Gen. Math. Notes, Vol. 20, No. 2, February 2014, pp.1-11
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Solution of General Fractional
Oscillation Relaxation Equation by
Adomian’s Method
Fitiavana Anjara1 and Joëlson Solofoniaina2
1
Département de Mathématiques et d’Informatique
Facutlé des Sciences, Université d’Antananarivo
Ambohitsaina, Antananarivo, Madagascar
E-mail: mr.njarxa@gmail.com
2
Département de Mathématiques et d’Informatique
Facutlé des Sciences, Université d’Antananarivo
Ambohitsaina, Antananarivo, Madagascar
E-mail: jsolofo@univ-antananarivo.mg
(Received: 17-7-13 / Accepted: 25-12-13)
Abstract
We show the efficiency of Adomian decomposition method to deal with the
General Fractional Oscillation Relaxation Equation, a generalization of oscillation and relaxation equations, under nonhomogeneous initial value conditions.
The analytical solution is obtained in compact and elegant forms in terms of
generalized Mittag-Leffler functions.
Keywords: Bagley-Torvik equation, Basset problem, Caputo fractional
derivative, Fractional decomposition method, Mittag-Leffler function.
1
Introduction
Fractional calculus has been the subject of intensive research since its first
international conference in 1974. Nowadays, it founds numerous applications
in different areas of applied sciences and engineering especially by the introduction of fractional differential equations which allow a better description of
nonhomogenous natural phenomena.
2
Fitiavana Anjara et al.
Whereas solving these kinds of equations is difficult by classical methods
like Laplace transform method. In recent times, several new techniques including analytical decomposition [15] have been proposed to obtain analytical
or approximate analytical solution of fractional differential equations.
In this paper, we adopt the Adomian’s method to solve a more general
2-order fractional differential equation, the so called General Fractional Oscillation Relaxation Equation. We get its solution in terms of series of generalized
Mittag-Leffler functions.
The outline of this work is as follow. We begin in Section 2 by giving
some useful notions related to Fractional calculus and Adomian decomposition
method (ADM). In Section 3, the fractional oscillation relaxation equation is
solved by using ADM. Section 4 is devoted to numerical illustrations of the
second order case such the resolution of Basset problem and Bagley-Torvik
equation. Concluding remarks are given in Section 5.
2
Preliminaries
In this section, we recall some necessary results relative to Fractional calculus
and to Adomian’s Method.
2.1
Fractional Calculus
Let α > 0 be an arbitrary real and f (t) a sufficiently well-behaved function.
Following [10], the Riemann-Liouville fractional integral of order α is defined
by
1 Zt
(t − τ )α − 1 f (τ ) dτ, t > 0,
(1)
J α f (t) =
Γ(α) 0
its left inverse, the Riemann-Liouville fractional derivative of same order in
the form of
Dα f (t) = Dm J m−α f (t)

1
dm Z t
f (τ )



dτ, m − 1 < α < m


m
 Γ(m − α) dt
0 (t − τ )α+1−m
=



dm



f (t),
α=m
dtm
(2)
(3)
and the Caputo fractional derivative of order α by
D∗α f (t) = J m−α Dm f (t)
(4)
3
Solution of General Fractional...
=













Z t
f (m) (τ )
1
dτ, m − 1 < α < m
Γ(m − α) 0 (t − τ )α+1−m
(5)
m
d
f (t),
dtm
α = m ∈ IN.
For the Caputo derivative, we have the following composition rule with the
Riemann-Liouville fractional integral for β > α > 0 and m − 1 < α ≤ m
J β [D∗α f (t)] = J β−α f (t) −
m−1
X
p=0
f (p) (0)
tβ−α+p
.
Γ(β − α + p + 1)
(6)
These results are helpful to deal with fractional differential equations by
Adomian decomposition method. The solution of such equations often involves
special functions like Mittag-Leffler type functions. We would recall the definition of the Mittag-Leffler function, for α ∈ C,
l <(α) > 0, β ∈ C,
l <(β) > 0,
γ ∈ C,
l <(γ) > 0,
∞
X
zk
.
(7)
Eα (z) =
k=0 Γ(αk + 1)
Wiman [16] introduced a generalization of the Mittag-Leffler function in the
general form,
∞
X
zk
.
(8)
Eα, β (z) =
k=0 Γ(αk + β)
Another generalization of (7) was proposed by Prabhakar [14] in the form,
γ
(z) =
Eα,β
∞
X
(γ)k z k
,
k=0 k! Γ(αk + β)
(9)
where Γ(z), z ∈ C,
l is the Gamma Euler function and (γ)k the Pochhammer
symbol defined by
(γ)0 = 1, (γ)k = γ(γ + 1)...(γ + k − 1) =
Γ(γ + k)
, γ 6= 0.
Γ(γ)
The generalized Mittag-Leffler function (9) is an entire function of order
h
i−ρ
ρ = [<(α)]−1 and type σ = ρ1 {<(α)}<(α) . For some particular values
of parameters, we have
1
1
Eα (z) = Eα,1
(z), Eα,β (z) = Eα,β
(z),
(10)
α
φ(α, β; z) = 1 F1 (α; β; z) = Γ(β)E1,β
(z)
(11)
where φ(α, β; z) is the Kummer’s confluent hypergeometric function.
4
2.2
Fitiavana Anjara et al.
Adomian Decomposition Method
The technique presented by Adomian [4, 5, 6] consists of splitting the given
equation into linear and nonlinear parts. Then the solution is decomposed in a
series of functions where the nonlinear contribution is obtained in the form of
“Adomian’s polynomials” from its expansion into power series. It was proven
that the series solution converge accurately [7, 9, 1, 2, 11].
To illustrate the method, we consider the following general nonlinear system
Lu(t) + Ru(t) + N u(t) = g(t)
u(0)
= u0
(12)
where L is the highest order derivative which assumed to be invertible, R the
remaining linear part, N represents a nonlinear operator and g a well-behaved
function.
Applying the inverse operator L−1 to both side of (12), we have
u(t) = f (t) − L−1 Ru(t) − L−1 N u(t)
(13)
where f (t) = u0 + L−1 g(t)
The next step is to introduce the series form of the general solution and of
the nonlinear operator into eq.(13),
u=
∞
X
un
n=0
and
Nu =
∞
X
An
(14)
n=0
The polynomials (An ) in (u1 , . . . , un ) are the Adomian’s polynomials generated by
"
!#
∞
X
1 dn
i
N
λ ui
(15)
An =
n! dλn
i=0
λ=0
Therefore, by identification, we obtain the successive terms of the series
solution by the following recurrent relation
3
u0
=
f (t)
−1
un+1 = −L (Run ) − L−1 (An )
(16)
General Solution
Fractional differential equations are main application of the Fractional calculus.
These types of equations appear frequently in many physical and technical
areas [13].
In this section, we deal with 2-order fractional differential equation by the
Adomian decomposition method. These kind of equations are called general
5
Solution of General Fractional...
fractional oscillation relaxation equation as a fractional generalization of oscillation and relaxation equations. We apply some basic transformation and
integration to obtain the solution in elegant form.
Let
βi ∈ IR,
β0 = 0 < β1 < β2 ≤ 2
mi −1<βi ≤mi , mi ∈ IN, i = 0, 1, 2.
Consider the following form of general fractional oscillation relaxation equation
λ D β2 u(t) + λ D β1 u(t) + λ u(t) = f (t)
0
1 ∗
2 ∗
(17)
(p) +
u (0 )p=0, 1, m2 −1 = kp ,
with D∗βi u(t) denotes the Caputo fractional derivative of order βi of the field
variable u(t), λi and kp are real constants, λ2 6= 0.
For the fractional Adomian’s method, we choose for the linear operator
the Caputo fractional derivative of order β2 : L = D∗β2 , and the inverse the
Riemann-Liouville fractional integral of same order: L−1 = J β2 .
In virtue of the composition rule (6), applying L−1 to both sides of the
Eq.(17) leads

λ2 u(t) −
mX
2 −1
p=0
mX
1 −1
p=0

tp
 + λ1 J β2 −β1 u(t)−
kp
Γ(p + 1)
"
tβ2 −β1 +p
kp
+ λ0 J β2 u(t) = J β2 f (t).
Γ(β2 − βi + p + 1)
#
(18)
Then,
1
i −1
X
tβ2 −βi +p
1 β2
λi β2 −βi
λi mX
kp
J
u(t) (19)
+ J f (t) −
u(t) =
Γ(β2 − βi + p + 1) λ2
i=0 λ2
i=1 λ2 p=0
2
X
Setting u(t) =
P∞
u0
un+1
n=0
un and by identification, we get the iteration process
i −1
λi mX
tβ2 −βi +p
1
kp
+ J β2 f (t)
Γ(β2 − βi + p + 1) λ2
i=1 λ2 p=0
!
λ1 β2 −β1 λ0 β2
= −
J
+ J
(un ).
λ2
λ2
=
2
X
(20)
Thus,
!
u1
λ1 β2 −β1 λ0 β2
= −
J
+ J
(u0 )
λ2
λ2
u2 =
..
.
λ1 β2 −β1 λ0 β2
J
+ J
λ2
λ2
q
uq = (−1)
!2
(u0 )
λ1 β2 −β1 λ0 β2
J
+ J
λ2
λ2
(21)
!q
(u0 )
6
Fitiavana Anjara et al.
Applying the multinomial theorem [3] yields
uq = (−1)q
X
r+s=q
r≥0, s≥0
q!
r!s!
λ0
λ2
!r
λ1
λ2
!s
J (β2 −β1 )s+β2 r (u0 )
(22)
Reconstituting the decomposition series, the exact solution of Eq.(17) reads
u(t) =
=
∞
X
uq
q=0
∞
X
(−1)
q
q=0
X
r+s=q
r≥0, s≥0
q!
r!s!
λ0
λ2
!r
λ1
λ2
!s
J (β2 −β1 )s+β2 r (u0 )
(23)
Next, we substitute the value of u0 (20) and have
∞
i −1
X
X
q!
λi mX
kp (−1)q
u(t) =
r+s=q r!s!
q=0
i=1 λ2 p=0
2
X
r≥0, s≥0
∞
X
1 X
q!
+
(−1)q
λ2 q=0
r+s=q r!s!
λ0
λ2
!r
λ1
λ2
λ0
λ2
!r
λ1
λ2
!s
t ζi,p
Γ(ξi,p )
(24)
!s
J
γ+β2 r
f (t)
r≥0, s≥0
where γ = (β2 − β1 )s + β2 , ζi,p = γ + β2 r − βi + p and ξi,p = ζi,p + 1.
According to the definition of the Riemann-Liouville fractional integral, we
can write
!s
∞
∞
i −1
X
X
(r + s)! λ1
λi mX
kp (−1)s
tγ−βi +p ×
u(t) =
λ
r!s!
λ
2
2
p=0
s=0
r=0
i=1 !
)
r
∞
∞
X
λ0
tβ2 r
1 X
(r + s)!
−
+
(−1)s
λ!2 Γ(γ !
+ β(2 r − βi + p + 1)
λ2 s=0
r=0 ) r!s!
s
r
Z t
λ1
λ0
1
−
(t − τ )γ+β2 r−1 f (τ )dτ .
λ2
λ2
Γ(γ + β2 r) 0
2
X
(
(25)
Therefore
s
∞
∞
i −1
X
X
(r + s)!
λi mX
s λ1
u(t) =
kp (−1)
tγ−βi +p
×
λ2 p=0 s=0
λ2
r! s!
r=0
i=1
"
#r
!)
!s
∞
λ0
tβ2 r
1 Z tX
s λ1
−
+
(−1)
λ2 Γ(γ + β2 r − βi + p + 1)
λ2 0 s=0
λ2
2
X
(
γ−1
(t − τ )
!
(∞
X
"
(r + s)!
λ0
−
λ2
r=0 r! s!
#r
(t − τ )β2 r
f (τ )dτ.
Γ(γ + β2 r)
)
(26)
7
Solution of General Fractional...
So
!s
∞
i −1
X
λ1
λi mX
λ0
u(t) =
kp (−1)s
tγ−βi +p Eβs+1
− tβ2
2 , γ−βi +p+1
λ2
λ2
p=0
s=0 !
i=1 λ2
!
s
Z tX
∞
1
λ0
λ1
+
− {t − τ }β2 f (τ )dτ
(−1)s
(t − τ )γ−1 Eβs+1
2, γ
λ2 0 s=0
λ2
λ2
2
X
=
mX
2 −1
(
∞
X
!s
λ1
−
λ2
kp
p=0
s=0
∞
1 −1
X
λ1 mX
+
kp
λ2 p=0
s=0
Z tX
∞
t
λ1
−
λ2
λ1
−
λ2
!)
(β2 −β1 )s+p
Eβs+1
2 , (β2 −β1 )s+p+1
λ0
− tβ2
λ2
!
!s
(β2 −β1 )(s+1)+p
t
Eβs+1
2 , (β2 −β1 )(s+1)+p+1
!s
λ0
− tβ2
λ2
!
!
λ0
− τ β2 f (t − τ )dτ
τ
λ2
0 s=0
(27)
ρ
where Eα,
(z)
is
the
generalized
Mittag-Leffler
function
(9).
β
Finally, introducing the fractional Green’s function G2 (t)
1
+
λ2
(β2 −β1 )s+β2 −1
∞
λ1
1 X
−
G2 (t) =
λ2 s=0
λ2
Eβs+1
2 , (β2 −β1 )s+β2
!s
!
(β2 −β1 )s+β2 −1
t
Eβs+1
2 , (β2 −β1 )s+β2
λ0
− tβ2 .
λ2
(28)
we obtain an elegant form of the analytical solution of eq.(17)
u(t) =
2
X
i=1
λi
mX
i −1
kp J 1+p Dβi {G2 (t)} + G2 ∗ f (t)
(29)
p=0
with J α , Dα and h∗g(t) denote respectively the Riemann-Liouvile fractional
integral of order α, the Riemann-Liouvile fractional derivative of order α and
the Laplace convolution defined by
h ∗ g(t) =
Z t
f (t − τ )g(τ )dτ
(30)
0
We summarize above results by the following theorem,
Theorem 3.1 Let 0 < β1 < β2 ≤ 2, mi − 1 < βi ≤ mi , mi ∈ IN, λi and kp
be real constants, λ2 6= 0, i = 0, 1, 2.
The solution of the initial value problem for the general fractional oscillation
relaxation equation
λ2 D∗β2 + λ1 D∗β1 u(t) + λ0 u(t) = f (t)
u(p) (0+ )p=0, 1, m2 −1 = kp
(31)
8
Fitiavana Anjara et al.
writes
u(t) =
mX
2 −1
∞
X
λ1
−
λ2
kp
p=0
s=0
∞
1 −1
X
λ1 mX
+
kp
λ2 p=0
s=0
Z tX
∞
1
+
λ2
=
2
X
i=1
0 s=0
λi
mX
i −1
λ1
−
λ2
!s
t
λ1
−
λ2
(β2 −β1 )s+p
Eβs+1
2 , (β2 −β1 )s+p+1
λ0
− tβ2
λ2
!
!s
(β2 −β1 )(s+1)+p
t
Eβs+1
2 , (β2 −β1 )(s+1)+p+1
!s
λ0
− tβ2
λ2
!
!
τ
(β2 −β1 )s+β2 −1
Eβs+1
2 , (β2 −β1 )s+β2
λ0
− τ β2 f (t − τ )dτ
λ2
kp J 1+p Dβi {G2 (t)} + G2 ∗ f (t)
p=0
(32)
where
∞
1 X
λ1
G2 (t) =
−
λ2 s=0
λ2
4
!s
!
(β2 −β1 )s+β2 −1
t
Eβs+1
2 , (β2 −β1 )s+β2
λ0
− tβ2 .
λ2
(33)
Illustrations
For the applications, we study below the Basset problem [12] and the BagleyTorvik equation [8], two special cases of the general fractional relaxation oscillation equation.
4.1
Basset Problem
This classical problem of Fluid dynamics concerns the unsteady motion of a
spherical particle accelerating in a viscous fluid under the action of the gravity.
The motion is governed by the composite fractional relaxation equation [12, 10]
du
+ aD∗α u(t) + u(t) = 1
dt
(34)
9
with 0 < α < 1, a = β α > 0, β = 1+2χ
and χ = ρρfp . β and χ are related to the
densities ρf , ρp of the fluid and particle.
For the application of Adomian’s method, we consider the generalized Basset problem where α = 34 , χ = 34 , u(0) = 1 namely
3
du
9
+
dt
2
4
3
D∗4 u(t) + u(t) = 1
(35)
9
Solution of General Fractional...
By the theorem 3.1, we have
u(t) =
3 X
9 4 ∞
2
s=0
∞ X
9
−
2
s=0
9
−
2
3s
4
t
1
s
4
3s
4
1
t 4 (s+1) E1,s+1
(−t)+
1
(s+1)+1
4
E1,s+1
(−t)
1
s+1
4
+
Z tX
∞ 9
−
2
0 s=0
3s
4
1
τ 4 s E1,s+1
(−τ )dτ.
1
s+1
4
After simple calculations, we get
3
∞ X
3
9
9 4 s 1 s s+1
t 4 E1, 1 s+1 (−t) +
u(t) =
−
4
2
2
s=0
3
∞
s
X
9 4 1 s s+1
+t
−
t 4 E1, 1 s+2 (−t).
4
2
s=0
4
t
1
4
∞ X
s=0
9
−
2
3s
4
1
(−t)
t 4 s E1,s+1
1
s+ 5
4
4
(36)
4.2
Bagley-Torvik Equation
The Bagley-Torvik equation arises in the modelling of the motion of a rigid
plate immersed in a Newtonian fluid [8]. It is a composite fractional oscillation
equation [10]
3
d2 u
(37)
λ2 2 + λ1 D∗2 u(t) + λ0 u(t) = f (t).
dt
For numerical application, we set λ2 = λ1 = λ0 = 1, u(0) = u0 (0) = 1 and
f (t) = 1 + t namely
3
d2 u
2
+
D
u(t) + u(t) = 1 + t.
∗
dt2
Applying the theorem 3.1 holds
u(t) =
∞
X
1
2
s=0
+
+
∞
X
∞
X
3
(−1)s t 2 (s+1) E2,s+1
(−t2 ) + t 2
1
(s+1)+1
1
s
2
(−1)s t E2,s+1
(−t2 ) + t
1
s+1
s=0
Z tX
∞
2
∞
X
(38)
1
(−1)s t 2 s E2,s+1
(−t2 )
1
(s+1)+2
2
s=0
1
s
2
(−1)s t E2,s+1
(−t2 )
1
s+2
2
s=0
1
(−1)s τ 2 s+1 E2,s+1
(−τ 2 )(1 + t − τ )dτ,
1
s+2
0 s=0
2
hence,
u(t) =
∞
X
s=0
∞
X
+t
2
1
s
2
(−1)s t E2,s+1
(−t2 ) + t
1
s+2
s=0
∞
X
2
+t
1
1
(−1)s t 2 s E2,s+1
(−t2 ) + t 2
1
s+1
2
∞
X
s=0
∞
X
3
2
2
1
2
2
1
s
2
(−1)s t E2,s+1
(−t2 )
1
s+ 5
s=0
∞
X
3
(−1)s t 2 s E2,s+1
(−t2 ) + t
1
s+3
s=0
1
(−1)s t 2 s E2,s+1
(−t2 )
1
s+ 3
2
1
2
(−1)s t 2 s E2,s+1
(−t2 ).
1
s+4
s=0
2
(39)
10
5
Fitiavana Anjara et al.
Conclusion
In this paper, we have applied the so-called Adomian’s method for solving
the general fractional oscillation relaxation equations. We get the same general exact solution as the Laplace transform technique in terms of generalized
Mittag-Leffler functions. All results prove the effectiveness of the Adomian
decomposition method to deal with fractional differential equations.
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Solution of General Fractional...
11
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