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Bhrawy and Alghamdi Boundary Value Problems 2012,
2012:62http://www.boundaryvalueproblems.com/content/2012/1/62
RESEARCH Open Access
A shifted Jacobi-Gauss-Lobatto collocationmethod for solving
nonlinear fractionalLangevin equation involving two
fractionalorders in different intervalsAli H Bhrawy1,2* and
Mohammed A Alghamdi1
*Correspondence:[email protected] of
Mathematics,Faculty of Science, King AbdulazizUniversity, Jeddah
21589, SaudiArabia2Department of Mathematics,Faculty of Science,
Beni-SuefUniversity, Beni-Suef, Egypt
AbstractIn this paper, we develop a Jacobi-Gauss-Lobatto
collocation method for solving thenonlinear fractional Langevin
equation with three-point boundary conditions. Thefractional
derivative is described in the Caputo sense. The
shiftedJacobi-Gauss-Lobatto points are used as collocation nodes.
The main characteristicbehind the Jacobi-Gauss-Lobatto collocation
approach is that it reduces such aproblem to those of solving a
system of algebraic equations. This system is written ina compact
matrix form. Through several numerical examples, we evaluate
theaccuracy and performance of the proposed method. The method is
easy toimplement and yields very accurate results.
Keywords: fractional Langevin equation; three-point boundary
conditions;collocation method; Jacobi-Gauss-Lobatto quadrature;
shifted Jacobi polynomials
1 IntroductionMany practical problems arising in science and
engineering require solving initial andboundary value problems of
fractional order differential equations (FDEs), see [, ]
andreferences therein. Several methods have also been proposed in
the literature to solveFDEs (see, for instance, [–]). Spectral
methods are relatively new approaches to pro-vide an accurate
approximation to FDEs (see, for instance, [–]).In this work, we
propose the shifted Jacobi-Gauss-Lobatto collocation (SJ-GL-C)
method to solve numerically the following nonlinear Langevin
equation involving twofractional orders in different intervals:
Dν(Dμ + λ
)u(x) = f
(x,u(x)
), < μ ≤ , < ν ≤ ,x ∈ I = [,L], ()
subject to the three-point boundary conditions
u() = s, u(x) = s, u(L) = s, x ∈ ],L[, ()
where Dνu(x) ≡ u(ν)(x) denotes the Caputo fractional derivative
of order ν for u(x), λ is areal number, s, s, s are given constants
and f is a given nonlinear source function.
© 2012 Bhrawy and Alghamdi; licensee Springer. This is an Open
Access article distributed under the terms of the Creative Com-mons
Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and repro-duction in
any medium, provided the original work is properly cited.
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The existence and uniqueness of solution of Langevin equation
involving two fractionalorders in different intervals ( < μ ≤ ,
< ν ≤ ) have been studied in [], and for otherchoices of ν and
μ, see [, ].Fractional Langevin equation is one of the basic
equations in the theory of the evolution
of physical phenomena in fluctuating environments and provides a
more flexible modelfor fractal processes as compared with the usual
ordinary Langevin equation. Moreover,fractional generalized
Langevin equation with external force is used to model
single-filediffusion. This equation has been the focus of many
studies, see, for instance, [–].Due to high order accuracy,
spectral methods have gained increasing popularity for
several decades, especially in the field of computational fluid
dynamics (see, e.g., []and the references therein). Collocation
methods have become increasingly popular forsolving differential
equations; also, they are very useful in providing highly accurate
so-lutions to nonlinear differential equations [–]. Bhrawy and
Alofi [] proposed thespectral shifted Jacobi-Gauss collocation
method to find the solution of the Lane-Emdentype equation.
Moreover, Doha et al. [] developed the shifted Jacobi-Gauss
collocationmethod for solving nonlinear high-order multi-point
boundary value problems. To thebest of our knowledge, there are no
results on Jacobi-Gauss-Lobatto collocation methodfor three-point
nonlinear Langevin equation arising in mathematical physics. This
par-tially motivated our interest in such a method.The advantage of
using Jacobi polynomials for solving differential equations is
obtaining
the solution in terms of the Jacobi parameters α and β (see
[–]). Some special casesof Jacobi parameters α and β are used for
numerically solving various types of differentialequations (see
[–]).The main concern of this paper is to extend the application of
collocation method to
solve the three-point nonlinear Langevin equation involving two
fractional orders in dif-ferent intervals. It would be very useful
to carry out a systematic study on Jacobi-Gauss-Lobatto collocation
method with general indexes (α,β > –). The fractional
Langevinequation is collocated only at (N – ) points; for suitable
collocation points, we use the(N – ) nodes of the shifted
Jacobi-Gauss-Lobatto interpolation (α,β > –). These equa-tions
together with the three-point boundary conditions generate (N + )
nonlinear alge-braic equations which can be solved usingNewton’s
iterativemethod. Finally, the accuracyof the proposed method is
demonstrated by test problems.The remainder of the paper is
organized as follows. In the next section, we introduce
some notations and summarize a few mathematical facts used in
the remainder of thepaper. In Section , the way of constructing the
Gauss-Lobatto collocation technique forfractional Langevin equation
is described using the shifted Jacobi polynomials; and in Sec-tion
the proposed method is applied to some types of Langevin equations.
Finally, someconcluding remarks are given in Section .
2 PreliminariesIn this section, we give some definitions and
properties of the fractional calculus (see, e.g.,[, , ]) and Jacobi
polynomials (see, e.g., [–]).
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Definition . The Riemann-Liouville fractional integral operator
of order μ (μ ≥ ) isdefined as
Jμf (x) =
�(μ)
∫ x(x – t)μ–f (t)dt, μ > ,x > ,
Jf (x) = f (x).()
Definition . The Caputo fractional derivative of order μ is
defined as
Dμf (x) = Jm–μDmf (x) =
�(m –μ)
∫ x(x – t)m–μ–
dm
dtmf (t)dt,
m – < μ ≤ m,x > ,()
wherem is an integer number and Dm is the classical differential
operator of orderm.
For the Caputo derivative, we have
Dμxβ =
⎧⎪⎨⎪⎩, for β ∈N and β < �μ�,
�(β + )�(β + –μ)
xβ–μ, for β ∈N and β ≥ �μ� or β /∈N and β > �μ.()
We use the ceiling function �μ� to denote the smallest integer
greater than or equal toμ and the floor function �μ to denote the
largest integer less than or equal to μ. AlsoN = {, , . . .} and N
= {, , , . . .}. Recall that for μ ∈ N , the Caputo differential
operatorcoincides with the usual differential operator of an
integer order.Let α > –, β > – and P(α,β)k (x) be the
standard Jacobi polynomial of degree k. We have
that
P(α,β)k (–x) = (–)kP(α,β)k (x), P
(α,β)k (–) =
(–)k�(k + β + )k!�(β + )
,
P(α,β)k () =�(k + α + )k!�(α + )
.()
Besides,
DmP(α,β)k (x) = –m �(m + k + α + β + )
�(k + α + β + )P(α+m,β+m)k–m (x). ()
Let w(α,β)(x) = ( – x)α( + x)β , then we define the weighted
space Lw(α,β) (–, ) as usual,equipped with the following inner
product and norm:
(u, v)w(α,β) =∫ –u(x)v(x)w(α,β)(x)dx, ‖v‖w(α,β) = (v, v)
w(α,β) .
The set of Jacobi polynomials forms a complete Lwα,β (–,
)-orthogonal system, and
∥∥P(α,β)k ∥∥w(α,β) = h(α,β)k = α+β+�(k + α + )�(k + β + )(k + α
+ β + )�(k + )�(k + α + β + ) . ()
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Let L > , then the shifted Jacobi polynomial of degree k on
the interval (,L) is definedby P(α,β)L,k (x) = P
(α,β)k (
xL – ).
By virtue of (), we have that
P(α,β)L,j () = (–)j �(j + β + )�(β + ) j!
. ()
Next, letw(α,β)L (x) = (L–x)αxβ , then we define the weighted
space Lw(α,β)L(,L) in the usual
way, with the following inner product and norm:
(u, v)w(α,β)L=
∫ L
u(x)v(x)w(α,β)L (x)dx, ‖v‖w(α,β)L = (v, v)w(α,β)L
.
The set of shifted Jacobi polynomials is a complete Lw(α,β)L
(,L)-orthogonal system. More-
over, due to (), we have
∥∥P(α,β)L,k ∥∥w(α,β)L =(L
)α+β+h(α,β)k = h
(α,β)L,k . ()
For α = β one recovers the shifted ultraspherical polynomials
(symmetric shifted Jacobipolynomials) and for α = β = ∓ , α = β = ,
the shifted Chebyshev of the first and secondkinds and shifted
Legendre polynomials respectively; and for the nonsymmetric shifted
Ja-cobi polynomials, the two important special cases α = –β = ±
(shifted Chebyshev poly-nomials of the third and fourth kinds) are
also recovered.
3 Shifted Jacobi-Gauss-Lobatto collocationmethodIn this section,
we derive the SJ-GL-C method to solve numerically the following
modelproblem:
Dν(Dμ + λ
)u(x) = f (x,u), < μ ≤ , < ν ≤ ,x ∈ I = (,L), ()
subject to the three-point boundary conditions
u() = s, u(x) = s, u(L) = s, x ∈ ],L[, ()
where Dνu(x) ≡ u(ν)(x) denotes the Caputo fractional derivative
of order ν for u(x), λ is areal number, s, s, s are given constants
and f (x,u) is a given nonlinear source function.For the existence
and uniqueness of solution of ()-(), see [].The choice of
collocation points is important for the convergence and efficiency
of the
collocation method. For boundary value problems, the
Gauss-Lobatto points are com-monly used. It should be noted that
for a differential equation with the singularity at x = in the
interval [,L] one is unable to apply the collocation method with
Jacobi-Gauss-Lobatto points because the two assigned abscissas and
L are necessary to use as a twopoints from the collocation nodes.
Also, a Jacobi-Gauss-Radau nodes with the fixed nodex = cannot be
used in this case. In fact, we use the collocation method with
Jacobi-Gauss-Lobatto nodes to treat the nonlinear Langevin
differential equation; i.e., we collo-cate this equation only at
the (N – ) Jacobi-Gauss-Lobatto points (,L). These equations
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together with three-point boundary conditions generate (N +)
nonlinear algebraic equa-tions which can be solved.Let us first
introduce some basic notation that will be used in the sequel. We
set
SN (,L) = span{P(α,β)L, (x),P
(α,β)L, (x), . . . ,P
(α,β)L,N (x)
}. ()
We next recall the Jacobi-Gauss-Lobatto interpolation. For any
positive integerN , SN (,L)stands for the set of all algebraic
polynomials of degree at most N . If we denote byx(α,β)N ,j (x
(α,β)L,N ,j), ≤ j ≤ N , and � (α,β)N ,j (� (α,β)L,N ,j ), ( ≤ i
≤ N ), to the nodes and Christoffel num-
bers of the standard (shifted) Jacobi-Gauss-Lobatto quadratures
on the intervals (–, ),(,L) respectively. Then one can easily show
that
x(α,β)L,N ,j =L(x(α,β)N ,j +
), ≤ j ≤ N ,
�(α,β)L,N ,j =
(L
)α+β+�
(α,β)N ,j , ≤ j ≤ N .
For any φ ∈ SN+(,L),
∫ L
w(α,β)L (x)φ(x)dx =(L
)α+β+ ∫ –( – x)α( + x)βφ
(L(x + )
)dx
=(L
)α+β+ N∑j=
�(α,β)N ,j φ
(L(x(α,β)N ,j +
))
=N∑j=
�(α,β)L,N ,j φ
(x(α,β)L,N ,j
).
()
We introduce the following discrete inner product and norm:
(u, v)w(α,β)L ,N=
N∑j=
u(x(α,β)L,N ,j
)v(x(α,β)L,N ,j
)�
(α,β)L,N ,j , ‖u‖w(α,β)L ,N =
√(u,u)w(α,β)L ,N
, ()
where x(α,β)L,N ,j and �(α,β)L,N ,j are the nodes and the
corresponding weights of the shifted Jacobi-
Gauss-quadrature formula on the interval (,L) respectively.Due
to (), we have
(u, v)w(α,β)L ,N= (u, v)w(α,β)L
, ∀uv ∈ SN–. ()
Thus, for any u ∈ SN (,L), the norms ‖u‖w(α,β)L ,N and
‖u‖w(α,β)L coincide.Associating with this quadrature rule, we
denote by IP
(α,β)L
N the shifted Jacobi-Gauss in-terpolation,
IP(α,β)L
N u(x(α,β)L,N ,j
)= u
(x(α,β)L,N ,j
), ≤ k ≤ N .
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The shifted Jacobi-Gauss collocation method for solving ()-() is
to seek uN (x) ∈SN (,T), such that
Dμ+νuN(x(α,β)L,N–,k
)+ λDνuN
(x(α,β)L,N–,k
)= f
(x(α,β)L,N–,k ,uN
(x(α,β)L,N–,k
)), k = , , . . . ,N – .
()
uN () = s, uN (x) = s, uN (L) = s, x ∈ ],L[. ()
We now derive an efficient algorithm for solving ()-(). Let
uN (x) =N∑j=
ajP(α,β)L,j (x), a = (a,a, . . . ,aN )T . ()
We first approximate u(x),Dμ+νu(x) and Dμu(x), as Eq. (). By
substituting these approx-imations in Eq. (), we get
N∑j=
aj(Dμ+νP(α,β)L,j (x) + λD
μP(α,β)L,j (x))= f
(x,
N∑j=
ajP(α,β)L,j (x)
). ()
Here, the fractional derivative of order μ in the Caputo sense
for the shifted Jacobi poly-nomials expanded in terms of shifted
Jacobi polynomials themselves can be representedformally in the
following theorem.
Theorem . Let P(α,β)L,j (x) be a shifted Jacobi polynomial of
degree j, then the fractionalderivative of order ν in the Caputo
sense for P(α,β)L,j (x) is given by
DνP(α,β)L,j (x) =∞∑i=
Qν(j, i,α,β)P(α,β)L,i (x), j = �ν�, �ν� + , . . . , ()
where
Qν(j, i,α,β) =j∑
k=�ν�
(–)j–kLα+β–ν+�(i + β + )�(j + β + )�(j + k + α + β + )hi�(i + α
+ β + )�(k + β + )�(j + α + β + )�(k – ν + )(j – k)!
×i∑
l=
(–)i–l�(i + l + α + β + )�(α + )�(l + k + β – ν + )�(l + β +
)�(l + k + α + β – ν + )(i – l)!l!
.
Proof This theorem can be easily proved (see Doha et al. []).In
practice, only the first (N + )-terms shifted Jacobi polynomials
are considered, with
the aid of Theorem . (Eq. ()), we obtain from () that
N∑j=
aj
( N∑i=
Qμ+ν(j, i,α,β)P(α,β)L,i (x) + λN∑i=
Qμ(j, i,α,β)P(α,β)L,i (x)
)
= f
(x,
N∑j=
ajP(α,β)L,j (x)
).
()
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Also, by substituting Eq. () in Eq. () we obtain
N∑j=
ajP(α,β)L,j () = s,
N∑j=
ajP(α,β)L,j (x) = s,
N∑j=
ajP(α,β)L,j (L) = s.
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
()
To find the solution uN (x), we first collocate Eq. () at the (N
–) shifted Jacobi-Gauss-Lobatto notes, yields
N∑j=
aj
( N∑i=
Qμ+ν(j, i,α,β)P(α,β)L,i(x(α,β)L,N–,k
)+ λ
N∑i=
Qμ(j, i,α,β)P(α,β)L,i(x(α,β)L,N–,k
))
= f
((x(α,β)L,N–,k
),
N∑j=
ajP(α,β)L,j(x(α,β)L,N–,k
)), ≤ k ≤ N – .
()
Next, Eq. (), after using () and (), can be written as
N∑j=
(–)j�(j + β + )�(β + )j!
aj = s,
N∑j=
( j∑i=
(–)j–i�(j + β + )�(j + i + α + β + )
�(i + β + )�(j + α + β + )(j – i)!i!Lixi
)aj = s,
N∑j=
( j∑i=
(–)j–i�(j + β + )�(j + i + α + β + )
�(i + β + )�(j + α + β + )(j – i)!i!
)aj = s.
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
()
The scheme ()-() can be rewritten as a compact matrix form. To
do this, we intro-duce the (N + )× (N + ) matrix A with the entries
akj as follows:
akj =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
N∑i=
Qμ+ν(j, i,α,β)P(α,β)L,i(x(α,β)L,N–,k
),
≤ k ≤ N – , �μ + ν� ≤ j ≤ N ,(–)j
�(j + β + )�(β + )j!
, k =N – , ≤ j ≤ N ,j∑
i=
(–)j–i�(j + β + )�(j + i + α + β + )
�(i + β + )�(j + α + β + )(j – i)!i!Lixi, k =N – , ≤ j ≤ N ,
j∑i=
(–)j–i�(j + β + )�(j + i + α + β + )
�(i + β + )�(j + α + β + )(j – i)!i!, k =N , ≤ j ≤ N ,
, otherwise.
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Also, we define the (N + )× (N + ) matrix B with the
entries:
bkj =
⎧⎪⎪⎨⎪⎪⎩
N∑i=
Qμ(j, i,α,β)P(α,β)L,i(x(α,β)L,N–,k
), ≤ k ≤N – , �μ� ≤ j ≤ N ,
, otherwise,
and the (N – )× (N + ) matrix C with the entries:
ckj = P(α,β)T ,j(x(α,β)T ,N–,k
), ≤ k ≤ N – , ≤ j ≤ N .
Further, let a = (a,a, . . . ,aN )T , and
F(a) =(f(x(α,β)T ,N–,,uN
(x(α,β)T ,N–,
)), . . . , f
(x(α,β)T ,N–,N–,uN
(x(α,β)T ,N–,N–
)), s, s, s
)T ,where uN (x(α,β)T ,N–,k) is the kth component of Ca. Then we
obtain from ()-() that
(A + λB)a = F(a),
or equivalently
a = (A + λB)–F(a). ()
Finally, from (), we obtain (N + ) nonlinear algebraic equations
which can be solvedfor the unknown coefficients aj by using any
standard iteration technique, like Newton’siteration method.
Consequently, uN (x) given in Eq. () can be evaluated. �
Remark . In actual computation for fixed μ, ν and λ, it is
required to compute (A +λB)– only once. This allows us to save a
significant amount of computational time.
4 Numerical resultsTo illustrate the effectiveness of the
proposedmethod in the present paper, two test exam-ples are carried
out in this section. Comparison of the results obtained by various
choicesof Jacobi parameters α and β reveal that the present method
is very effective and conve-nient for all choices of α and β .We
consider the following two examples.
Example Consider the nonlinear fractional Langevin equation
D
(D
+
)u(x) =
(tan– u(x) + cosx
), in I = (, ), ()
subject to three-point boundary conditions:
u() = , u(.) = , u() = . ()
The analytic solution for this problem is not known. In Table we
introduce the ap-proximate solution for ()-() using SJ-GL-C method
at α = β = and N = . The
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Table 1 Approximate solution of ()-() using SJ-GL-C method for N
= 12
x Approximate solution
0.1 0.008374370.2 0.01013560.3 0.008114270.4 0.004308770.5
–9.994× 10–20
x Approximate solution
0.6 –0.003646020.7 –0.005853570.8 –0.006157270.9 –0.004212871.0
6.098× 10–19
Figure 1 Comparing the approximate solutions at N = 4,6, 8, 16,
for Example 1.
approximate solutions at α = β = – and a few collocation points
(N = ,, , ) of thisproblem are depicted in Figure . The approximate
solution atN = agrees very well withthe approximate solution at N =
; this means the numerical solution converges fast asN
increases.
Example In this example we consider the following nonlinear
fractional Langevin dif-ferential equation
Dν(Dμ +
)u(x) = u(x) + eu(x) + g(x), ν ∈ (, ),μ ∈ (, ), ()
subject to the following three-point boundary conditions:
u() = , u(
)=
,
–(
)–μ–ν, u() = , ()
where
g(x) = –ex–x–xμ+ν –(x – x – xμ+ν
) + x–μ–ν�( –μ – ν)
–,x–μ–ν
�( –μ – ν)
–xμ�( + μ + ν)
�( +μ)+
(x–ν
�( – ν)–,x–ν
�( – ν)–xμ�( + μ + ν)
�( + μ)
).
The exact solution of this problem is u(x) = –xν+μ + x – x.
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Table 2 Maximum absolute error of u – uN using SJ-GL-C method
for α = β = 0
N α β ν = 1.5, μ = 0.5 ν = 1.8, μ = 0.8 ν = 1.999, μ = 0.999
8 0 0 2.09× 10–4 4.91× 10–5 1.07× 10–716 1.39× 10–5 4.02× 10–7
3.99× 10–1024 3.25× 10–6 5.87× 10–8 2.33× 10–11
Table 3 Maximum absolute error of u – uN using SJ-GL-C method
for α = β = –1/2
N α β ν = 1.5, μ = 0.5 ν = 1.8, μ = 0.8 ν = 1.999, μ = 0.999
8 –12–12 3.64× 10–4 1.15× 10–4 2.83× 10–7
16 9.66× 10–6 1.16× 10–6 1.01× 10–924 1.99× 10–6 8.35× 10–8
7.15× 10–11
Figure 2 Approximate solution for ν = 1.2, 1.4, 1.6, 1.8, 2, μ =
1 with 14 nodes and the exact solutionat ν = 2 and μ = 1, for
Example 2.
Numerical results are obtained for different choices of ν ,μ, α,
β , andN . In Tables and we introduce the maximum absolute error,
using the shifted Jacobi collocation methodbased on Gauss-Lobatto
points, with two choices of α, β , and various choices of ν , μ,and
N .The approximate solutions are evaluated for ν = ., ., ., ., , μ
= , α = β =
and N = . The results of the numerical simulations are plotted
in Figure . In Fig-ure , we plotted the approximate solutions at
fixed ν = , and various choices of μ =., ., ., ., with α = β = andN
= . It is evident fromFigure and Figure that, asν andμ approach
close to and , the numerical solution by shifted
Jacobi-Gauss-Lobattocollocation method with α = β = for fractional
order differential equation approaches tothe solution of integer
order differential equation.In the case of < ν ≤ , μ = with α =
β = , and N = , the results of the numerical
simulations are shown in Figure . In Figure , we plotted the
approximate solutions forν = , < μ ≤ with α = β = , andN = . In
fact, the approximate solutions obtained bythe present method at
< ν ≤ , < μ ≤ with N = are shown in Figure and Figure to make
it easier to show that; as ν and μ approach to their integer
values, the solution offractional order Langevin equation
approaches to the solution of integer order Langevindifferential
equation.
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Figure 3 Approximate solution for μ = 0.2, 0.4, 0.6, 0.8, 1, ν =
2 with 14 nodes and the exact solutionat ν = 2 and μ = 1, for
Example 2.
Figure 4 Approximate solution for 1 < ν ≤ 2, μ = 1 with 12
nodes, for Example 2.
5 ConclusionAn efficient and accurate numerical scheme based on
the Jacobi-Gauss-Lobatto colloca-tion spectral method is proposed
for solving the nonlinear fractional Langevin equation.The problem
is reduced to the solution of nonlinear algebraic equations.
Numerical ex-amples were given to demonstrate the validity and
applicability of themethod. The resultsshow that the SJ-GL-C method
is simple and accurate. In fact, by selecting a few colloca-tion
points, excellent numerical results are obtained.
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Figure 5 Approximate solution for 0
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doi:10.1016/j.apm.2011.12.031
doi:10.1186/1687-2770-2012-62Cite this article as: Bhrawy and
Alghamdi: A shifted Jacobi-Gauss-Lobatto collocation method for
solving nonlinearfractional Langevin equation involving two
fractional orders in different intervals. Boundary Value Problems
20122012:62.
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A shifted Jacobi-Gauss-Lobatto collocation method for solving
nonlinear fractional Langevin equation involving two fractional
orders in different intervalsAbstractKeywords
IntroductionPreliminariesShifted Jacobi-Gauss-Lobatto
collocation methodNumerical resultsConclusionCompeting
interestsAuthors' contributionsAcknowledgementsReferences