๐ถ1 and ๐ถ2 are constants independent of ๐ and 0 โค ๐ โค ๐.
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
๐๐(๐ฅ) = ๐ฆ(๐ฅ) โ ๐ฆ๐(๐ฅ), be the error function of the Chebyshev-Legendre spectral approximation
to ๐ฆ(๐ฅ). 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 ๐, the Chebyshev-Legendre spectral approximations ๐ฆ๐(๐ฅ)
converge to exact solution in ๐ฟ2-norm, i.e.
โ๐๐โ๐ฟ2 (๐ผ) = โ๐ฆ โ ๐ฆ๐โ๐ฟ2(๐ผ) โ 0.
Proof. Assume that ๐ฆ๐(๐ฅ) is obtained by using the Chebyshev-Legendre spectral method Eq. (1)
together with initial conditions Eq. (2). Then, we have
โ๐๐ ๐ท(๐)๐ฆ๐(๐ก)
๐
๐=0
= ๐๐(๐(๐ก)) + ๐๐ (โซ ๐(. , ๐ ) ๐ท๐ผ๐ฆ๐(๐ ) ๐๐
1
0
), (38)
such that ๐๐ is the Lagrange interpolation polynomial operator defined for Legendre polynomial.
With ๐ times integration from Eq. (38), we obtain
โ๐๐ โซ โซ โฆโซ โซ ๐ฆ๐(๐)(๐ก1)
๐ก2
0
๐๐ก1๐๐ก2 โฆ๐๐ก๐
๐ก3
0
๐ก๐
0
๐ก
0
๐
๐=0
= โซ โซ โฆโซ โซ ๐๐(๐(๐ก1))๐ก2
0
๐๐ก1๐๐ก2 โฆ๐๐ก๐
๐ก3
0
๐ก๐
0
๐ก
0
+โซ โซ โฆโซ โซ ๐๐ (โซ ๐(๐ก1, ๐ ) ๐ท๐ผ๐ฆ๐(๐ ) ๐๐
1
0
)๐ก2
0
๐๐ก1๐๐ก2 โฆ๐๐ก๐
๐ก3
0
๐ก๐
0
๐ก
0
. (39)
By virtue of Lemma 4.1, we can convert each of the multiple integral to single integral, so we
have
๐๐๐ฆ๐(๐ก) + ๐(๐ก) +โโซ๐๐
(๐โ ๐ โ 1)! (๐ก โ ๐ )๐โ๐โ1 ๐ฆ๐(๐ ) ๐๐
๐ก
0
๐โ1
๐=0
=โซ(๐ก โ ๐ )๐โ1
(๐ โ 1)! ๐๐(๐(๐ )) ๐๐
๐ก
0
+โซ(๐ก โ ๐ )๐โ1
(๐โ 1)! ๐๐ (โซ ๐(๐ , ๐ 1) ๐ท
๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
) ๐๐ ,๐ก
0
(40)
where ๐ is a polynomial of degree ๐ with the initial condition coefficient. Similarly, from Eq.
(1), we get
๐๐๐ฆ(๐ก) + ๐(๐ก) +โโซ๐๐
(๐ โ ๐ โ 1)! (๐ก โ ๐ )๐โ๐โ1 ๐ฆ(๐ ) ๐๐
๐ก
0
๐โ1
๐=0
= โซ(๐ก โ ๐ )๐โ1
(๐ โ 1)! ๐(๐ ) ๐๐
๐ก
0
+โซ(๐ก โ ๐ )๐โ1
(๐โ 1)! โซ ๐(๐ ,๐ 1) ๐ท
๐ผ๐ฆ(๐ 1) ๐๐ 1
1
0
๐๐ .๐ก
0
(41)
By subtracting Eq. (40) from Eq. (41), we obtain
๐๐๐๐(๐ก) +โโซ๐๐
(๐โ ๐ โ 1)! (๐ก โ ๐ )๐โ๐โ1 ๐๐(๐ ) ๐๐
๐ก
0
๐โ1
๐=0
= โซ(๐ก โ ๐ )๐โ1
(๐ โ 1)! ๐๐๐(๐(๐ )) ๐๐
๐ก
0
+โซ(๐ก โ ๐ )๐โ1
(๐โ 1)! ๐๐๐(๐พ๐ผ๐ฆ(๐ )) ๐๐ ,
๐ก
0
(42)
such that
๐๐๐(๐(๐ )) = ๐(๐ ) โ ๐๐(๐(๐ )),
๐พ๐ผ๐ฆ(๐ ) = โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ(๐ 1) ๐๐ 1
1
0
,
๐๐๐(๐พ๐ผ๐ฆ(๐ )) = ๐พ๐ผ๐ฆ(๐ ) โ ๐๐(๐พ๐ผ๐ฆ๐(๐ ))
= ๐พ๐ผ๐ฆ(๐ ) โ๐พ๐ผ๐ฆ๐(๐ ) + ๐พ๐ผ๐ฆ๐(๐ ) โ ๐๐(๐พ๐ผ๐ฆ๐(๐ ))
= ๐พ๐ผ๐๐(๐ ) โ ๐๐๐(๐พ๐ผ๐ฆ๐(๐ )). 0 โค ๐ โค ๐ก.
From Eq. (42), we can obtain
|๐๐(๐ก)| โคโ |๐๐
โ๐๐ (๐โ ๐ โ 1)!|
๐โ1
๐=0
โซ |(๐ก โ ๐ )๐โ๐โ1 ๐๐(๐ )|๐๐ ๐ก
0
+1
|๐๐ |(๐โ 1)! โซ |(๐ก โ ๐ )๐โ1 ๐๐๐(๐(๐ ))|๐๐
๐ก
0
+1
|๐๐|(๐โ 1)! โซ |(๐กโ ๐ )๐โ1 ๐๐๐(๐พ๐ผ๐ฆ(๐ ))|๐๐
๐ก
0
โค ๐ถ2โซ |๐๐(๐ )|๐ก
0
๐๐ + ๐ถ3โซ |๐๐๐(๐(๐ ))|๐ก
0
๐๐ + ๐ถ4โซ |๐๐๐(๐พ๐ผ๐ฆ(๐ ))|๐ก
0
๐๐ . (43)
Applying Lemma 4.2 leads to
|๐๐(๐ก)| โค exp(โซ ๐ถ2๐๐ ๐ก
0
) (๐ถ3โซ |๐๐๐(๐(๐ ))|๐ก
0
๐๐ + ๐ถ4โซ |๐๐๐(๐พ๐ผ๐ฆ(๐ ))|๐ก
0
๐๐ )
โค ๐ถ5โซ |๐๐๐(๐(๐ ))|๐ก
0
๐๐ + ๐ถ6โซ |๐๐๐(๐พ๐ผ๐ฆ(๐ ))|๐ก
0
๐๐ . (44)
Equivalently, by using the ๐ฟ2 โnorm, we get
โ๐๐โ๐ฟ2 (๐ผ) โค ๐ถ5 โโซ |๐๐๐(๐(๐ ))|๐ก
0
๐๐ โ๐ฟ2 (๐ผ)
+ ๐ถ6 โโซ |๐๐๐(๐พ๐ผ๐ฆ(๐ ))|๐ก
0
๐๐ โ๐ฟ2(๐ผ)
. (45)
Bu using Lemma 4.3, the above inequality reduce to
โ๐๐โ๐ฟ2(๐ผ) โค ๐ถ7 โ๐๐๐(๐(๐ ))โ๐ฟ2 (๐ผ)+๐ถ8 โ๐๐๐(๐พ๐ผ๐ฆ(๐ ))โ๐ฟ2 (๐ผ)
. (46)
On the other hand, we have
๐๐๐(๐พ๐ผ๐ฆ(๐ )) = โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ(๐ 1) ๐๐ 1
1
0
โ๐๐ (โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
)
=โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ(๐ 1) ๐๐ 1
1
0
โโซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
+โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
โ ๐๐ (โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
)
=โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐๐(๐ 1) ๐๐ 1
1
0
+๐ธ(๐ ), (47)
where
๐ธ(๐ ) = โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
โ ๐๐ (โซ ๐(๐ , ๐ 1) ๐ท๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
).
Therefor
โ๐๐๐(๐พ๐ผ๐ฆ(๐ ))โ๐ฟ2(๐ผ)โค โโซ ๐(. , ๐ 1) ๐ท
๐ผ๐๐(๐ 1) ๐๐ 1
1
0
โ๐ฟ2(๐ผ)
+ โ๐ธ(๐ )โ๐ฟ2 (๐ผ) . (48)
According to relation (37) and Lemma 4.3, we have
โ๐ธ(๐ )โ๐ฟ2 (๐ผ) โค ๐ถ2๐โ๐ โโซ ๐(. , ๐ 1) ๐ท
๐ผ๐ฆ๐(๐ 1) ๐๐ 1
1
0
โ๐ป๐:๐ (๐ผ)
โค ๐ถ2๐ถโ๐โ๐โ๐ท๐ผ๐ฆโ๐ป๐:๐ (๐ผ) (49)
In the other hand, because of linear operator ๐ท๐ผ โถ โ๐ โ โ๐ is continuous and bounded [31], thus
there exists a constant ๐ถโโ โฅ 0 such that
โ๐ท๐ผ๐ฆ๐โ๐ป๐:๐(๐ผ) โค ๐ถโโ โ๐ฆ๐โ๐ป๐:๐(๐ผ) , ๐ โ โ ๐๐๐ ๐ โค ๐. (50)
Therefore, by combining two recent relations, we get
โ๐ธ(๐ )โ๐ฟ2 (๐ผ) โค ๐ถ2๐ถโ๐ถโโ ๐โ๐โ๐ฆ๐โ๐ป๐:๐(๐ผ)
= ๐ถ9 ๐โ๐ โ๐ฆ โ ๐๐โ๐ป๐:๐ (๐ผ)
โค ๐ถ9 ๐โ๐ (โ๐ฆโ๐ป๐:๐ (๐ผ) + โ๐๐โ๐ป๐ :๐(๐ผ) ). (51)
Now, by using relation(36), we proceed with the above inequality as
โ๐ธ(๐ )โ๐ฟ2 (๐ผ) โค ๐ถ9 ๐โ๐ (โ๐ฆโ๐ป๐:๐(๐ผ) + โ๐๐โ๐ป๐:๐(๐ผ))
โค ๐ถ9 ๐โ๐ (โ๐ฆโ๐ป๐ :๐(๐ผ) + โ๐๐โ๐ป1:๐(๐ผ))
โค ๐ถ9 ๐โ๐ (โ๐ฆโ๐ป๐ :๐(๐ผ) +๐ถ1๐
32โ๐|๐ฆ|๐ป๐:๐(๐ผ))
โค ๐ถ9 ๐โ๐โ๐ฆโ๐ป๐:๐ (๐ผ) + ๐ถ10 ๐
32โ2๐ |๐ฆ|๐ป๐ :๐ (๐ผ) . (52)
Similarly, from Lemma 4.3 and relation (50), we obtain
โโซ ๐(. , ๐ 1) ๐ท๐ผ๐๐(๐ 1) ๐๐ 1
1
0
โ๐ฟ2(๐ผ)
โค ๐ถ11 โ๐ท๐ผ๐๐โ๐ป๐:๐(๐ผ)
โค ๐ถ11 โ๐๐โ๐ป1:๐(๐ผ)
โค ๐ถ12 ๐32โ๐|๐ฆ|๐ป๐:๐(๐ผ) . (53)
At last, combining(48),(52), and (53) gives
โ๐๐๐(๐พ๐ผ๐ฆ(๐ ))โ๐ฟ2 (๐ผ)โค ๐ถ9 ๐
โ๐โ๐ฆโ๐ป๐:๐(๐ผ) + ๐ถ13 ๐32โ2๐ |๐ฆ|๐ป๐:๐(๐ผ) . (54)
In a similar manner with relation (37), we may write
โ๐๐๐(๐(๐ ))โ
๐ฟ2 (๐ผ)โค ๐ถ2๐
โ๐|๐|๐ป๐:๐(๐ผ) . (55)
Finally, by substituting (54) โ (55) in (46), the following relation can be obtained
โ๐๐โ๐ฟ2(๐ผ) โค ๐ถ7 ๐ถ2๐โ๐|๐|๐ป๐ :๐(๐ผ) +๐ถ8 (๐ถ9 ๐
โ๐โ๐ฆโ๐ป๐:๐ (๐ผ) + ๐ถ13 ๐32โ2๐ |๐ฆ|๐ป๐ :๐ (๐ผ))
โค ๐พ1 ๐โ๐(|๐|๐ป๐:๐(๐ผ) + โ๐ฆโ๐ป๐ :๐ (๐ผ))+ ๐พ2๐
32โ2๐
|๐ฆ|๐ป๐:๐ (๐ผ) .
The above inequality proves that the approximation is convergent in ๐ฟ2 โnorm. 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]
๐ฆโฒ(๐ฅ) = 14 (1 โ๐ก
2.5 ฮ(1.5))+โซ ๐ฅ๐ ๐ท
12๐ฆ(๐ ) ๐๐
1
0
,
with the initial condition: ๐ฆ(0) = 0, and exact solution ๐ฆ(๐ฅ) = 14๐ฅ.
We have solved this example using Chebyshev-Legendre spectral method and approximations
are obtained as follows:
๐ = 0: ๐ฆ0(๐ฅ) = 0,
๐ = 1: ๐ฆ1(๐ฅ) = 14๐ฅ,
๐ = 2: ๐ฆ2(๐ฅ) = 14๐ฅ,
๐ = 3: ๐ฆ3(๐ฅ) = 14๐ฅ,
and so on. Therefore, we obtain ๐ฆ(๐ฅ) = 14๐ฅ which is the exact solution of the problem.
Example 5.2. Our second example is the following fractional integro-differential equation [20]
๐ฆโฒ(๐ฅ) = ๐(๐ฅ)+ โซ ๐ฅ2๐ 2 ๐ท14๐ฆ(๐ ) ๐๐
1
0
,
with the initial condition
๐ฆ(0) = 0,
where ๐(๐ฅ) = 8๐ฅ3 โ3
2๐ฅ1
2 โ (48
6.75 ฮ(4.75)โ
ฮ(2.75)
4.25 ฮ(2.25))๐ฅ2, and ๐ฆ(๐ฅ) = 2๐ฅ4 โ ๐ฅ
3
2 is the exact
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
๐(๐ก) 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 ๐ฟ2 norm
Example 5.3. Consider the following fractional integro-differential equation
2๐ฆโฒโฒ(๐ฅ)+ ๐ฆโฒ(๐ฅ) = (9โ๐ โ 12
โฯ)๐ฅ2 + 36๐ฅ + 8+ โซ ๐ฅ2โ๐ ๐ท
32๐ฆ(๐ ) ๐๐
1
0
,
with the initial conditions
๐ฆ(0) = 0,
๐ฆโฒ(0) = 8.
Taking ๐ = 4, by implementing the Chebyshev-Legendre spectral method, we get the numerical
solution as follow
๐ฆ4 = 8๐ฅ + 1.003417 ร 10โ12๐ฅ2 +3๐ฅ3 + 1.652105 ร 10โ13๐ฅ4 โ 8๐ฅ + 3๐ฅ3.
The approximate solution ๐ฆ4 for this fractional integro-differential equation tends rapidly to exact
solution, i.e. ๐ฆ(๐ฅ) = 8๐ฅ + 3๐ฅ3.
Example 5.4. Let us consider the following fractional integro-differential equation
3๐ฆ(3)(๐ฅ) โ ๐ฆโฒโฒ(๐ฅ) + ๐ฆ(๐ฅ) = (7 โ32
15โ๐)๐๐ฅ +3๐ฅ๐๐ฅ + โซ ๐๐ฅโ๐ ๐ท
12๐ฆ(๐ ) ๐๐
1
0
,
with the initial conditions
๐ฆ(0) = 0,
๐ฆโฒ(0) = 1,
๐ฆโฒโฒ(0) = 2.
The exact solution of this fractional integro-differential is ๐ฆ(๐ฅ) = ๐ฅ๐๐ฅ . We have reported the
obtained numerical results for ๐ = 4 and 8 in Table 1. Also, in Figure 3, we plot the resulting
errors versus the number ๐ of the steps. This figure shows the exponential rate of convergence
predicted by the proposed method.
t Proposed method
at ๐ = 4 Proposed method
at ๐ = 8 Exact solution
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 ๐ฟ2 โ ๐๐๐๐. 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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