Hindawi Publishing Corporation Journal of Computational Methods in Physics Volume 2014, Article ID 381074, 10 pages http://dx.doi.org/10.1155/2014/381074
Research Article Solving Fractional Diffusion Equation via the Collocation Method Based on Fractional Legendre Functions Muhammed Syam and Mohammed Al-Refai Department of Mathematical Sciences, United Arab Emirates University, Al Ain, UAE Correspondence should be addressed to Muhammed Syam;
[email protected] Received 10 April 2014; Revised 24 June 2014; Accepted 1 July 2014; Published 24 July 2014 Academic Editor: Mikhail Tokar Copyright Β© 2014 M. Syam and M. Al-Refai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented.
1. Introduction We consider the generalized time-fractional diffusion equation of the form π·π‘πΌ π’ (π₯, π‘) = π (π₯, π‘) π·π₯2 π’ (π₯, π‘) + π (π₯, π‘) , π₯ β (β1, 1) , π‘ β (0, π) ,
(1)
with initial and boundary conditions π’ (β1, π‘) = β1 (π‘) ,
π’ (1, π‘) = β2 (π‘) ,
π’ (π₯, 0) = π (π₯) , (2)
where π, π β πΆ1 ([β1, 1] Γ [0, π]), π β πΆ[β1, 1], β1 , β2 β πΆ[0, π], π > 0, and 0 < πΌ β€ 1. For πΌ = 1, the fractional diffusion equation is reduced to a conventional diffusion-reaction equation which is well studied, so we focus on 0 < πΌ < 1. Some existence and uniqueness results of Problem (1)-(2) were established in [1]. In recent years, great interests were devoted to the analytical and numerical treatments of fractional differential equations (FDEs). Usually, FDEs appear as generalizations to existing models with integer derivative and they also present new models for some physical problems [2, 3]. In general, FDEs do not possess exact solutions in closed forms, and, therefore, numerical methods such as the variational iteration (VIM) [4, 5], the homotopy analysis method (HAM) [6, 7], and the Adomian decomposition method (ADM) [8, 9] have
been implemented for several types of FDEs. Also, the maximum principle and the method of lower and upper solutions have been extended to deal with FDEs and obtain analytical and numerical results [10, 11]. The Tau method, the pseudospectral method, and the wavelet method based on the Legendre polynomials have been implemented for several types of FDEs [12β14]. Kazem et al. [12] have constructed the Legendre functions of fractional order and discussed some of their properties. The resulting Legendre function operational and product matrices, together with the Tau method, have been implemented to solve linear and nonlinear fractional differential equations. The effectiveness of the approach has been examined through several examples. In [13], a fractional diffusion equation is considered, where the fractional derivative of order 1 < πΌ β€ 2 refers to the spatial variable π₯. The Legendre pseudospectral method is implemented to solve the problem, where the solution is expanded with regular Legendre polynomials. As a result, a system of linear equation has been obtained and integrated using the finite difference method. However, in solving fractional differential equations of order πΌ using series expansions, it is common and more efficient to expand the solution with fractional functions of ππΌ the form βπ π=0 ππ π₯ . Rawashdeh [14] has implemented the Legendre wavelets method for integrodifferential equations with fractional order. The Legendre collocation method has been implemented for wide classes of differential equations and the effectiveness
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of the method is illustrated [15]. In the recent work, we intend to apply the collocation method based on the shifted fractional Legendre functions to integrate the Problem (1)-(2). To the best of our knowledge, the method has not been developed to integrate fractional diffusion equations of the form (1)-(2). We organize this paper as follows. In Section 2, we present basic definitions and results of fractional derivative. In Section 3, we present the numerical technique for solving Problem (1)-(2). In Section 4, we present some numerical results to illustrate the efficiency of the presented method. Finally we conclude with some comments in Section 5.
In this section, we present the definition and some preliminary results of the Caputo fractional derivative, as well as the definition of the fractional-order Legendre functions and their properties. Definition 1. A real function π(π‘), π‘ > 0, is said to be in the space πΆπ , π β R, if there exists a real number π > π, such that π(π‘) = π‘π π1 (π‘), where π1 (π‘) β πΆ[0, β), and it is said to be in the space πΆππ if π(π) β πΆπ , π β N. Definition 2. The left Riemann-Liouville fractional integral of order πΏ β₯ 0, of a function π β πΆπ , π β₯ β1, is defined by π‘ { 1 β« (π‘ β π )πΏβ1 π (π ) ππ , πΏ > 0, πΌ π (π‘) = { Ξ (πΏ) 0 πΏ = 0. {π (π‘) ,
σΈ
((1 β π₯2 ) πΏσΈ π (π₯)) + π (π + 1) πΏ π (π₯) = 0,
Among the properties of the Legendre polynomials we list the following [19]: 1
0, π=πΜΈ
(1) β«β1 πΏ π (π₯)πΏ π (π₯)ππ₯ = (2/(2π+1))πΏππ , where πΏππ = {1, π=π},
(3)
(3) πΏ π (Β±1) = (Β±1)π . In order to use these polynomials on the interval [0, 1], we define the shifted Legendre polynomials by ππ (π₯) = πΏ π (2π₯β1). Using the change of variable π§ = 2π₯β1, ππ (π₯) has the following properties: 1
(1) β«0 ππ (π₯)ππ (π₯)ππ₯ = (1/(2π + 1))πΏππ , (2) ππ+1 (π₯) = ((2π + 1)/(π + 1))(2π₯ β 1)ππ (π₯) β (π/(π + 1))ππβ1 (π₯), for π β©Ύ 1, (3) ππ (0) = (β1)π and ππ (1) = 1. The analytic closed form of the shifted Legendre polynomials of degree π is given by π
ππ (π₯) = β (β1)π+π
Definition 3. For πΏ > 0, π β 1 < πΏ β€ π, π β N, π‘ > 0, and π , the left Caputo fractional derivative is defined by π β πΆβ1 π·πΏ π (π‘) π‘ 1 { β« (π‘ β π )πβ1βπΏ π(π) (π ) ππ , π β 1 < πΏ < π, { = { Ξ (π β πΏ) 0 { (π) πΏ = π, {π (π‘) , (4)
where Ξ is the well-known Gamma function. The Caputo derivative defined in (4) is related to the Riemann-Liouville fractional integral, πΌπΏ , of order πΏ β R+ , by π·πΏ π(π‘) = πΌπβπΏ π(π) (π‘). The Caputo fractional derivative satisfies the following properties for π β πΆπ , π β₯ β1, and πΌ β₯ 0 (see [16]): πΌ πΌ
(1) π· πΌ π(π‘) = π(π‘), (π) π (2) πΌπΌ π·πΌ π(π‘) = π(π‘) β βπβ1 π=0 π (0)(π‘ /π!), (3) π·πΌ π = 0, where π is constant,
(4) (5)
π₯ β [β1, 1] . (5)
(2) πΏ π+1 (π₯) = ((2π + 1)/(π + 1))π₯πΏ π (π₯) β (π/(π + 1))πΏ πβ1 (π₯), for π β©Ύ 1,
2. Preliminaries
πΏ
Definition 4. The Legendre polynomials {πΏ π (π₯) : π = 0, 1, 2, . . .} are the eigenfunctions of the Sturm-Liouville problem:
0, πΎ 0 is using series expansion of the form βππ=0 ππ π‘πΌπ . For this reason, we define the fractional-order Legendre function by πΉππΌ (π‘) = ππ (π‘πΌ ). Using the properties of the shifted Legendre polynomials, it is easy to verify that [20] (1) ((π‘ β π‘1+πΌ )πΉππΌσΈ (π‘))σΈ + π(π + 1)πΌ2 π‘πΌβ1 πΉππΌ (π‘) = 0, π‘ β (0, 1),
πΌ (2) πΉππΌ (π‘) = ((2π+1)/(π+1))(2π‘πΌ β1)πΉππΌ (π‘)β(π/(π+1))πΉπβ1 (π‘), for π β©Ύ 1,
(3) πΉ0πΌ (π‘) = 1 and πΉ1πΌ (π‘) = 2π‘πΌ β 1, (4) πΉππΌ (0) = (β1)π and πΉππΌ (1) = 1. In addition, {πΉππΌ (π‘) : π = 0, 1, 2, . . .} are orthogonal functions with respect to the weight function π€(π‘) = π‘πΌβ1 on (0, 1) with 1
β« πΉππΌ (π‘) πΉππΌ (π‘) π€ (π‘) ππ‘ = 0
1 πΏ . + (2π 1) πΌ ππ
(7)
The closed form of πΉππΌ (π‘) is given by π
πΉππΌ (π‘) = β (β1)π+π π=0
(π + π)! π‘πΌπ . (π β π)! (π!)2
(8)
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Using properties (4) and (5) of the Caputo fractional derivative, we have π
π·πΌ πΉππΌ (π‘) = β (β1)π+π π=1
Ξ (ππΌ + 1) (π + π)! π‘(πβ1)πΌ . (π β π)!(π!)2 Ξ ((π β 1) πΌ + 1) (9)
Table 1: Error for Example 1. πΌ 0.5 0.9 0.99 0.9999
The following result is important, since it facilitates applying the collection method. Theorem 5. Let π’ β πΆ1 [0, 1] and let π’σΈ σΈ (π‘) be a piecewise continuous function on [0, 1]. Then, (i) π’(π‘) can be represented by infinite series expansion as πΌ π’(π‘) = ββ π=0 π’π πΉπ (π‘), where 1
π’π = (2π + 1) πΌ β« π’ (π‘) πΉππΌ (π‘) π€ (π‘) ππ‘. 0
(10)
Table 2: Error for Example 2. πΌ 0.5 0.9 0.99 0.9999
1
Proof. (i) Since π’ β πΆ1 [β1, 1] and π’σΈ σΈ (π₯) is a piecewise continuous function on [β1, 1], ββ π=0 Vπ Lπ (π₯) converges uniformly to π’(π₯) on [β1, 1], where {Vπ } can be computed by the orthogonality relation of the Legendre polynomials; see [11]. Since π : [0, 1] β [β1, 1], defined by π(π‘) = 2π‘πΌ β 1, is a πΌ bijection continuous function, the infinite series ββ π=0 π’π πΉπ (π‘) converges uniformly to π’(π‘) on [0, 1], where the value of π’π follows from the orthogonality relation of {πΉππΌ (π‘) : π = 0, 1, 2, . . .} with respect to the weight function π€(π‘) = π‘πΌβ1 on [0, 1], which completes the proof. (ii) Let ππ (π‘) = βππ=0 π’π πΉππΌ (π‘) for π = 0, 1, 2, . . .. From Part (i), ππ (π‘) converges uniformly to π’(π‘) on [0, 1]. Since π’ β πΆ1 [0, 1] and π’σΈ σΈ (π‘) is a piecewise continuous function on [0, 1], π π ( Lim π (π‘)) = Lim ( ππ (π‘)) , π β β ππ‘ ππ‘ π β β π
Er 7.2 β 10β10 3.3 β 10β11 4.1 β 10β12 7.2 β 10β13
Expand the solution π’(π₯, π‘) in terms of the fractional-order Legendre function as follows: π+2
(πΌ) πΌ πΌ (ii) ββ π=0 π’π πΉπ (π‘) converges uniformly on [0, 1] to π· π’(π‘), β (πΌ) = βπ=π+1 πππ π’π and πππ = (2π + where π’π
1)πΌ β«0 π·πΌ πΉππΌ (π‘)πΉππΌ (π‘)π€(π‘)ππ‘, for π = 0, 1, 2, . . . and π = π + 1, π + 2, . . . .
Er 4.4 β 10β9 8.9 β 10β10 6.1 β 10β10 5.2 β 10β12
ππ (π₯, π‘) = β π’π (π‘) Lπ (π₯) . Thus, π
π·π₯2 ππ (π₯, π‘) = β π’π(2) (π‘) Lπ (π₯) ,
and (π/ππ‘)ππ (π‘) converges uniformly to (π/ππ‘)π’(π‘) on π₯ [0, 1]. Thus, β«0 (ππσΈ (π‘)/(π₯ β π‘)πΌ )ππ‘ converges uniformly to π₯
β«0 (π’σΈ (π‘)/(π₯ β π‘)πΌ )ππ‘ on [0, 1] which gives the result of the second part. The value of πππ follows from the orthogonality relation of {πΉππΌ (π‘) : π = 0, 1, 2, . . .} with respect to the weight function π€(π‘) = π‘πΌβ1 on [0, 1].
(13)
π=0
where π’π(2) = (π+(1/2)) βπ=π+2,π+π even [π(π+1)βπ(π+1)]π’π for π = 0, 1, . . . , π. Therefore, for ππ(π₯, π‘), the residual is given by π
(ππ (π₯, π‘)) β π·π‘πΌ ππ (π₯, π‘) β π (π₯, π‘) π·π₯2 ππ (π₯, π‘) β π (π₯, π‘) . (14) Orthogonalize the residual with respect to the Dirac delta function as follows: β¨π
(ππ (π₯, π‘)) , πΏ (π₯ β π₯π )β© (15)
1
= β« π
(ππ (π₯, π‘)) πΏ (π₯ β π₯π ) ππ₯ = 0, 0
(11)
(12)
π=0
for π = 0 : π,
where π₯π are the collocation points. We choose the collocation points to be the roots of LσΈ π+2 . Therefore, (15) leads to the elementwise equation: π·π‘πΌ ππ (π₯π , π‘) β π (π₯π , π‘) π·π₯σΈ σΈ ππ (π₯π , π‘) + π (π₯π , π‘) = 0, (16) for π = 0 : π, or π+2
β [π·π‘πΌ π’π (π‘)] Lπ (π₯π )
π=0
3. Collocation Method In this section, we use the fractional-order Legendre collocation method to discretize Problem (1)-(2). For simplicity, we assume that π(π₯, π‘) = π(π₯) and π = 1. If π =ΜΈ 1, we use the change of variable π = π‘/π to make the π‘-domain (0,1).
π
β π (π₯π , π‘) β π’π(2) (π‘) Lπ (π₯π ) β π (π₯π , π‘) = 0, π=0
for π = 0 : π.
(17)
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π₯ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Error in FDM 1.7 β 10β6 3.1 β 10β6 4.0 β 10β6 4.6 β 10β6 4.8 β 10β6 4.6 β 10β6 4.0 β 10β6 3.1 β 10β6 1.7 β 10β6
πΌ = 0.98 Er 3.1 β 10β11 4.2 β 10β11 4.7 β 10β11 4.9 β 10β11 5.2 β 10β11 3.9 β 10β11 3.2 β 10β11 2.8 β 10β11 2.1 β 10β11
Error in FDM 1.4 β 10β6 2.5 β 10β6 3.3 β 10β6 3.8 β 10β6 4.0 β 10β6 3.8 β 10β6 3.3 β 10β6 2.5 β 10β6 1.4 β 10β6
Er 2.9 β 10β11 2.9 β 10β11 3.1 β 10β11 3.3 β 10β11 3.6 β 10β11 3.3 β 10β11 3.1 β 10β11 2.2 β 10β11 1.9 β 10β11
Table 4: Results of the method in [18] and the proposed method. πΌ
Number of mesh points in [18] 10 20 40 80 10 20 40 80 10 20 40 80
0.25
0.5
0.75
πI 3.5 β 10β4 1.1 β 10β4 3.5 β 10β5 1.1 β 10β5 1.3 β 10β3 4.8 β 10β4 1.7 β 10β4 6.1 β 10β5 3.8 β 10β3 1.6 β 10β3 6.9 β 10β4 2.9 β 10β4
π’0(2) (π‘)
[ (2) ] [π’1 (π‘)] ] π(2) (π‘) = [ [ .. ] , [ . ] (2) [π’π (π‘)]
(18)
π (π₯0 , π‘) [ π (π₯ , π‘) ] 1 ] [ πΉ (π‘) = [ ]. .. ] [ . , π‘) π (π₯ ] [ π Thus, we can rewrite (17) in the matrix form as π΄ 1 π·π‘πΌ π (π‘) β π΄π΄ 2 π(2) (π‘) β πΉ (π‘) = 0,
πIII 1.9 β 10β4 5.0 β 10β5 1.3 β 10β5 3.5 β 10β6 6.1 β 10β4 1.9 β 10β4 5.8 β 10β5 1.9 β 10β5 1.9 β 10β3 7.4 β 10β4 2.9 β 10β4 1.2 β 10β4
0 π (π₯0 ) [ 0 π (π₯ 1) [ =[ . . .. [ .. 0 [ 0
Let π’0 (π‘) [ π’ (π‘) ] [ 1 ] π (π‘) = [ . ] , [ .. ] [π’π+2 (π‘)]
πII 2.4 β 10β4 7.2 β 10β5 2.2 β 10β5 6.5 β 10β6 8.9 β 10β4 3.1 β 10β4 1.1 β 10β4 3.8 β 10β5 2.7 β 10β3 1.1 β 10β3 4.7 β 10β4 2.0 β 10β4
(19)
L0 (π₯0 ) L1 (π₯0 ) [ [ L0 (π₯1 ) L1 (π₯1 ) [ π΄2 = [ .. .. [ [ . . [L0 (π₯π) L1 (π₯π)
β
β
β
β
β
β
Er 2.3 β 10β11
4.1 β 10β12
6.7 β 10β11
0 0 .. .
] ] ], ] d β
β
β
π (π₯π)]
β
β
β
Lπ (π₯0 )
] β
β
β
Lπ (π₯1 ) ] ] ]. .. ] ] d . β
β
β
Lπ (π₯π)]
Since π’π(2) = (π + (1/2)) βπ=π+2,π+π even [π(π + 1) β π(π + 1)]π’π for π = 0, 1, . . . , π, it is easy to see that π(2) (π‘) = π΄ 3 π(π‘), where π΄ 3 is (π+1)Γ(π+3) matrix. Therefore, System (19) becomes π΄ 1 π·π‘πΌ π (π‘) β π΄π΄ 2 π΄ 3 π (π‘) β πΉ (π‘) = 0.
where L0 (π₯0 ) L1 (π₯0 ) [ [ L0 (π₯1 ) L1 (π₯1 ) [ π΄1 = [ .. .. [ [ . . [L0 (π₯π) L1 (π₯π)
β
β
β
Lπ+2 (π₯0 )
] β
β
β
Lπ+2 (π₯1 ) ] ] ], .. ] ] d . β
β
β
Lπ+2 (π₯π)]
(20)
(21)
Now, we study the boundary conditions on the variable π₯. From (12), one can see that π+2
π+1
π=0
π=0
ππ (Β±1, π‘) = β π’π (π‘) Lπ (Β±1) = β (Β±1)π π’π (π‘) ,
(22)
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which implies that
Thus, π΄ 4 π (π‘) = [
π
β1 (π‘)
],
π·π‘πΌ ππ (π₯, π‘) = β ππ(πΌ) (π‘) πΉππΌ (π‘) ,
(23)
β2 (π‘)
π+1
1 β1 1 β
β
β
(β1) ]2Γ(π+2) . From Systems (21) where π΄ 4 = [ β1 1 1 1 1 β
β
β
1 and (23), we obtain the following fractional system:
π΅1 π·π‘πΌ π (π‘) β π΅2 π (π‘) = π
(π‘) ,
(24)
where ππ(πΌ) 1
=
βπ+1 π=π+1 πππ ππ , and πππ
=
for π = 0, 1, 2, . . . , π and π = π + 1, π + 2, . . . , π + 1. Therefore, for ππ(π‘), the residual is given by π
(ππ (π‘)) β π΅1 π·π‘πΌ ππ (π‘) β π΅2 ππ (π‘) β π
(π‘) .
π΅1 = [
π΄1 ], 02Γ(π+2)
π΅2 = [
π΄π΄ 2 π΄ 3 ], π΄4
1
β¨π
(ππ (π‘)) , πΏ (π‘ β π‘π )β© = β« π
(ππ (π‘)) πΏ (π‘ β π‘π ) ππ‘ = 0, 0
for π = 0 : π, (34)
From (2) and (12), we see that π+2
π (π₯) = ππ (π₯, 0) = β π’π (0) Lπ (π₯) .
(26)
where π‘π are the collocation points. We choose the collocation points to be
π=0
Using the orthogonality property of the Legendre polynomials, we get 1
β«β1 π (π₯) Lπ (π₯) ππ₯ 1
β«β1 L2π (π₯) ππ₯
=
(33)
Orthogonalize the residual with respect to the Dirac delta function as follows: (25)
πΉ (π‘) [ ] π
(π‘) = [β1 (π‘)] . [β2 (π‘)]
(2π +
1)πΌ β«0 π·πΌ πΉππΌ (π‘)πΉππΌ (π‘)π€(π‘)ππ‘,
where
π’π (0) =
(32)
π=0
2π + 1 1 β« π (π₯) Lπ (π₯) ππ₯ 2 β1 for π = 0 : π + 2. (27)
Therefore, our initial fractional value problem becomes π΅1 π·π‘πΌ π (π‘) β π΅2 π (π‘) = π
(π‘) ,
(28)
π (0) = π΅,
(29)
{(
π‘0 + 1 βπΌ π‘1 + 1 βπΌ π‘ + 1 βπΌ ) ,( ) ,...,( π ) }, 2 2 2
(35)
where π‘0 , π‘1 , . . . , π‘π are the roots of LσΈ π+1 (π‘). Therefore, (34) leads to the elementwise equation: π΅1 π·π‘πΌ ππ (π‘π ) β π΅2 ππ (π‘π ) β π
(π‘π ) = 0,
for π = 0 : π (36)
or π
π+1
π=0
π=0
π΅1 β ππ(πΌ) (π‘) πΉππΌ (π‘) β π΅2 β ππ πΉππΌ (π‘) β π
(π‘π ) = 0,
(37)
for π = 0 : π. Let
where 1 1 β« π (π₯) L0 (π₯) ππ₯ 2 β1 3 1 β« π (π₯) L1 (π₯) ππ₯ 2 β1
[ [ [ [ [ [ π΅=[ [ [ [ [ [ 2π + 5 [
2
1
.. .
β« π (π₯) Lπ+2 (π₯) ππ₯ β1
] ] ] ] ] ] ]. ] ] ] ] ]
π0
[ [ π1 π=[ [ .. [ . [ππβΌ +1
(30)
π=0
]
[ ] [π(πΌ) ] [ 1 ] ] =[ [ . ], [ . ] [ . ] (πΌ) [ππ ]
(38)
Thus, we can rewrite (37) in the matrix form as
π+1
ππ (π‘) = β ππ πΉππΌ (π‘) .
π
(πΌ)
π
(π‘0 ) ] [ [ π
(π‘1 ) ] [ Ξ=[ . ] ]. [ .. ] [π
(π‘π)]
]
To solve the System (28)-(29), we use the fractional-order Legendre collocation method. Approximate the solution π(π‘) in terms of the fractional-order Legendre function as follows:
] ] ], ] ]
π0(πΌ)
πΆ3 πΆ1 π(πΌ) β πΆ4 πΆ2 π β Ξ = 0,
(31) where
(39)
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π΅1 0 β
β
β
[ 0 π΅1 β
β
β
[ πΆ1 = [ .. .. [. . d [ 0 0 β
β
β
π΅2 0 β
β
β
[ 0 π΅2 β
β
β
[ πΆ2 = [ .. .. [. . d [ 0 0 β
β
β
πΉ0πΌ (π‘0 )πΌπ+3 πΉ1πΌ (π‘0 )πΌπ+3 [ πΌ [ πΉ0 (π‘1 )πΌπ+3 πΉ1πΌ (π‘1 )πΌπ+3 [ πΆ3 = [ .. .. [ [ . . πΌ πΌ (π‘ )πΌ πΉ (π‘ )πΌπ+3 πΉ π π+3 π 0 1 [ πΉ0πΌ (π‘0 )πΌπ+3
0 0] ] .. ] .]
,
π΅1 ](π+3)(π+1)Γ(π+3)(π+1) 0 0] ] .. ] .]
,
π΅2 ](π+3)(π+2)Γ(π+3)(π+2) πΌ β
β
β
πΉπ (π‘0 )πΌπ+3
(40)
] πΌ β
β
β
πΉπ (π‘1 )πΌπ+3 ] ] , ] .. ] ] d . πΌ β
β
β
πΉπ (π‘π)πΌπ+3 ](π+3)(π+1)Γ(π+3)(π+1)
πΌ πΉ1πΌ (π‘0 )πΌπ+3 β
β
β
πΉπ+1 (π‘0 )πΌπ+3
] [ πΌ πΌ [ πΉ0 (π‘1 )πΌπ+3 πΉ1πΌ (π‘1 )πΌπ+3 β
β
β
πΉπ+1 (π‘1 )πΌπ+3 ] ] πΆ4 = [ . ] [ .. .. .. ] [ . . d . πΌ πΌ πΌ [πΉ0 (π‘π)πΌπ+3 πΉ1 (π‘π)πΌπ+3 β
β
β
πΉπ+1 (π‘π)πΌπ+3 ](π+3)(π+1)Γ(π+3)(π+2) (πΌ) Since ππ(πΌ) = βπ+1 = π=π+1 πππ ππ , it is easy to see that π πΆ5 π, where
0πΌπ+3 π10 πΌπ+3 π20 πΌπ+3 π30 πΌπ+3 [0πΌ [ π+3 0πΌπ+3 π21 πΌπ+3 π31 πΌπ+3 [ π΄3 = [ [0πΌπ+3 0πΌπ+3 0πΌπ+3 π32 πΌπ+3 [ .. .. .. .. [ . . . . [0πΌπ+3 0πΌπ+3 0πΌπ+3 0πΌπ+3 1
and πππ = (2π + 1)πΌ β«0 π·πΌ πΉππΌ (π‘)πΉππΌ (π‘)π€(π‘)ππ‘ for π = 0, 1, 2, . . . , π and π = π + 1 : π + 1. Therefore, System (19) becomes πΆ3 πΆ1 πΆ5 π β πΆ4 πΆ2 π β Ξ = 0.
(42)
Now, we study the initial condition on the variable π₯. From (29), one can see that π+1
π΅ = π (0) = β
π=0
π½ ππ πΉπ
π+1
π
(0) = β (β1) ππ ,
(43)
π=0
which implies that πΆ6 π = π΅,
(41)
β
β
β
ππππΌπ+3 π(π+1)ππΌπ+3 ]
where πΆ3 πΆ1 πΆ5 β πΆ4 πΆ2 ], Ξ©=[ πΆ6
Ξ Ξ¨ = [ ]. π΅
(46)
Finally, we use Mathematica to solve the above linear system.
4. Numerical Results In this section, we implement the proposed numerical technique for four examples. Example 1. Consider the fractional diffusion equation:
(44)
where πΆ6 = [βπΌπ+3 πΌπ+3 β πΌπ+3 πΌπ+3 β
β
β
(β1)π+1 πΌπ+3 ](π+3)Γ(π+3)(π+2) . From Systems (21) and (23), we obtain the following linear system: Ξ©π = Ξ¨,
β
β
β
ππ0 πΌπ+3 π(π+1)0 πΌπ+3 β
β
β
ππ1 πΌπ+3 π(π+1)1 πΌπ+3 ] ] ] β
β
β
ππ2 πΌπ+3 π(π+1)2 πΌπ+3 ] ], ] .. .. ] d . .
(45)
π·π‘πΌ π’ (π₯, π‘) = π·π₯2 π’ (π₯, π‘) ,
π₯ β (β1, 1) , π‘ β (0, 1) , 0 < πΌ < 1,
π’ (Β±1, π‘) = Β±πΈπΌ (βπ‘πΌ ) sin (1) , π’ (π₯, 0) = sin π₯,
0 < π‘ < 1,
β1 < π₯ < 1, (47)
Journal of Computational Methods in Physics
7
πΌ = 0.5
πΌ = 0.7
0.5 0.0 β0.5
1.0 0.5 0.0
0.0
0.5 0.0 β0.5
1.0
0.5 0.0
0.0
β0.5
0.5
1.0
β0.5
0.5
β1.0
1.0
(a) πΌ = 0.9
β1.0
(b) πΌ = 0.99
0.5 0.0 β0.5
1.0 0.5 0.0
0.0
1.0
1.0 0.5 0.0
0.0
β0.5
0.5
0.5 0.0 β0.5
β0.5
0.5
β1.0
1.0
(c)
β1.0
(d)
Figure 1: The approximate solutions of Example 1.
The approximate solutions generated by the proposed method are presented in Figure 1, for different values of πΌ and π = π = 10. Figure 2 depicts the exact solution (red) and the approximate solution (green) for πΌ = 1, and π = π = 10. Define the error σ΅¨ σ΅¨ Er = max {σ΅¨σ΅¨σ΅¨σ΅¨π’ (π₯π , π‘π ) β π’app (π₯π , π‘π )σ΅¨σ΅¨σ΅¨σ΅¨
0.5 0.0 β0.5
0.0
: π₯π = β1 + 0.01 (π β 1) , π‘π = 0.01 (π β 1) , (50)
1.0 0.5
0.5
π : 1 : 201, π = 1 : 101} ,
0.0 β0.5
1.0 β1.0
Figure 2: Exact solution (red) and approximate solution (green) of Example 1.
where πΈπΌ (π‘) is the Mittag-Leffler function. Since π·π‘πΌ πΈπΌ (βπ‘πΌ ) = βπΈπΌ (βπ‘πΌ ) for 0 < πΌ < 1, it is easy to see that the exact solution
is
π’ (π₯, π‘) = πΈπΌ (βπ‘πΌ ) sin π₯.
(48)
Example 2. Consider the fractional diffusion equation: π·π‘πΌ π’ (π₯, π‘) = π·π₯2 π’ (π₯, π‘) β 2π‘3 +
π’ (Β±1, π‘) = π‘ , (49)
6 π‘3βπΌ π₯2 , Ξ (4 β πΌ)
π₯ β (β1, 1) , π‘ β (0, 1) , 0 < πΌ < 1, 3
The exact solution for πΌ = 1 is π’ (π₯, π‘) = πβπ‘ sin π₯.
where π’app (π₯, π‘) is the approximate solution generated by the proposed method for π = π = 10. Table 1 presents the error for different values of πΌ.
π’ (π₯, 0) = 0,
0 < π‘ < 1, β1 < π₯ < 1,
(51)
8
Journal of Computational Methods in Physics πΌ = 0.5
πΌ = 0.7
0.4 0.3 0.2 0.1 0.0 0.0
1.0 0.5 0.0
0.4 0.3 0.2 0.1 0.0 0.0
β0.5
0.5
1.0 0.5 0.0 β0.5
0.5
1.0 β1.0
1.0 β1.0
(a)
(b)
πΌ = 0.9
πΌ = 0.99
0.4 0.3 0.2 0.1 0.0 0.0
1.0 0.5 0.0 β0.5
0.5
0.4 0.3 0.2 0.1 0.0 0.0
1.0 0.5 0.0 β0.5
0.5
1.0 β1.0
1.0 β1.0 (c)
(d)
Figure 3: The approximate solutions of Example 2.
Table 2 presents the error for different values of πΌ. Example 3. Consider the fractional diffusion equation presented in [17]: 0.4 0.3 0.2 0.1 0.0 1.0
0.0
0.5
0.5 0.0 β0.5
1.0 β1.0
Figure 4: Exact solution (red) and approximate solution (green) for Example 2.
where π’(π₯, π‘) = π‘3 π₯2 is the exact solution. The approximate solutions generated by the proposed method are presented in Figure 3 for different values of πΌ and π = π = 6. Figure 4 depicts the exact solution (red) and the approximate solution (green) for πΌ = 1 and π = π = 6.
π·π‘πΌ π’ (π₯, π‘) = π·π₯2 π’ (π₯, π‘) +
2π₯ (1 β π₯) π‘2βπΌ + 2 (π‘2 + 1) , Ξ (3 β πΌ)
π₯ β (0, 1) , π‘ β (0, 1) , 0 < πΌ < 1, π’ (0, π‘) = π’ (1, π‘) = 0, π’ (π₯, 0) = π₯ (π₯ β 1) ,
0 < π‘ < 1, β1 < π₯ < 1. (52)
The exact solution is π’ (π₯, π‘) = π₯ (π₯ β 1) (π‘2 + 1) .
(53)
To apply the proposed method, we shall do the following change of variable π = 2π₯ β 1. In this case, the π₯-domain becomes [β1, 1]. The approximate solutions generated by the proposed method and the exact solution are presented in Figure 5 for πΌ = 0.92 and πΌ = 0.98 at π‘ = 0.01 and π = π = 10. Table 3 presents a comparison between the error in our
Journal of Computational Methods in Physics
9
πΌ = 0.92
0.25
πΌ = 0.98
0.25
0.20
0.20
0.15
0.15
0.10
0.10
0.05
0.05 0.2
0.4
0.6
0.8
1.0
Proposed method Exact solution
0.2
0.4
0.6
0.8
1.0
Proposed method Exact solution (a)
(b)
Figure 5: Exact and approximate solutions of Example 3.
results and the ones obtained by the finite difference method (FDM) [17] for πΌ = 0.92, 0.98 and π‘ = 0.01. Example 4. Consider the fractional diffusion equation presented in [18]: π·π‘πΌ π’ (π₯, π‘) = π·π₯2 π’ (π₯, π‘) + sin (ππ₯) (π2 π‘2 +
2π‘2βπΌ ), Ξ (3 β πΌ)
π₯ β (0, 1) , π‘ β (0, 1) , 0 < πΌ < 1, π’ (0, π‘) = π’ (1, π‘) = 0, π’ (π₯, 0) = 0,
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
0 < π‘ < 1,
β1 < π₯ < 1. (54)
The exact solution is π’ (π₯, π‘) = sin (ππ₯) π‘2 .
them in this paper. These examples show the efficiency and the accuracy of the proposed method, where in few terms we achieved accuracy up to 10β10 . In Examples 3 and 4, we compare our results with the ones obtained by FDM in [17, 18]. Both examples show that the proposed method works more efficiently and accurately than the methods in [17, 18].
(55)
To apply the proposed method, we will do the following change of variable π = 2π₯ β 1. In this case, the π₯-domain becomes [β1, 1]. To make a comparison with the results of [18], assume that πI , πII , and πIII are the errors in [18] using uniform mesh, quasiuniform mesh, and nonuniform mesh for π = 1. Let πpro be the error in the proposed method for π = 1 and π = π = 10. Results are presented in Table 4.
5. Conclusion In this paper, we use series expansion based on the shifted fractional Legendre functions to solve fractional diffusions equations of Caputoβs type. We write the coefficients of the fractional derivative in terms of the shifted fractional Legendre functions as indicated in Theorem 5 and give explicit relationship between them. Then, we use the collocation method to compute these coefficients. To the best of our knowledge, the method has not been developed to integrate fractional diffusion equations of the form (1)-(2). We test the proposed technique for several examples and present four of
Acknowledgment The authors would like to express their sincere appreciation to United Arab Emirates University for the financial support of Grant no. 21S074.
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