Hindawi Publishing Corporation î e ScientiďŹc World Journal Volume 2014, Article ID 982413, 9 pages http://dx.doi.org/10.1155/2014/982413
Research Article Leapfrog/Finite Element Method for Fractional Diffusion Equation Zhengang Zhao1 and Yunying Zheng2 1 2
Department of Fundamental Courses, Shanghai Customs College, Shanghai 201204, China School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, China
Correspondence should be addressed to Zhengang Zhao;
[email protected] Received 18 January 2014; Accepted 17 February 2014; Published 3 April 2014 Academic Editors: C. Li, A. Sikorskii, and S. B. Yuste Copyright Š 2014 Z. Zhao and Y. Zheng. 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. We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove an đż2 -error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.
1. Introduction Fractional calculus and fractional partial differential equations (FPDEs) have many applications in various aspects such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics [1]. And a lot of attention has recently been paid to the problem of the numerical approximation of FPDEs. Generally speaking, the finite difference method and the finite element method are the two main means to solve FPDEs. Recently, some typical fractional difference methods have been utilized to solve FPDEs numerically [2â4]. On the other hand, the finite element method has also been used to find the variational solution of FPDEs [5â14]. But there are still some interesting schemes that can be constructed to enhance the convergence order by using the finite difference/finite element mixed method. In this paper, we use the explicit leapfrog difference/ Galerkin finite element mixed method to numerically solve
the space fractional diffusion equation in order to get a higher convergence order. The fractional diffusion equation as a typical kind of fractional partial differential equation [15] is a generalization of the classical diffusion equation, which can be used to better characterize anomalous diffusion phenomena. Besides, the spatial fractional diffusion equation usually describes 2đź the L´evy flights. The operator RL đˇ2đź đ,đĽ ( RL đˇđĽ,đ ) is commonly referred to the left (right) sided L´evy stable distribution, where the underlying stochastic process is L´evy đź-stable flights; see [16â18]. And a more general form đ
1 â
RL đˇ2đź đ,đĽ + is widely used for mathematical modelling and đ
2 â
RL đˇ2đź đĽ,đ numerical computation. Here, we mainly focus on constructing and analyzing a kind of efficient numerical schemes for approximately solving space fractional diffusion equation. The considered problem reads as follows: for 1/2 < đź < 1, đđĄ đ˘ (đĽ, đĄ) â Îđź (đ â
đ˘ (đĽ, đĄ)) = đ (đĽ, đĄ) , đ˘ (đĽ, đĄ) = đ (đĄ) ,
đĽ â Ί, đĄ â [0, đ] ,
đĽ â đΊ, đĄ â [0, đ] ,
đ˘ (đĽ, 0) = đ (đĽ) ,
đĽ â Ί, (1)
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where Ί = [đ, đ], time đ > 0. Here the spatial fractional 2đź differential operator Îđź is denoted by đ
1 â
RL đˇ2đź đ,đĽ +đ
2 â
RL đˇđĽ,đ , where 0 ⤠đ
1 , đ
2 ⤠1, and đ
1 + đ
2 = 1. When đź = 1, the problem models a Brownian diffusion process. And đ is a source term, đ is a positive constant. The rest of this paper is constructed as follows. In Section 2, the preliminary knowledge of fractional derivative and the generalized fractional derivative spaces are defined. And some related properties are further discussed. The approximate system of the equation, existence and uniqueness of the weak solution, and the error estimates of the fully discrete scheme for (1) are studied in Section 3. In Section 4, numerical examples are presented to demonstrate the efficiency of the theoretical results derived in Section 3.
Lemma 3 (see [5]). Let Ί = [đ, đ] be bounded and đź > 0. Then đ˘ â đżđ (Ί) satisfies óľŠóľŠ óľŠóľŠ (đ â đ)đź óľŠóľŠ RL đˇâđź óľŠóľŠ đ ⤠đ˘(đĽ) đ,đĽ óľŠ óľŠđż (Ί) Î (đź + 1) âđ˘âđżđ (Ί) ,
(4)
óľŠóľŠ óľŠóľŠ (đ â đ)đź óľŠóľŠ RL đˇâđź óľŠóľŠ đ ⤠đ˘(đĽ) đĽ,đ óľŠ óľŠđż (Ί) Î (đź + 1) âđ˘âđżđ (Ί) .
Lemma 4 (fractional integration by parts, see [20]). The relation âŤ
đ
đ
âđź RL đˇđ,đĽ đ˘ (đĽ)
đ
â
V (đĽ) đđĽ = ⍠đ˘ (đĽ) â
đ
âđź RL đˇđĽ,đ V (đĽ) đđĽ
(5)
is valid under the assumption that đ˘ (đĽ) â đżđ (Ί) ,
2. Generalized Fractional Derivative Spaces In this section, we first give the definition of fractional derivatives. There are several definitions for the fractional derivatives, but Riemann-Liouville derivative is one of the most often used fractional derivatives, which is a reasonable generalization of the classical derivative [1, 19â22]. Then we define the generalized fractional derivative spaces by using Riemann-Liouville derivative, which is extended from the đż2 sense to the đżđ sense. Definition 1. The đźth order left and right Riemann-Liouville integrals of function đ˘(đĽ) are defined in a finite interval (đ, đ) as follows: đĽ đ˘ (đ ) 1 đđ , ⍠Π(đź) đ (đĽ â đ )1âđź
âđź RL đˇđ,đĽ đ˘ (đĽ)
=
âđź RL đˇđĽ,đ đ˘ (đĽ)
đ đ˘ (đ ) 1 = đđ , ⍠Π(đź) đĽ (đ â đĽ)1âđź
(2)
1 1 + ⤠1 + đź, đ đ
Corollary 5 (see [20]). The formula âŤ
đ
đ 1 ⍠(đĽ â đ)đâđźâ1 đ˘ (đ) đđ, Î (đ â đź) đđĽđ đ
=
đź RL đˇđĽ,đ đ˘ (đĽ)
(â1)đ đđ đ = ⍠(đ â đĽ)đâđźâ1 đ˘ (đ) đđ, Î (đ â đź) đđĽđ đĽ
đź đ
đż đˇđ,đĽ đ˘ (đĽ)
đ
â
V (đĽ) đđĽ = ⍠đ˘ (đĽ) â
đ
đź đ
đż đˇđĽ,đ V (đĽ) đđĽ
(7)
âđź đ is valid under the assumption that đ˘(đĽ) â đ
đż đˇđ,đĽ (đż (Ί)), âđź đ V(đĽ) â đ
đż đˇđĽ,đ (đż (Ί)), 1/đ + 1/đ ⤠1 + đź, where the function đ âđź space đ
đż đˇâđź đ,đĽ (đż (Ί)) = {đ(đĽ) | đ(đĽ) = đ
đż đˇđ,đĽ đ(đĽ), đ(đĽ) â đ âđź đ đż (Ί)}, đ
đż đˇđĽ,đ (đż (Ί)) = {đ(đĽ) | đ(đĽ) = đ
đż đˇâđź đĽ,đ đ(đĽ), đ(đĽ) â đżđ (Ί)}.
Corollary 6 (see [13]). One can further give the following corollary: đ
2đź đ
đż đˇđ,đĽ đ˘ (đĽ)
đ
â
V (đĽ) đđĽ = âŤ
đ
đź đ
đż đˇđ,đĽ đ˘ (đĽ)
â
đź đ
đż đˇđĽ,đ V (đĽ) đđĽ
(8)
under the assumption that đ˘(đĽ) â âđź đ đ
đż đˇđĽ,đ (đż (Ί)), 1/đ + 1/đ ⤠1 + đź.
â2đź đ đ
đż đˇđ,đĽ (đż (Ί)),
V(đĽ) â
đ Note that the above assumption đ˘(đĽ) â RL đˇâ2đź đ,đĽ (đż (Ί)) âđź đ implies đ˘(đĽ) â RL đˇđ,đĽ (đż (Ί)) one can prove that by using Lemma 3.
đĽ
đź RL đˇđ,đĽ đ˘ (đĽ)
đ
đ
đ
Definition 2. The đźth order left and right Riemann-Liouville derivatives of function đ˘(đĽ) defined in a finite interval (đ, đ) are given as
(6)
đ ⼠1, đ ⼠1,
with đ ≠ 1, đ ≠ 1 in the case 1/đ + 1/đ = 1 + đź.
⍠where đź > 0.
V (đĽ) â đżđ (Ί) ,
(3)
Corollary 7 (see [13]). Consider đ
⍠in which đ â 1 < đź < đ â đ+ . Obviously, they are the integer derivatives of the left and right fractional integrals, respectively. Now, we give some lemmas and corollaries which are necessary to define the generalized fractional derivative spaces.
đ
2đź đ
đż đˇđĽ,đ đ˘ (đĽ)
â
V (đĽ) đđĽ = âŤ
đ
đ
đź đ
đż đˇđĽ,đ đ˘ (đĽ)
under the assumption that đ˘(đĽ) â âđź đ đ
đż đˇđ,đĽ (đż (Ί)), 1/đ + 1/đ ⤠1 + đź.
â
đź RL đˇđ,đĽ V (đĽ) đđĽ
(9) â2đź đ đ
đż đˇđĽ,đ (đż (Ί)),
V(đĽ) â
Note that, from the definition of the function space đ we can get that if đ˘(đĽ) â RL đˇâđź đ,đĽ (đż (Ί)),
âđź đ RL đˇđ,đĽ (đż (Ί)),
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đź then đ˘(đĽ) = RL đˇâđź đ,đĽ đ(đĽ), and RL đˇđ,đĽ đ˘(đĽ) = đ(đĽ), where đ đ đ(đĽ) â đż (Ί), such that đ˘ â đż (Ί), which is obtained by Lemma 3. And RL đˇđźđ,đĽ đ˘(đĽ) â đżđ (Ί) naturally holds. So, by the above idea, we define the following fractional derivative spaces from the đż2 sense to the đżđ sense, which will be proved to be equivalent with the fractional Sobolev spaces under some certain conditions.
Definition 8. Define the following norms of the left (with đź,đ symbol đđż ) fractional derivative space and the right (with đź,đ symbol đđ
) fractional derivative space in a bounded interval Ί = [đ, đ] as follows correspondingly, where 1 < đ < +â: đź,đ
đđż (Ί) ⥠{đ˘ â đżđ (Ί) :
đź RL đˇđ,đĽ đ˘ (đĽ)
â đżđ (Ί)}
(10)
equipped with seminorm óľŠ óľŠ đź đ˘(đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) |đ˘|đđżđź,đ (Ί) = óľŠóľŠóľŠóľŠ RL đˇđ,đĽ
(11)
1/đ
[đź]
,
đż
đ=0
đź,đ
đđ
(Ί) ⥠{đ˘ â đżđ (Ί) :
RL
From [6], we can get the following lemma, which is true in the đż2 sense. đź,2 đź,2 đź Lemma 11. The spaces đđż,0 (Ί), đđ
,0 (Ί), đťđ,0 (Ί), and đź đť0 (Ί) are equal to equivalent seminorms and norms, where đťđź (Ί) is the fractional Sobolev space in terms of the Fourier transform.
Therefore, in this paper we always use đť0đź when đ = 2, to denote the fractional derivative space equipped with the norm â â
âđź which can be any one of (12), (15), and (18), and đťâđź (Ί) is denoted as the dual space of đť0đź (Ί), with norm â â
ââđź . Moreover, we can present some new properties about norms for the above left and right fractional derivative spaces in the đżđ sense.
(12) (1)
đź đˇđĽ,đ đ˘ â đżđ (Ί)}
âđź đ
đż đˇđ,đĽ đ˘(đĽ)
: đżđ (Ί) â đżđ (Ί) is a bounded linear
âđź đ
đż đˇđĽ,đ đ˘(đĽ)
: đżđ (Ί) â đżđ (Ί) is a bounded linear
đź đ
đż đˇđ,đĽ đ˘(đĽ)
: đđż (Ί) â đżđ (Ί) is a bounded linear
đź đ
đż đˇđĽ,đ đ˘(đĽ)
: đđ
(Ί) â đżđ (Ί) is a bounded linear
âđź đ
đż đˇđ,đĽ đ˘(đĽ)
: đżđ (Ί) â đđż (Ί) is a bounded linear
âđź đ
đż đˇđĽ,đ đ˘(đĽ)
: đżđ (Ί) â đđ
(Ί) is a bounded linear
operator;
(13)
equipped with seminorm
(2)
operator;
óľŠ óľŠ đź đ˘(đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) |đ˘|đđ
đź,đ (Ί) = óľŠóľŠóľŠóľŠ RL đˇđĽ,đ
(14) (3)
and norm
operator; [đź]
.
đ
đ=0
(15)
(4)
operator;
Definition 9. Define the symmetric fractional derivative space (with symbol đťđđź ) in a bounded interval Ί = [đ, đ] in the đż2 sense đťđđź (Ί) ⥠{ đ˘ â đż2 (Ί)
(5)
operator; (6)
operator. :âŤ
RL
đ
đź,đ
1/đ
óľŠ óľŠđ đ âđ˘âđđ
đź,đ (Ί) = ( â óľŠóľŠóľŠóľŠđˇđ đ˘óľŠóľŠóľŠóľŠđżđ (Ί) + |đ˘|đđź,đ (Ί) )
đ
đź,đ
Lemma 12. Let đź > 0 and Ί = [đ, đ] â R be bounded. Then the following mapping properties hold:
and norm óľŠ óľŠđ đ âđ˘âđđżđź,đ (Ί) = ( â óľŠóľŠóľŠóľŠđˇđ đ˘óľŠóľŠóľŠóľŠđżđ (Ί) + |đ˘|đđź,đ (Ί) )
đź,đ
Definition 10. Define the spaces đđż,0 (Ί), đđ
,0 (Ί), and đź đťđ,0 (Ί) as the closures of đś0â (Ί) under their respective norms.
đź đˇđ,đĽ đ˘ (đĽ) â
đź RL đˇđĽ,đ đ˘ (đĽ) đđĽ
â đż2 (Ί)} (16)
đź,đ
đź,đ
đź,đ
Proof. Properties (1) and (2) follow directly from Lemma 3. Property (3) follows directly from the definition of đź,đ đź,đ đđż (Ί) and đđ
(Ί) as
equipped with seminorm |đ˘|đťđđź (Ί)
óľ¨óľ¨ đ óľ¨ = óľ¨óľ¨óľ¨óľ¨âŤ óľ¨óľ¨ đ
đˇđź RL đ,đĽ
đ˘ (đĽ) â
đˇđź RL đĽ,đ
óľ¨óľ¨1/2 óľ¨ đ˘ (đĽ) đđĽóľ¨óľ¨óľ¨óľ¨ óľ¨óľ¨
(17)
[đź]
óľŠđ óľŠ óľŠđ óľŠ â¤ ( â óľŠóľŠóľŠóľŠđˇđ đ˘óľŠóľŠóľŠóľŠđżđ (Ί) + óľŠóľŠóľŠóľŠ RL đˇđźđ,đĽ đ˘ (đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) )
and norm âđ˘âđťđđź (Ί)
óľŠóľŠ óľŠ đź óľŠóľŠ đˇđ,đĽ đ˘ (đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) óľŠ RL
đ=0
[đź]
1/2
óľŠ óľŠ2 = ( â óľŠóľŠóľŠóľŠđˇđ đ˘óľŠóľŠóľŠóľŠđż2 (Ί) + |đ˘|2đťđź (Ί) ) đ đ=0
.
(18)
Property (4) follows similarly.
1/đ
(19) .
4
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Property (5) follows from the definition of đđż (Ί) and the semigroup property of fractional operator, óľŠóľŠ óľŠóľŠ óľŠ đź,đ óľŠóľŠ RL đˇâđź đ,đĽ đ˘ (đĽ)óľŠ óľŠđđż (Ί) óľŠ [đź]
óľŠ = ( â óľŠóľŠóľŠóľŠđˇđ â
đˇâđź RL đ,đĽ
đ=0
óľŠ + óľŠóľŠóľŠóľŠ RL đˇđźđ,đĽ â
Therefore, (25) is true. And the following inequality certainly holds: óľŠ óľŠóľŠ óľŠđ . âđ˘âđżđ (Ί) ⤠đśóľŠóľŠóľŠóľŠ RL đˇđźâđ (28) đ,đĽ đ˘(đĽ)óľŠ óľŠđż (Ί) So, we get that
óľŠđ đ˘ (đĽ)óľŠóľŠóľŠóľŠđżđ (Ί)
óľŠóľŠđ âđź óľŠđ ) RL đˇđ,đĽ đ˘ (đĽ)óľŠ óľŠđż (Ί)
1/đ
(20)
óľŠ óľŠóľŠ óľŠóľŠ óľŠ óľŠđ óľŠóľŠ RL đˇđ đ,đĽ đ˘ (đĽ)óľŠóľŠóľŠ đ ⤠óľŠóľŠóľŠ RL đˇđ đ,đĽ RL đˇđźâđ đ,đĽ đ˘ (đĽ)óľŠ óľŠđż (Ί) óľŠ óľŠđż (Ί) óľŠ óľŠóľŠ óľŠ = óľŠóľŠóľŠ RL đˇđźđ,đĽ đ˘(đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) .
(29)
Therefore, (26) holds. [đź]
óľŠ óľŠđ đ đâđź = ( â óľŠóľŠóľŠóľŠ RL đˇđ,đĽ đ˘(đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) + âđ˘âđżđ (Ί) )
1/đ
.
3. Error Estimates of the Leapfrog/Finite Element Scheme
đ=0
Using Lemma 3, there exist constants đđ , đ = 0, . . . , [đź] such that óľŠđ óľŠóľŠ đ đâđź óľŠóľŠ đˇđ,đĽ đ˘(đĽ)óľŠóľŠóľŠóľŠđżđ (Ί) ⤠đśđ âđ˘âđżđ (Ί) . (21) óľŠ RL Therefore, we obtain the bound [đź]
1/đ
óľŠóľŠ óľŠ âđź óľŠóľŠ đˇđ,đĽ đ˘(đĽ)óľŠóľŠóľŠóľŠđđź,đ (Ί) ⤠(1 + â đđ ) óľŠ RL đż đ=0
âđ˘âđżđ (Ί) .
(22)
In this section, we firstly give a fully discrete scheme, where we use the leapfrog difference method in the temporal direction and the finite element method in the spatial direction and then analyze the error estimate. Let đâ denote a uniform partition on Ί, with grid parameter â. For đ â đ, let đđ (Ί) denote the space of polynomials on Ί with degree not greater than đ. Then we define đâ as the finite element space on đâ with the basis of the piecewise polynomials of order đ â đ+ ; that is, đâ = {V â đ ⊠đś (Ί) : V|đˇ â đđ (đˇ) , âđˇ â đâ } ,
Property (6) follows similarly. Corollary 13. Consider đź,đ
đźâ[đź],đ
(Ί) ,
đź,đ
đźâ[đź],đ
(Ί)
đđż,0 (Ί) ół¨â đđż,0
đđ
,0 (Ί) ół¨â đđ
,0
(23)
in which đˇ is the unit of đâ . The following property of finite element spaces is necessary for our subsequent analysis [23]: for đ˘ â đťđ+1 (Ί), 0 ⤠đ ⤠đ + 1, there exists V â đâ such that âđ˘ â Vâđ ⤠đśâđ+1âđ âđ˘âđ+1 .
for 1 ⤠đ < â. And if 1 ⤠đ ⤠đ < â, one has đź,đ
đź,đ
đđ
,0 (Ί) ół¨â đżđ (Ί) .
(24)
Lemma 15 (discrete Gronwallâs lemma, see [24]). Let ÎđĄ, đť and đđ , đđ , đđ , đžđ (for integer đ ⼠0) be nonnegative numbers such that
It is obviously true by using the norms of fractional derivative spaces and imbedding theorems for đżđ (Ί). Lemma 14. Let Ί = [đ, đ] â R be bounded. Then for đ˘ â đź,đ đđż,0 (Ί), one has đź,đ âđ˘âđżđ (Ί) ⤠đśâđ˘âđđż,0 (Ί) ,
(25)
đź,đ
đ
đ
đ
đ=0
đ=0
đ=0
đđ + ÎđĄ â đđ ⤠ÎđĄ â đžđ đđ + ÎđĄ â đđ + đť,
(26)
Proof. If đ˘ â đđż,0 (Ί) by using Lemmas 3 and 12, we have that óľŠ óľŠóľŠ âđź đź óľŠđ âđ˘âđżđ (Ί) = óľŠóľŠóľŠóľŠ RL đˇđ,đĽ RL đˇđ,đĽ đ˘ (đĽ)óľŠ óľŠđż (Ί) (27) óľŠ (đ â đ)đź óľŠóľŠ óľŠóľŠ RL đˇđźđ,đĽ đ˘ (đĽ)óľŠóľŠóľŠ đ . ⤠óľŠđż (Ί) Î (đź + 1) óľŠ
(32)
for đ ⼠0. Suppose that ÎđĄđžđ < 1 for all đ, and set đđ = (1 â ÎđĄđžđ )â1 ; then đ
đ
đ
đ=0
đ=0
đ=0
đđ + ÎđĄ â đđ ⤠exp (ÎđĄ â đđ đžđ ) {ÎđĄ â đđ + đť}
and for 0 < đ < đź, one has đ ,đ đź,đ âđ˘âđđż,0 (Ί) ⤠đśâđ˘âđđż,0 (Ί) .
(31)
The Gronwallâs lemma is also needed for the error analysis.
đź,đ
đđż,0 (Ί) ół¨â đđż,0 (Ί) ,
(30)
(33)
for đ ⼠0. In the following, we give the fully discrete scheme of (1). Let ÎđĄ denote the step size for đĄ so that đĄđ = đÎđĄ, đ = 1, 2, . . . , đ â 1. For notational convenience, we denote đ˘đ := đ˘(â
, đĄđ ) and đđĄ đ˘đ :=
đ˘đ+1 â đ˘đâ1 . 2ÎđĄ
(34)
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Let đ˘âđ of (1) be the finite element solution at time đĄ = đĄđ of the following fully discrete scheme: (đđĄ đ˘âđ , V) â (Îđź (đ â
đ˘âđ ) , V) = â¨đđ , V⊠,
âV â đâ ;
(35)
that is, (đ˘âđ+1 â đ˘âđâ1 , V) â 2ÎđĄ (Îđź (đ â
đ˘âđ ) , V) = 2ÎđĄâ¨đđ , VâŠ,
(36)
âV â đâ ,
where (â
, â
) is denoted by an đż2 inner product and (Îđź (đ â
đź đź (đ â
đ˘âđ ), RL đˇđźđĽ,đ V) + đ
2 â
( RL đˇđĽ,đ (đ â
đ˘âđ ), V) = đ
1 â
( RL đˇđ,đĽ đ đź đź đ đ˘â ), RL đˇđ,đĽ V). For brevity, we always use (Î (đâ
đ˘â ), V) instead of the right hand side of this equation. Lemma 16. For a sufficient small step size ÎđĄ > 0, there exists a unique solution đ˘âđ â đâ satisfying (36). (đ˘âđ , đ˘âđ )/2ÎđĄ
đź
đ˘âđ ), đ˘âđ )
â (Î (đ â
Proof. Firstly, we prove that is positive, which is one of the sufficient conditions for the existence and uniqueness of đ˘âđ . For ÎđĄ > 0 chosen sufficiently small, we have that
(đ˘âđ , đ˘âđ ) óľŠ óľŠ2 (37) â (Îđź (đ â
đ˘âđ ) , đ˘âđ ) ⼠đśóľŠóľŠóľŠđ˘âđ óľŠóľŠóľŠđź . 2ÎđĄ Besides, by using the fractional Poincare-Friedrichs formula, we can easily get the continuity of (đ˘âđ , đ˘âđ )/2ÎđĄ â (Îđź (đ â
đ˘âđ ), đ˘âđ ). Hence, by using the Lax-Milgram theorem, we have that (36) is uniquely solvable for đ˘âđ . Now, we carry out the error analysis for the fully discrete problem. The following norms are also used in the analysis: óľŠ óľŠ âđ˘ââ,đ = max óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠđ , 1â¤đâ¤đ âđ˘â0,đź
đ
óľŠ óľŠ2 = ( â óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠđź đđĄ)
(38)
1/2
.
đ=1
Theorem 17. Assume that (1) has a solution đ˘ satisfying đ˘ â đż2 (0, đ; đťđź ⊠đťđ+1 (Ί)), đ˘đĄ â đż2 (0, đ; đťđ+1 (Ί)), and đ˘đĄđĄđĄ â đż2 (0, đ; đż2 (Ί)), with đ˘0 â đťđ+1 (Ί). đ˘âđ is the solution of (36), and đ˘â1 is computed in such a way that óľŠ óľŠóľŠ 1 óľŠóľŠđ˘ (đĽ) â đ˘ (đĽ, ÎđĄ)óľŠóľŠóľŠ ⤠đś(ÎđĄ)2 . (39) óľŠ óľŠ Then, there exists a constant đś0 independent of â and ÎđĄ, such that if ÎđĄ â
ââ2đź ⤠đś0 ,
(40)
then the finite element approximation (36) is convergent to the solution of (1) on the interval (0, đ) as ÎđĄ, â â 0. And the approximation solution đ˘â satisfies the following error estimates: óľŠ óľŠ óľŠóľŠ đ+1 óľŠ óľŠ 2óľŠ óľŠóľŠđ˘ â đ˘â óľŠóľŠóľŠ0,đź ⤠đś (â óľŠóľŠóľŠđ˘đĄ óľŠóľŠóľŠ0,đ+1 + (ÎđĄ) óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠóľŠ0,0 (41) + âđ+1âđź âđ˘â0,đ+1 ) ; óľŠóľŠ óľŠ óľŠ đ+1 óľŠ óľŠ 2óľŠ óľŠóľŠđ˘ â đ˘â óľŠóľŠóľŠâ,0 ⤠đś (â óľŠóľŠóľŠđ˘đĄ óľŠóľŠóľŠ0,đ+1 + (ÎđĄ) óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠóľŠ0,0 +âđ+1âđź âđ˘â0,đ+1 + âđ+1 âđ˘ââ,đ+1 ) .
Proof. In order to estimate (41) and (42), we first discuss the error at đĄ = đĄđ , đ = 1, 2, . . . , đ â 1. Let đ˘đ = đ˘(â
, đĄđ ) represent the solution of (1), define đđ = đ˘đ âđ˘âđ , and for đđ â đâ , define Îđ and đ¸đ as Îđ = đ˘đ â đđ , đ¸đ = đđ â đ˘âđ . So, we have đđ = Îđ + đ¸đ . Obviously the true solution of this problem (1) đ˘đ also satisfies (đđĄ đ˘đ , V) â (Îđź (đ â
đ˘đ ) , V) (43) = â¨đđ , V⊠â (đ˘đĄđ â đđĄ đ˘đ , V) , âV â đâ . Therefore, subtracting (36) from (43) gives (đđĄ đđ , V) â (Îđź (đ â
đđ ) , V) = (đđĄ đ˘đ â đ˘đĄđ , V) ,
âV â đâ ; (44)
that is, (đđ+1 â đđâ1 , V) â 2ÎđĄ (Îđź (đ â
đđ ) , V) = 2ÎđĄ (đđĄ đ˘đ â đ˘đĄđ , V) ,
âV â đâ .
Substituting đđ+1 = Îđ+1 + đ¸đ+1 , V = đ¸đ+1 + đ¸đâ1 into (45) leads to (đ¸đ+1 â đ¸đâ1 , đ¸đ+1 + đ¸đâ1 ) â 2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đ+1 + đ¸đâ1 ) = â (Îđ+1 â Îđâ1 , đ¸đ+1 + đ¸đâ1 )
(46)
+ 2ÎđĄ (Îđź (đ â
Îđ ) , đ¸đ+1 + đ¸đâ1 ) + 2ÎđĄ (đđĄ đ˘đ â đ˘đĄđ , đ¸đ+1 + đ¸đâ1 ) . 2
After adding âđ¸đ â to both sides of (46), we obtain the identity óľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠ2 + óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠ2 â 2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đ+1 ) óľŠóľŠ óľŠ óľŠ óľŠóľŠ óľŠ2 óľŠ óľŠ2 óľŠ = óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ + 2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đâ1 ) â (Îđ+1 â Îđâ1 , đ¸đ+1 + đ¸đâ1 )
(47)
+ 2ÎđĄ (Îđź (đ â
Îđ ) , đ¸đ+1 + đ¸đâ1 ) + 2ÎđĄ (đđĄ đ˘đ â đ˘đĄđ , đ¸đ+1 + đ¸đâ1 ) . Define now the quantity đ´đ+1 , for 1 ⤠đ ⤠đ â 1, by óľŠ2 óľŠ óľŠ2 óľŠ đ´đ+1 = óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠ â 2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đ+1 ) . We can rewrite (47) as
(48)
đ´đ+1 = đ´đ â (Îđ+1 â Îđâ1 , đ¸đ+1 + đ¸đâ1 ) + 2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đâ1 ) + 2ÎđĄ (Îđź (đ â
Îđ ) , đ¸đ+1 + đ¸đâ1 ) + 2ÎđĄ (Îđź (đ â
đ¸đâ1 ) , đ¸đ )
(42)
(45)
+ 2ÎđĄ (đđĄ đ˘đ â đ˘đĄđ , đ¸đ+1 + đ¸đâ1 ) .
(49)
6
The Scientific World Journal
Denoting
For the third term of the right hand side, one has 2ÎđĄ (Îđź (đ â
Îđ ) , đ¸đ+1 + đ¸đâ1 )
đš (đ¸đâ1 , đ¸đ , đ¸đ+1 ) = â (Îđ+1 â Îđâ1 , đ¸đ+1 + đ¸đâ1 )
= 2ÎđĄ â
đ
1 ( RL đˇđźđ,đĽ (đ â
Îđ ) ,
+ 2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đâ1 )
+ 2ÎđĄ (Î (đ â
đ¸
đâ1
đ
đź RL đˇđ,đĽ
(đ¸đ+1 + đ¸đâ1 ))
(54)
óľŠ óľŠ óľŠ óľŠ â¤ đś4 ÎđĄóľŠóľŠóľŠÎđ óľŠóľŠóľŠđź â
óľŠóľŠóľŠóľŠđ¸đ+1 + đ¸đâ1 óľŠóľŠóľŠóľŠđź
),đ¸ )
+ 2ÎđĄ (đđĄ đ˘đ â đ˘đĄđ , đ¸đ+1 + đ¸đâ1 ) ,
óľŠ óľŠ2 ⤠đś5 ÎđĄ â
â2(đ+1âđź) óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠđ+1 óľŠ2 óľŠ óľŠ2 óľŠ + đś6 ÎđĄ (óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠđź + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠđź ) ,
then (49) can be abbreviated as đ´đ+1 = đ´đ + đš (đ¸đâ1 , đ¸đ , đ¸đ+1 ) .
(51)
in which óľŠóľŠ óľŠ đóľŠ đóľŠ đ+1âđź óľŠ óľŠóľŠđ˘đ óľŠóľŠóľŠ . óľŠóľŠđ â
Î óľŠóľŠóľŠđź ⤠âđââ â
óľŠóľŠóľŠÎ óľŠóľŠóľŠđź ⤠đś5 â óľŠ óľŠđ+1 For the fourth term of the right hand side, one has
= 2ÎđĄ â
đ
1 ( RL đˇđźđ,đĽ đ â
đ¸đâ1 ,
đź đ RL đˇđĽ,đ đ¸ )
+ 2ÎđĄ â
đ
2 ( RL đˇđźđĽ,đ đ â
đ¸đâ1 ,
2ÎđĄ (Îđź (đ â
đ¸đ ) , đ¸đâ1 )
(56)
And for the term 2ÎđĄ(đđĄ đ˘đ â đ˘đĄđ , đ¸đ+1 + đ¸đâ1 ), by using the Cauchy-Schwarz inequality, we obtain
đź đâ1 )) RL đˇđ,đĽ đ¸
óľŠ2 óľŠ óľŠ óľŠ2 ⤠đś1 ÎđĄóľŠóľŠóľŠđ¸đ óľŠóľŠóľŠđź + đś2 ÎđĄóľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠđź , (Îđ+1 â Îđâ1 , đ¸đ+1 + đ¸đâ1 )
đź đ RL đˇđ,đĽ đ¸ )
óľŠ2 óľŠ óľŠ óľŠ2 ⤠đś7 ÎđĄ â
óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠđź + đś8 ÎđĄ â
óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠđź .
đź đâ1 ) RL đˇđĽ,đ đ¸
+ đ
2 ( RL đˇđźđĽ,đ (đ â
đ¸đ ) ,
(55)
2ÎđĄ (Îđź (đ â
đ¸đâ1 ) , đ¸đ )
We now estimate each term in đš(đ¸đâ1 , đ¸đ , đ¸đ+1 ). For the second term of the right hand side, one has
= 2ÎđĄ (đ
1 ( RL đˇđźđ,đĽ (đ â
đ¸đ ) ,
(đ¸đ+1 + đ¸đâ1 ))
+ 2ÎđĄ â
đ
2 ( RL đˇđźđĽ,đ (đ â
Îđ ) ,
+ 2ÎđĄ (Îđź (đ â
Îđ ) , đ¸đ+1 + đ¸đâ1 ) (50) đź
đź RL đˇđĽ,đ
2ÎđĄ (đđĄ đ˘đ â đ˘đĄđ , đ¸đ+1 + đ¸đâ1 ) = (đ˘đ+1 â đ˘đâ1 â 2ÎđĄ â
đ˘đĄđ , đ¸đ+1 + đ¸đâ1 )
(52)
óľŠ óľŠ óľŠ óľŠ â¤ óľŠóľŠóľŠóľŠÎđ+1 â Îđâ1 óľŠóľŠóľŠóľŠ â
óľŠóľŠóľŠóľŠđ¸đ+1 + đ¸đâ1 óľŠóľŠóľŠóľŠ óľŠ óľŠ óľŠ óľŠ = 2ÎđĄ óľŠóľŠóľŠđđĄ Îđ óľŠóľŠóľŠ â
óľŠóľŠóľŠóľŠđ¸đ+1 + đ¸đâ1 óľŠóľŠóľŠóľŠ óľŠ2 óľŠ óľŠ2 óľŠ óľŠ óľŠ2 ⤠ÎđĄóľŠóľŠóľŠđđĄ Îđ óľŠóľŠóľŠ + ÎđĄ (óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ )
(57)
óľŠ đ+1 óľŠ2 óľŠ đâ1 óľŠ2 óľŠ đ óľŠóľŠ2 ⤠đś9 (ÎđĄ)5 óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠ + ÎđĄ (óľŠóľŠóľŠóľŠđ¸ óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸ óľŠóľŠóľŠóľŠ ) , where by Taylorâs theorem óľŠóľŠ đ+1 óľŠ đ óľŠ óľŠóľŠ . óľŠóľŠđ˘ â đ˘đâ1 â 2ÎđĄ â
đ˘đĄđ óľŠóľŠóľŠ ⤠đś9 (ÎđĄ)3 óľŠóľŠóľŠóľŠđ˘đĄđĄđĄ óľŠ óľŠ óľŠ Hence, summing from đ = 1 to đ â 1, one has
óľŠ2 óľŠ óľŠ2 óľŠ óľŠ óľŠ2 ⤠đś3 ÎđĄ â
â2đ+2 óľŠóľŠóľŠđ˘đĄđ óľŠóľŠóľŠđ+1 + ÎđĄ (óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ ) ,
(58)
đâ1
đ´đ â đ´đâ1 ⤠â đš (đ¸đâ1 , đ¸đ , đ¸đ+1 ) ;
(59)
đ=1
where
that is, đ´đ ⤠đ´đâ1
đ
óľŠ2 óľŠ â ÎđĄóľŠóľŠóľŠđđĄ Îđ óľŠóľŠóľŠ
đâ1
óľŠ óľŠ2 óľŠ óľŠ2 + đś10 â ÎđĄ (â2đ+2 óľŠóľŠóľŠđ˘đĄđ óľŠóľŠóľŠđ+1 + â2đ+2â2đź óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠđ+1
đ=0
óľŠóľŠ óľŠ2 óľŠ 1 đĄđ đÎ óľŠóľŠ = â ÎđĄóľŠóľŠóľŠóľŠ ⍠1 đđĄóľŠóľŠóľŠóľŠ óľŠ ÎđĄ đĄđâ1 đđĄ óľŠóľŠ đ=1 óľŠ
đ=1
đ
đ
đĄđ đĄđ đÎ 1 2 ⤠â ÎđĄ( ) ⍠(⍠1đđĄ) (⍠đđĄ) đđĽ ÎđĄ Ί đĄđâ1 đĄđâ1 đđĄ đ=1
óľŠ óľŠ2 ⤠đś3 â2đ+2 óľŠóľŠóľŠđ˘đĄ óľŠóľŠóľŠ0,đ+1 .
óľŠ đ óľŠóľŠ2 +(ÎđĄ)4 óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠ )
(53)
đâ1 óľŠ2 óľŠ óľŠ2 óľŠ óľŠ2 óľŠ + đś11 â ÎđĄ (óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠđź đ=1
óľŠ2 óľŠ óľŠ2 óľŠ + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠđź + óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠđź ) .
(60)
The Scientific World Journal
7
We now show that, under our stability assumption (40), 2 2 đ´đ+1 is positive and comparable to âđ¸đ â + âđ¸đ+1 â . To this âđź end, we use the inverse inequality âVâđź ⤠đś12 â âVâ, V â đâ , and this yields óľ¨ óľ¨ 2ÎđĄ óľ¨óľ¨óľ¨óľ¨â (Îđź (đ 1 đ¸đ , đ¸đ+1 ))óľ¨óľ¨óľ¨óľ¨ óľŠ2 óľŠ óľŠ2 óľŠ â¤ ÎđĄ (óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠđź + óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠđź ) (61) óľŠ2 óľŠ óľŠ2 óľŠ â¤ đś12 ÎđĄ â
ââ2đź (óľŠóľŠóľŠđ¸đ óľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ ) .
By using the discrete Gronwallâs Lemma 15, we have đ óľŠóľŠ đóľŠóľŠ2 óľŠóľŠđ¸ óľŠóľŠ + đś14 â ÎđĄóľŠóľŠóľŠóľŠđ¸đ óľŠóľŠóľŠóľŠ2đź óľŠ óľŠ đ=1
óľŠ óľŠ2 ⤠đś17 óľŠóľŠóľŠóľŠđ¸1 óľŠóľŠóľŠóľŠ
(66)
óľŠ óľŠ2 + đś15 (â2đ+2 óľŠóľŠóľŠđ˘đĄ óľŠóľŠóľŠ0,đ+1 + â2đ+2â2đź âđ˘â20,đ+1 óľŠ óľŠ2 + (ÎđĄ)4 óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠóľŠ0,0 ) ,
Hence, if ÎđĄ â
ââ2đź is sufficiently small such that đś12 ÎđĄ â
ââ2đź ⤠đś13 ⤠1, we get óľŠ2 óľŠ óľŠ2 óľŠ (1 â đś13 ) (óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đóľŠóľŠóľŠóľŠ ) (62) óľŠ2 óľŠ óľŠ2 óľŠ â¤ đ´đ ⤠(1 + đś13 ) (óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đóľŠóľŠóľŠóľŠ ) . So we have that
2
2
where denoting đđ = âđ¸đâ , đđ = âđ¸đ âđź , đť 2 đś15 (â2đ+2 âđ˘đĄ â20,đ+1 + â2đ+2â2đź âđ˘đ âđ+1 + (ÎđĄ)4 âđ˘đĄđĄđĄ â20,0 ).
=
Now denoting đş(ÎđĄ, â) = â2đ+2 âđ˘đĄ â20,đ+1 +â2đ+2â2đź âđ˘â20,đ+1 +(ÎđĄ)4 âđ˘đĄđĄđĄ â20,0 and using the condition (39), we get that đ
óľŠ óľŠ2 âđ¸â20,đź = â ÎđĄóľŠóľŠóľŠđ¸đ óľŠóľŠóľŠđź ⤠đś18 (đ + 1) đş (ÎđĄ, â) .
đ
óľŠóľŠ đâ1 óľŠóľŠ2 óľŠóľŠ đóľŠóľŠ2 óľŠóľŠđ¸ óľŠóľŠ + óľŠóľŠđ¸ óľŠóľŠ + đś14 â ÎđĄóľŠóľŠóľŠóľŠđ¸đ óľŠóľŠóľŠóľŠ2đź óľŠ óľŠ óľŠ óľŠ
(67)
đ=1
đ=1
By using the interpolation property and the following result
óľŠ óľŠ2 óľŠ óľŠ2 ⤠óľŠóľŠóľŠóľŠđ¸1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸0 óľŠóľŠóľŠóľŠ
óľŠóľŠ óľŠ óľŠóľŠđ˘ â đ˘â óľŠóľŠóľŠ0,đź ⤠âđ¸â0,đź + âÎâ0,đź ,
đâ1
óľŠ óľŠ2 óľŠ óľŠ2 + đś15 â ÎđĄ (â2đ+2 óľŠóľŠóľŠđ˘đĄđ óľŠóľŠóľŠđ+1 + â2đ+2â2đź óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠđ+1
(63)
đ=1
óľŠ đ óľŠóľŠ2 + (ÎđĄ)4 óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠ ) đâ1
óľŠ + đś16 â ÎđĄ (óľŠóľŠóľŠóľŠđ¸
đ+1 óľŠ óľŠ2
đ=1
óľŠ óľŠóľŠ + óľŠóľŠóľŠđ¸ óľŠ óľŠ
óľŠóľŠ ) . óľŠ
⤠đş (ÎđĄ, â) + â2đ+2 âđ˘â2â,đ+1 ,
Therefore, we obtain
4. Numerical Examples for Piecewise Linear Polynomials
đ=1
óľŠ óľŠ2 óľŠ óľŠ2 ⤠óľŠóľŠóľŠóľŠđ¸1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸0 óľŠóľŠóľŠóľŠ óľŠ2 2đ+2â2đź óľŠ óľŠóľŠđ˘đ óľŠóľŠóľŠ2 óľŠóľŠđ˘đĄ óľŠóľŠóľŠ0,đ+1 + â óľŠ óľŠđ+1
+ đś15 (â
(64)
óľŠ óľŠ2 + (ÎđĄ)4 óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠóľŠ0,0 ) đâ1 óľŠ2 óľŠ óľŠ2 óľŠ + đś16 â ÎđĄ (óľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸đâ1 óľŠóľŠóľŠóľŠ ) . đ=1
Hence, đ óľŠóľŠ đóľŠóľŠ2 óľŠóľŠđ¸ óľŠóľŠ + đś14 â ÎđĄóľŠóľŠóľŠóľŠđ¸đ óľŠóľŠóľŠóľŠ2đź óľŠ óľŠ đ=1
óľŠ óľŠ2 óľŠ óľŠ2 ⤠óľŠóľŠóľŠóľŠđ¸1 óľŠóľŠóľŠóľŠ + óľŠóľŠóľŠóľŠđ¸0 óľŠóľŠóľŠóľŠ óľŠ óľŠ2 óľŠ óľŠ2 + đś15 (â2đ+2 óľŠóľŠóľŠđ˘đĄ óľŠóľŠóľŠ0,đ+1 + â2đ+2â2đź óľŠóľŠóľŠđ˘đ óľŠóľŠóľŠđ+1 đâ1 óľŠ2 óľŠ óľŠ óľŠ2 + (ÎđĄ) óľŠóľŠóľŠđ˘đĄđĄđĄ óľŠóľŠóľŠ0,0 ) + đś16 â ÎđĄóľŠóľŠóľŠóľŠđ¸đ+1 óľŠóľŠóľŠóľŠ . 4
đ=1
(69)
which yields estimate (42).
đ óľŠóľŠ đâ1 óľŠóľŠ2 óľŠóľŠ đóľŠóľŠ2 óľŠóľŠđ¸ óľŠóľŠ + óľŠóľŠđ¸ óľŠóľŠ + đś14 â ÎđĄóľŠóľŠóľŠóľŠđ¸đ óľŠóľŠóľŠóľŠ2đź óľŠ óľŠ óľŠ óľŠ
2đ+2 óľŠ óľŠ
estimate (41) follows. Using estimate (66) and approximation properties, we have óľŠ2 óľŠóľŠ 2 2 óľŠóľŠđ˘ â đ˘â óľŠóľŠóľŠâ,0 ⤠âđ¸ââ,0 + âÎââ,0
đâ1 óľŠ óľŠ2
(68)
(65)
Let đâ denote a uniform partition on Ί = [đ, đ] and đâ the space of continuous piecewise linear functions on đâ ; that is, đ = 1. Then we use the Galerkin finite element method for the spatial variables. After the spatial discretization, we get classical ODEs systems with variables đ˘âđ , đ = 1, 2, . . . , đ/ÎđĄ. In order to satisfy the condition (39) in Theorem 17, we use the two-order Runge-Kutta method to compute the variable đ˘â1 . In this section, we present numerical calculations which support the error estimates in Theorem 17. If we suppose ÎđĄ = đśâ2đź , then we have the convergence rate óľŠ óľŠóľŠ óľŠóľŠđ˘(đĄđ+1 ) â đ˘âđ+1 óľŠóľŠóľŠ âź O (â2âđź ) , óľŠ0,đź óľŠ óľŠóľŠ óľŠ 2âđź óľŠóľŠđ˘(đĄđ+1 ) â đ˘âđ+1 óľŠóľŠóľŠ óľŠ óľŠâ,0 âź O (â ) .
(70)
Example 1. (i) Let đ˘ (đĽ, đĄ) = đĄ2 đĽ (1 â đĽ)
(71)
8
The Scientific World Journal Table 1: The experiential error results and convergence rates of Example 1 (i).
â 1/4 1/8 1/16 1/32 1/64
âđ˘ â đ˘â â0,0 1.0569 â
10â2 4.0416 â
10â3 3.8027 â
10â4 1.0910 â
10â4 4.8599 â
10â5
âđ˘ â đ˘â ââ,0 3.1077 â
10â2 6.5469 â
10â3 4.4162 â
10â4 1.4097 â
10â4 6.4241 â
10â5
cv. rate â 1.3869 3.4098 1.8014 1.1551
cv. rate â 2.2470 3.8899 1.6474 1.1338
Table 2: The experiential error results and convergence rates of Example 1 (ii). â 1/4 1/8 1/16 1/32 1/64
âđ˘ â đ˘â â0,0 5.6283 â
10â3 1.9379 â
10â3 7.1701 â
10â4 2.6932 â
10â4 1.0362 â
10â4
then đ˘ is the exact solution to the problem đź
đđĄ đ˘ (đĽ, đĄ) â Î (đ â
đ˘ (đĽ, đĄ)) = đ (đĽ, đĄ) , đ˘ (0, đĄ) = đ˘ (1, đĄ) = 0, đ˘ (đĽ, 0) = 0,
đĽ â Ί, đĄ â [0, đ] ,
đĄ â [0, đ] , đĽ â Ί, (72)
where đź = 0.8, đ
1 = đ
2 = 1/2, đ = 1, Ί = [0, 1], đ = 1, and đ (đĽ, đĄ) = 2đĄđĽ (1 â đĽ) â đĄ2 (
đĽ
1â2đź
âđ˘ â đ˘â ââ,0 9.3921 â
10â3 3.2521 â
10â3 1.2910 â
10â3 4.7917 â
10â4 1.9584 â
10â4
cv. rate â 1.5382 1.4344 1.4128 1.3780
cv. rate â 1.5301 1.4274 1.3353 1.2909
accurate explicit scheme, leapfrog difference method in time, and the finite element method in space. Under the suitably accurate initial conditions and the stability requirement that ÎđĄ â
ââ2đź be sufficiently small, the error analysis for the fully discrete scheme is discussed, which is an đż2 -error bound of finite element accuracy and of second order in time. Numerical examples are given to demonstrate the efficiency of the theoretical results.
Conflict of Interests 1â2đź
+ (1 â đĽ) 2Î (2 â 2đź)
+
đĽ
2â2đź
2â2đź
+ (1 â đĽ) 2Î (3 â 2đź)
).
The authors declare that there is no conflict of interests regarding the publication of this paper.
(73) The experiential error results and convergence rates are presented in Table 1. (ii) Let đ˘ (đĽ, đĄ) = đâđĄ đĽ (đĽ + 1)
(74)
be the exact solution to the problem đđĄ đ˘ (đĽ, đĄ) â Îđź (đ â
đ˘ (đĽ, đĄ)) = đ (đĽ, đĄ) , đ˘ (0, đĄ) = 0, đ˘ (1, đĄ) = 2đâđĄ đ˘ (đĽ, 0) = đĽ (đĽ + 1) ,
đĽ â Ί, đĄ â [0, đ] , đĄ â [0, đ] ,
Acknowledgments This work was partially supported by the Funding Scheme for Training Young Teachers in Shanghai Colleges under Grant no. zzhg12001, the National Natural Science Foundation of China under Grant no. 11301333, the Innovation Program of Shanghai Municipal Education Commission under Grant no. 14YZ165, and Anhui Natural Science Foundation under Grant no. 1408085MA14.
References
đĽ â Ί, (75)
where đź = 0.6, đ
1 = đ
2 = 1/2, đ = 1, Ί = [0, 1], đ = 1, and đ(đĽ, đĄ) is numerically obtained. The experiential error results and convergence rates are displayed in Table 2.
5. Conclusion In this paper, we study the finite element method for fractional diffusion equation. We use the simple, second order
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