The Legendre Spectral-Collocation method for a class of fractional integral equations A. Yousefi, S. Javadi, E. Babolian Department of Computer Science, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran
[email protected],
[email protected],
[email protected]
Abstract In this paper, we consider spectral-collocation method base on Legendre-Gauss-Lobatto point. We present a computational method for solving a class of fractional integral equation of the second kind. Then based on Legendre-Gauss-Lobatto point and using, we derive a system of algebraic equations. The method is illustrated by applications and the results obtained are compared with the exact solutions in open literature. The obtained numerical results show that our proposed method is efficient and accurate for fractional integral equations of second kind. In addition, we prove that the error of the approximate solution decay exponentially in πΏ2 norm. Keyword. Legendre Spectral-Collocation method, Fractional integral equation, Fractional order.
1. Introduction Fractional calculus has gained importance and popularity during the past three decades or so, due to mainly its demonstrated applications in numerous seemingly diverse fields of science and engineering. The advantage of fractional calculus becomes apparent in modeling mechanical and electrical properties of real material, as well as in the description of many fractional fields (see for details [1-2]). In recent years, there has been many studies of fractional integral and differential equations (for instance, see [3-13]). Spectral methods are an emerging area in the field of applied science 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 ([14-19]). In this paper, we are concerned with the numerical solutions of the following equation: (1) π¦(π₯) = π(π₯) πΌ πΌ {π(π₯)π¦(π₯)} + π(π₯), where πΌ Ξ± is the fractional integral of order 0 < πΌ < 1, π, π and π are the continuous real functions on [0, π], π¦ is the unknown function that must be determined. Existence and uniqueness of the solution of the Eq.(1), in additional to some analytical properties and important inequalities, are investigated [20]. The authors in [20] applied the product trapezoidal method based on Picard iteration for approximating solution at the mesh point. To the best of our knowledge, fractional derivative and integral are global, i.e. they are defined by an interval over the whole interval [0, π], and therefore global method, such as spectral methods, are better suited for this equation. We introduce a collocation method based on the Gauss-Legendre interpolation for solving Eq. (1). Inspired by the work of [21], we extend the approach to Eq. (1) and provide a rigorous convergence analysis for the Legendre spectral-
collocation method. We show that an approximate solution is convergent in πΏ2 and πΏβ norms. The results obtained in this paper may be considered as a generalization of the result in [20]. The structure of this paper is as follows: In section 2, some necessary deο¬nitions and mathematical tools of the fractional calculus which are required for our subsequent developments are introduced. In section 3, the Legendre spectral-collocation method of the fractional integral equation of second kind is obtained. After this section, we discuss about convergence analysis and then, some numerical experiments are presented in Section 5 until efficiency of Legendre spectral-collocation method. Some conclusions are given in section 6. 2. Basic Definitions and Notations For the concept of fractional integral, we will adopt Riemann-Liouville definition. In whole of this paper, integral operator will be Riemann-Liouville fractional integral operator. Definition 1.2 A real function π on [0, π] is said to be in the space πΆπ , π β β, if there exists a real number π(> π), such that π(π₯) = π₯ π π1 (π₯), where π1 β πΆ[π, π], and it is said to be in the space πΆππ iff π (π) β πΆπ , π β β. Definition 2.2 The Riemann-Liouville fractional integral operator of order πΌ β₯ 0, of a function π β πΆπ , π β₯ β1, is defined as π₯ 1 (2) πΌ πΌ π(π₯) = β« (π₯ β π‘)πΌβ1 π(π‘)ππ‘, πΌ > 0, π₯ β [0, π], Ξ(πΌ) 0 πΌ 0 π(π₯) = π(π₯), where Ξ(. ) is the Gamma function where
β
Ξ(πΌ) = β« π‘ πΌβ1 π βπ‘ ππ‘. 0
Some properties of the Riemann-Liouville fractional integral are (π) πΌ πΌ πΌπ½ = πΌ πΌ+π½ ,
(ππ) πΌ πΌ πΌπ½ = πΌπ½ πΌ πΌ , π€(π+1)
(πππ) πΌ πΌ (π₯ β π)π = (π₯ β π)πΌ+π , π€(πΌ+π+1) where πΌ, π½ β₯ 0 and π > β1. Let ππ1 ,π2 (π₯) = (1 β π₯)π1 (1 + π₯)π2 be a weight function in the usual sense, for π1 , π2 > β1. π ,π2 β }π=0
The Jacobi polynomials {π½π 1
are orthogonal with respect to ππ1,π2 over interval (β1,1).
The set of Jacobi polynomials is a complete πΏ2ππ1,π2 (β1,1) βorthogonal system, namely [17] 1
π ,π2 π1 ,π2 π1 ,π2 (π₯) π½π π
β« π½π 1 β1
where πΏππ is the Kronecker function, and
π ,π2
ππ₯ = πΎπ 1
πΏππ ,
(3)
π ,π πΎπ 1 2
2π1 +π2 +1 Ξ(π1 + 1)Ξ(π2 + 1) , Ξ(π1 + π2 + 2) = 2π1 +π2 +1 Ξ(π + π1 + 1)Ξ(π + π2 + 1) , {(2π + π1 + π2 + 1)n! Ξ(π + π1 + π2 + 1)
π = 0, π β₯ 1.
In particular, we find π ,π2
π½0 1
1 1 (π₯) = 1, π½1π1 ,π2 (π₯) = (π1 + π2 + 2)π₯ + (π1 β π2 ). 2 2
Let ππ [β1,1] the set of all polynomials of degree β€ π (π β₯ 0). Thus, if we denote by π ,π2
{π₯π 1
π ,π2 π
, ππ 1
}
π=0
the nodes and the corresponding Christoffel numbers of the standard Jacobi-
Gauss interpolation on Ξ = (β1,1), then we have 1
π ,π2
β«β1 π(π₯)ππ 1
π1 ,π2 π ,π (π₯)ππ₯ = βπ )ππ 1 2 , π=0 π(π₯π
βπ β π2πβ1 [β1,1].
(4)
π ,π
1 2 For any π’ β πΆ(Ξ), we denote by ππ₯,π βΆ πΆ(Ξ) β ππ the Jacobi-Gauss interpolation operator, such that
π ,π
π ,π2
1 2 ππ₯,π π’(π₯π 1
π ,π2
) = π’(π₯π 1
),
(5)
0 β€ π β€ π.
It is clear that π π1 ,π2 ππ₯,π
π ,π2
π’(π₯) = β π’π 1
π ,π2
π½π 1
(π₯),
(6)
π=0
where π ,π π’π 1 2
=
π
1 π ,π2
πΎπ 1
π ,π2
β π’(π₯π 1
π ,π2
) π½π 1
π ,π2
(π₯π 1
π ,π2
)ππ 1
.
π=0
In special case, if π1 = π2 = 0, then the Jacobi polynomial is reduced to the Legendre polynomial. Thus, we recall that the Legendre polynomials [17-18] {πΏπ (π₯); π = 0,1, β¦ } are defined on the Ξ with the following recurrence formula: 2π + 1 π πΏπ+1 (π₯) = π₯ πΏπ (π₯) β πΏ (π₯), π = 1,2, β¦, π+1 π + 1 πβ1 with πΏ0 (π₯) = 1 and πΏ1 (π₯) = π₯. By using the Rodrigues formula πΏπ (π₯) =
(β1)π 2π π!
π· π ((1 β π₯ 2 )π ),
the Legendre polynomial has the expansion as follow π [ ] 2
πΏπ (π₯) =
(2π β 2π)! 1 β(β1)π π π₯ πβ2π . π 2 2 π! (π β π)! (π β 2π)! π=0
The set of {πΏπ (π₯); π = 0,1, β¦ } is a complete orthogonal system in πΏ2 (Ξ), and we have
(7)
1
(πΏπ , πΏπ ) = β« πΏπ (π₯) πΏπ (π₯) ππ₯ = βπ πΏππ ,
(8)
β1
2
where πΏππ is the Kronecker delta symbol and βπ = 2π+1. Thus, for any π β πΏ2 (Ξ), we obtain β
π(π₯) = β ππ πΏπ (π₯) , π=0
1 1 ππ = β« π(π₯) πΏπ (π₯) ππ₯ . βπ β1
(9)
0,0 For simplicity, we let π₯π = π₯π0,0 , ππ = ππ0,0 and ππ₯,π = πππ₯, . Thus, according to Eq.(4), we have π
1
β« π(π₯)ππ₯ = β π(π₯π )ππ , β1
βπ β π2πβ1 [β1,1],
(10)
π=0
where π₯π (0 β€ π β€ π), i.e. are the zero of (1 β π₯ 2 )πΏβ² π (π₯), and ππ =
2 π (π+1) (πΏπ (π₯π ))
2
ππ (0 β€ π β€ π), i.e.
, are denoted to the nodes and Christoffel numbers of Legendre-Gauss-
Lobatto interpolation on the classical interval (β1,1), repectively. The norm and discrete inner product in πΏ2 (Ξ) are defined as π
(π’, π)π = β π’(π₯π ) π(π₯π ) ππ ,
1
βπ’βπ = (π’, π’)2 .
(11)
π=0
In the next section, we use some special techniques for converting Eq. (1) to a problem can be solved by the standard Legendre spectral collocation scheme. These topics will be presented in the next section. 3. The Spectral Collocation Method In this section, we shall propose a Legendre spectral collocation method for solving the Eq. (1). The Legendre spectral-collocation method takes advantage of both the Legendre polynomials and Legendre-Gauss-Lobatto interpolation nodes. The main idea is to use the suggested method consists of discretizing the fractional integral equation to obtain a system of algebraic equations with unknown coefficients. As a result, the solution of the obtained system can be easily solved. Again, let us to consider Eq. (1) by applying definitions 2.2 as follow π(π₯) π₯ π¦(π₯) = β« π(π )(π₯ β π )πΌβ1 π¦(π ) ππ + π(π₯), Ξ(πΌ) 0
π₯ β πΌ = [0, π].
Next, in enjambment we let Ξ = [β1,1]. For ease of analysis, we transfer the problem (1) to an equivalent problem defined in Ξ. More specifically, we apply the change of variable π‘ =
2π₯ π
β1
π
or π₯ = 2 (π‘ + 1), such that π‘ β Ξ. Then, we have
π π¦ ( (π‘ + 1)) = 2
π π ( 2 (π‘ + 1)) Ξ(πΌ)
β«
π (π‘+1) 2
0
π +π ( (π‘ + 1)) , 2
πΌβ1 π π(π ) ( (π‘ + 1) β π ) π¦(π ) ππ 2
(12)
π‘ β Ξ.
π
Moreover, to transfer the integral interval (0, 2 (π‘ + 1)) to (β1, π‘), we make the transformation π
π = 2 (π + 1). Then Eq. (12) can be written
π π π¦ ( (π‘ + 1)) = ( ) 2 2
πΌ
π π (2 (π‘ + 1)) Ξ(πΌ)
π +π ( (π‘ + 1)) , 2
π‘ π π β« π ( (π + 1)) (π‘ β π)πΌβ1 π¦ ( (π + 1)) ππ 2 2 β1
(13)
π‘ β Ξ.
Further, if we let π π(π‘) = π¦ ( (π‘ + 1)) , 2 π π΅(π‘) = π ( (π‘ + 1)) , 2
π π΄(π‘) = π ( (π‘ + 1)), 2 π πΉ(π‘) = π ( (π‘ + 1)), 2
(14)
then, Eq. (13) can be reduced to π πΌ π΄(π‘) π‘ π(π‘) = ( ) β« π΅(π) (π‘ β π)πΌβ1 π(π) ππ + πΉ(π‘), π‘ β Ξ = [β1,1]. 2 Ξ(πΌ) β1
(15)
At last, under the following linear transformation π = π(π‘, π) = the Eq. (15) becomes
π‘+1 π‘β1 π+ , 2 2
π β Ξ,
(16)
π πΌ π‘ + 1 πΌ π΄(π‘) 1 π(π‘) = ( ) ( ) β« π΅(π(π‘, π)) (1 β π)πΌβ1 π(π(π‘, π)) ππ + πΉ(π‘). 2 2 Ξ(πΌ) β1
(17)
The Legendre spectral collocation method for Eq. (17) is to seek π β ππ (Ξ) (π β₯ 1), such that 1 ππΌ πΌβ1,0 πΌ π(π‘) = Ξ± ππ‘,π {(π‘ + 1) π΄(π‘) β« (1 β π)πΌβ1 ππ,π (π΅(π(π‘, π)) π(π(π‘, π))) ππ} 4 Ξ(πΌ) β1
(18)
+ππ‘,π (πΉ(π‘)).
Now, we want to describe a numerical implementation for Eq. (18). To this end, we use the following relations π
π(π‘) = β π’π πΏπ (π‘), π=0 π πΌβ1,0 ππ‘,π (ππ,π ((π‘
π
+ 1)πΌ π΄(π‘)π΅(π(π‘, π)) π(π(π‘, π)))) β β πππ πΏπ (π‘) π½ππΌβ1,0 (π) .
(19)
π=0 π=0
Then by using (19) and then Eq. (3), a direct computation leads to the following equation as 1 ππΌ πΌβ1,0 β« (1 β π)πΌβ1 ππ₯,π (ππ,π ((π‘ + 1)πΌ π΄(π‘)π΅(π(π‘, π)) π(π(π‘, π)))) ππ Ξ± 4 Ξ(πΌ) β1 π
π
1 ππΌ (π‘) = Ξ± β β πππ πΏπ β« (1 β π)πΌβ1 π½ππΌβ1,0 (π) ππ 4 Ξ(πΌ) β1 π=0 π=0 π
ππΌ = Ξ± β ππ0 πΏπ (π‘) . 2 Ξ(πΌ + 1)
(20)
π=0
Applying Eqs. (3) to (6), di0 can be readily obtained as π
π
π1
π2
πΌ (2π + 1) πΌ ππ0 = )) π (π(π₯π1 , πππΌβ1,0 )) πΏπ (π₯π1 )ππ1 πππΌβ1,0 . β β(π₯π1 + 1) π΄(π₯π1 ) π΅ (π(π₯π1 , πππΌβ1,0 πΌ+1 2 2 2 2
Hence, by inserting Eqs. (19) and (20) in Eq. (18), we conclude π
π
ππΌ β π’π πΏπ (π‘) = Ξ± β ππ0 πΏπ (π‘) . 2 Ξ(πΌ + 1) π=0
π=0
(21)
Finally, by comparing the expansion coefficients of eq. (21), we can get ππΌ π’π = Ξ± π , 2 Ξ(πΌ + 1) π0
(22)
0 β€ π β€ π.
By replacing the above relation, we obtain the expression of π(π‘) accordingly.
4. Convergence Analysis In this section, we want to carry out the error analysis for the numerical method (18) under L2 (Ξ) and Lβ (Ξ). Here, we present some preparations as follow [14,17]:
L2ππ1,π2 (Ξ): the Jacobi-weighted L2 Hilbert space with the scalar product 1
(u, v) = β« π’(t) π£(t) π(t)dt , βu, v β L2Οq1,q2 (Ξ). β1
Therefore, the corresponding norm is 1
βπ’βπΏ2 π
π 1 ,π2
π»πππ1,π2 (π¬) = {π’ β πΏ2ππ1,π2 (π¬) :
ππ π’ ππ₯ π
= (π’, π’)2 .
β πΏ2ππ1,π2 (π¬), π = 0,1, β¦ π}: the Jacobi-weighted
Sobolev spaces with the norm and semi norm respectively π
βπ’β2π» ππ ,π π 1 2
2
= ββππ‘π π’βπΏ2
ππ1 ,π2
π=0
,
and
|π’|π» ππ ,π = βππ‘π π’β π1+π,π2+π π 1 2 π
π»πππ1,π2 by its inner product is Hilbert space. For the purpose of convenience, we write 0
πΏ2 (Ξ) = π»π0,0 , β. β2 = β. βπΏ2(Ξ) and β. ββ = β. βπΏβ(Ξ) . m
Lemma 4.1 [17] Assume that π’ β Hππ1,π2 with π1 , π2 > β1, and π β₯ 1. Then for any integer number π such that 0 β€ π β€ π β€ π + 1, exists the constant πΆ > 0 that the following relation holds: π1 ,π2 (π’))β βππ‘π (π’ β ππ‘,π
ππ1 +π ,π2 +π
β€ πΆπ πβπ βππ‘π π’βππ1+π ,π2+π .
(23)
In particular, for any u β H m (Ξ) with 1 β€ m β€ N + 1, the above relation reduced to the following result 3
βπ’ β ππ‘,π (π’)βπ» 1 (Ξ) β€ πΆπ 2βπ βππ‘π π’β2 .
(24)
Theorem 4.2 For sufficiently large π, the Legendre spectral-collocation approximation converges
to exact solution in πΏ2 -norm, i.e. βππ β2 = βπ β πβ2 β 0. Proof. Assume that π(x) is obtained by using the Legendre spectral-collocation method Eq. (1). Then, according to definition of ππ‘,π in last section, we will have βππ β2 β€ βπ β ππ‘,π (π)β + βππ‘,π (π) β πβ . 2 2
(25)
By using Lemma 4.1, we can get for any integer 1 β€ π β€ π + 1,
βπ β ππ‘,π (π)β2 β€ πΆπ βπ βππ‘π πβππ ,π .
(26)
The rest of our proof is to show βππ‘,π (π) β πβ2 β 0. To this end, along with definition π in Eq. (17), for π β₯ 1 we have ππΌ
π‘
ππ‘,π π(π‘) = 2πΌ Ξ(πΌ) ππ‘,π (π΄(π‘) β«β1(π‘ β π)πΌ πΉ(π΅(π), π(π)) ππ) + ππ‘,π (πΉ),
(27)
where, πΉ(π΅(π), π(π)) = π΅(π(π‘, π))π(π(π‘, π)). Similarly, we can write π‘ ππΌ π(π‘) = Ξ± ππ‘,π { π΄(π‘) β« (π‘ β π)πΌβ1 ππΌβ1,0 (πΉ(π΅(π), π(π))) ππ} + ππ‘,π (πΉ(π‘)), π,π 2 Ξ(πΌ) β1
(28)
where πΌβ1,0 πΌβ1,0 ππ,π π£(π) = ππ,π π£(π(π‘, π))| 2π π‘β1 . π= β
(29)
π‘+1 π‘+1
Before we proceed to the rest of proof, we present some relations that will be used later. By virtue of above definition and then using standard Jacobi-Gauss quadrature formula (4), we obtain π‘ 1+π‘ πΌ 1 β« (π‘ β π)πΌβ1 ππΌβ1,0 π£(π)ππ = ( ) β« (1 β π)πΌβ1 ππΌβ1,0 π£(π(π‘, π))ππ π,π π,π 2 β1 β1 π
1+π‘ πΌ =( ) β π£ (π(π‘, πππΌβ1,0 )) πππΌβ1,0 2 π=0 π
1+π‘ πΌ =( ) β π£(πππΌβ1,0 )πππΌβ1,0 . 2 π=0
(30)
Again, in the same scheme we have π‘
β« (π‘ β π) β1
πΌβ1
2
(ππΌβ1,0 π£(π)) π,π
π
1+π‘ πΌ ππ = ( ) β π£ 2 (πππΌβ1,0 )πππΌβ1,0 . 2
(31)
π=0
At last, by applying Lemma 4.1 and Eq. (29) together, for any integer π (1 β€ π β€ π + 1), we conclude that 2
π‘
πΌβ1,0 β«β1(π‘ β π)πΌβ1 (π£(π) β ππ,π π£(π)) ππ
=(
2 1+π‘ πΌ 1 ( ))) ) β« (1 β π)πΌβ1 (π£(π(π‘, π)) β ππΌβ1,0 π£ π π‘, π ππ ( π,π 2 β1
2 1+π‘ πΌ 1 π β€ πΆπ βπ ( ) β« (ππ π£(π(π‘, π))) (1 β π)πΌ+πβ1 (1 + π)π ππ 2 β1 π₯
2
= πΆπ βπ β« (πππ π£(π)) (π₯ β π)πΌ+πβ1 (1 + π)π ππ .
(32)
β1
Now, let us explain the rest of proof. By subtracting Eq. (28) from (27), we derive that ππ‘,π π(π‘) β π(π‘) =
π‘ ππΌ πΌβ1,0 (π‘ β π)πΌβ1 (πΉ(π΅(π), π(π)) β ππ,π Γ π β« (πΉ(π΅(π), π(π)))) ππ). (π΄(π‘) π‘,π 2πΌ π€(πΌ) β1
The above formula can be rewritten as ππ‘,π π(π‘) β π(π‘) π‘ ππΌ πΌβ1,0 = πΌ Γ ππ‘,π (π΄(π‘) β« (π‘ β π)πΌβ1 (πΉ(π΅(π), π(π)) β ππ,π (πΉ(π΅(π), π(π)))) ππ) 2 π€(πΌ) β1
+
ππΌ 2πΌ π€(πΌ) π‘
πΌβ1,0 Γ ππ‘,π (π΄(π‘) β« (π‘ β π)πΌβ1 ππ,π (πΉ(π΅(π), π(π)) β πΉ(π΅(π), π(π))) ππ ).
(33)
β1
We let πΈ1 (π‘) =
π‘ ππΌ πΌβ1,0 (π‘ β π)πΌβ1 (πΉ(π΅(π), π(π)) β ππ,π Γ π β« (πΉ(π΅(π), π(π)))) ππ), (π΄(π‘) π‘,π 2πΌ π€(πΌ) β1
πΈ2 (π‘) =
π‘ ππΌ πΌβ1,0 Γ ππ‘,π (π΄(π‘) β« (π‘ β π)πΌβ1 ππ,π (πΉ(π΅(π), π(π)) β πΉ(π΅(π), π(π))) ππ). 2πΌ π€(πΌ) β1
Next, by using standard Legendre-Gauss quadrature formula we obtain βπΈ1 β2 π
πΌ
=
2πΌ
1 2 2
π₯π
π πΌβ1 πΌβ1,0 {β ππ π΄(π₯π ) (β« (π₯π β π) (πΉ(π΅(π), π(π)) β ππ,π | (πΉ(π΅(π), π(π)))) ππ) } . π₯π π€(πΌ) β1 π=0
By using the Cauchy-Schwartz inequality and Eq. (32), we get βπΈ1 β2 π
π₯π π₯π ππΌ πΌβ1 πΌβ1 β€ max π΄(π‘) πΌ ππ β« (π₯π β π) {β ππ β« (π₯π β π) (πΉ(π΅(π), π(π)) π‘βΞ 2 π€(πΌ) β1 β1 π=0
1 2
2 πΌβ1,0 β ππ,π | (πΉ(π΅(π), π(π)))) ππ} π₯π
= max π΄(π‘) π‘βΞ
ππΌ 2πΌ π€(πΌ + 1)
π
πΌ
π₯π
Γ {β ππ (π₯π + 1) β« (π₯π β π)
πΌβ1
β1
π=0
2
1 2
πΌβ1,0 (πΉ(π΅(π), π(π)) β ππ,π | (πΉ(π΅(π), π(π)))) ππ} π₯π
ππΌ β€ πΆ max π΄(π‘) πΌ π βπ π‘βΞ 2 π€(πΌ + 1) π
πΌ
π₯π
2
Γ {β ππ (π₯π + 1) β« (πππ πΉ(π΅(π), π(π))) (π₯π β π)
1 2 πΌ+πβ1
(1 + π)π ππ}
β1
π=0
= πΆ1 π βπ βπππ πΉ(π΅(. ), π(. ))βππΌ+πβ1,π .
(34)
Similarly, by applying Eqs. (4) and (31) and the Cauchy-Schwartz inequality, we can get βπΈ2 β2 πΌ
=
2πΌ
π
1 2 2
π₯π
π πΌβ1 πΌβ1,0 ππ,π | (πΉ(π΅(π), π(π)) β πΉ(π΅(π), π(π))) ππ] } {β ππ [ π΄(π₯π ) β« (π₯π β π) π₯π π€(πΌ) β1
β€ max π΄(π‘) π‘βΞ
π=0
ππΌ πΌ
2 π€(πΌ)
π₯π
π
{β ππ β« (π₯π β π) π=0
Γ β« (π₯π β π) β1
π₯π
πΌβ1
ππ
β1
2 πΌβ1
1 2
(ππΌβ1,0 | (πΉ(π΅(π), π(π)) β πΉ(π΅(π), π(π)))) ππ} π,π π₯π
= max π΄(π‘) π‘βΞ
π
ππΌ 2πΌ π€(πΌ)
{β ππ
( π₯ π + 1) πΌ
π=0
πΌβ1,0
β πΉ (π΅(ππ
πΌ
π
β πππΌβ1,0 (πΉ (π΅(πππΌβ1,0 ), π(πππΌβ1,0 )) π=0
2
1 2
), π(πππΌβ1,0 ))) } .
(35)
On the other hand, Jenson's inequality for any convex function π on interval (π, π) is π(ππ‘1 + (1 β π)π‘2 ) β€ ππ(π‘1 ) + (1 β π)π(π‘2 ),
βπ β [0,1].
βπ‘1 , π‘2 β (π, π),
Thus, since π2 π‘ π‘ (π₯π + 1) = (π₯π + 1) ln2 (π₯π + 1) > 0, 2 ππ‘
π₯π is constant,
π‘
then, for any given π₯π β (β1,1), π(π‘) = (π₯π + 1) is convex function and according to Jenson's inequality on interval [0,1], we will have π(π‘) = π((1 β π‘) Γ 0 + π‘ Γ 1) β€ (1 β π‘)π(0) + π‘π(1).
(36)
Therefore, by applying the previous inequality, we yield the following inequality π
π
πΌ
β ππ (π₯π + 1) β€ β ππ {(1 β πΌ) + πΌ(π₯π + 1)} π=0
π=0 π
π
= (1 β πΌ) β ππ + πΌ β ππ (π₯π + 1) π=0
π=0
= (1 β πΌ) β« ππ‘ + πΌ β« (π₯ + 1)ππ‘ Ξ
= 2,
Ξ
βπΌ β [0,1].
(37)
Next, by using relation (37), Eq. (35) will be reduced to βπΈ2 β2 β€ max π΄(π‘) π‘βΞ
ππΌ 2πΌβ1 π€(πΌ + 1)
π₯π
Γ max { β« (π₯π β π) 0β€πβ€π
πΌβ1
β1
β€ max π΄(π‘) max π΅(π‘) π‘βΞ
π‘βΞ
2
1 2
πΌβ1,0 (ππ,π | (πΉ(π΅(π), π(π)) β πΉ(π΅(π), π(π)))) ππ} π₯π
ππΌ 2πΌβ1 π€(πΌ + 1)
π₯π
Γ max { β« (π₯π β π) 0β€πβ€π
πΌβ1
πΌβ1,0 (ππ,π | (π(π) β π(π))) ππ} π₯π
β1
π‘βΞ
ππΌ 2πΌβ1 π€(πΌ + 1)
π₯π
πΌβ1
β€ max π΄(π‘) max π΅(π‘) π‘βΞ
Γ max {(β« (π₯π β π) 0β€πβ€π
1 2
2
2
1 2
πΌβ1,0 (ππ,π | (π(π)) β π(π)) ππ) π₯π
β1
π₯π
πΌβ1
+ (β« (π₯π β π)
2
1 2
(38)
(π(π) β π(π)) ππ) }.
β1
At last, along with Lemma 4.1 and relations (32), (37), the previous result yields βπΈ2 β2 π₯π
2
β€ πΆ2 {πΆπ βπ max (β« (πππ π(π)) (π₯π β π) 0β€πβ€π
πΌ+πβ1
(1 + π)π ππ)
1 2
β1
1
1 2 2 2 + (β« (π(π) β π(π)) ππ) } πΌ β1
β€ πΆ2 πΆπ βπ βπππ πβππΌ+πβ1,π +
2πΆ2 βππ β. πΌ
(39)
Finally, relations (33) β (34), (39) prove that the approximation is convergent in πΏ2 βnorm. Hence the theorem is proved. 5. Numerical example To show efο¬ciency of our numerical method, the following examples which have exact solution are considered. Example 5.1. Consider the following fractional integral equation [20] 0.01
5
π‘
1
3
π‘3
π¦(π‘) = Ξ(0.5) π‘ 2 β«0 (π‘ β π )β2 π¦(π ) ππ + βπ(1 + π‘)β2 β 0.02 1+π‘ ,
π‘ β [0,1].
3
The corresponding exact solution is given by π¦(π‘) = βπ (1 + π‘)β2 . Figure 1 present the approximate and exact solutions which are found in very good agreement. In Figure 2, the numerical errors are plotted in πΏ2 βnorm for 2 β€ π β€ 24. As expected, an exponential rate of convergence is observed which confirmed our theoretical predictions in comparison of results in [20].
Figure 1. Comparison between exact solution and approximate solution of Example 5.1 for N = 10
Figure 2. The error function of Example 5.1 versus the number of interpolation operator Example 5.2. Our second example is the following fractional integral equation [20] π¦(π‘) =
1
π‘
1 2 1 8 β« π (π‘ β π )β3 π¦(π ) ππ + Ξ ( ) π‘ β π‘ 3, 2 0 3 40 27 Ξ ( ) 3
π‘ β [0,1].
2
The exact solution is π¦(π₯) = Ξ (3) π‘. We get the numerical solution of the above fractional integral equation as 2 π¦1 (π‘) = Ξ ( ) π‘, 3 2 π¦2 (π‘) = 1.21 Γ 10β2 + Ξ ( ) π‘, 3 2 π¦4 (π‘) = 2.15 Γ 10β9 + Ξ ( ) π‘ + 3.12 Γ 10β10 π‘ 2 + 2.39 Γ 10β7 π‘ 3 + 4.81 Γ 10β9 . 3 The numerical results of proposed method can be seen from Figure 3 and Figure 4. These results indicate that the spectral accuracy is obtained for this problem, although the given function π(π‘) is not very smooth.
Figure 3. Comparison between exact solution and approximate solution of Example 5.2
for π = 6
Figure 4. The error function of Example 5.1 versus the number of interpolation operator
These results indicate that the spectral accuracy is obtained by using Legendre-Collocation method for this problem. Example 5.3. Consider the following fractional integral equation [20] π‘ 1 1 π‘ π¦(π‘) = β β« (π‘ β π )β2 π¦(π ) ππ + 2β , Ξ(0.5) 0 π
π‘ β [0,1].
The exact solution is π¦(π‘) = 1 β et ππππ(βπ‘), such that ππππ is the complementary error function defined by ππππ(π‘) = 1 β πππ(π‘) =
2 βπ
β
2
β« π βπ ππ . π‘
We list the numerical results in Table 1 to be able to test the convergence rate of the suggested. In Figure 5, we can observe that our numerical solutions coincide closely with the exact ones. At a glance, we can find out the results of Legendre spectral-collocation method are satisfactory in the few steps.
t
Proposed method at π = 8 .0000000000 .2764215215 .3562117262 .4079815809 .4463937357 .4768434165 .5019754248 .5232972359 .5417539715 .5579785802 .5724164231
Proposed method at π = 10 .0000000000 .2764215613 .3562117272 .4079815886 .4463937452 .4768434157 .5019754315 .5232972687 .5417539771 .5579785884 .5724164243
Exact solution
0 .0000000000 0.1 .2764215616 0.2 .3562117278 0.3 .4079815886 0.4 .4463937458 0.5 .4768434159 0.6 .5019754315 0.7 .5232972687 0.8 .5417539772 0.9 .5579785885 1 .5724164241 Table 1. Comparison of approximate solutions and exact solutions for various numbers N
Figure 2. The error function of Example 5.1 versus the number of interpolation operator for π = 16
6. Conclusion In this paper we present a spectral Legendre-collocation approximation of a class of fractional integral equations of the second kind. The most important contribution of this paper is that the errors of approximations decay exponentially in πΏ2 β ππππ. We prove that our proposed method
is effective and has high convergence rate and better than some new methods which be define in this paper. During three numerical applications, we ensure that the proposed method is efficient and powerful numerical scheme for solving many fractional problems.
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