The fractional transport equation: an analytical
solution and a spectral approximation
by Chebyshev polynomials
Abdelouahab Kadem
Abstract. In this paper, we show that the fractional transport equation
can be reduced to a fractional linear differential equation system by using
Chebyshev polynomials. We give the analytical solution of this system
followed by the spectral approximation.
M.S.C. 2000: 65D32, 82D75, 40A10, 41A50.
Key words: Linear transport equation, isotropic scattering, Chebyshev spectral
method, discrete-ordinates method.
1
Introduction
The fractional derivatives technique has been widely employed for solving linear differential and integral equations, among them is cited the diffusion problems in curvilinear
coordinates [7, 12].
In our recent work we have presented a new method for solving a steady fractional transport equation in two dimensional case using the orthogonal properties of
Chebyshev polynomials and Walsh functions, in ordinary case a new approximation
for solving the one dimensional transport equation analytically, have been reported
where we are using the Chebyshev polynomials combined with the Sumudu transform
[4]. The approach is based on expansion of the angular flux in a truncated series of
Chebyshev polynomials in the angular variable. By replacing this development in
the transport equation, this which will result a first-order linear differential system
is solved for the spatial function coefficients by application of the Sumudu transform technique [11]. The inversion of the transformed coefficients is obtained using
Trzaska’s method [10] and the Heaviside expansion technique.
In order to obtain a new rule for the calculate the matrix exponential which arises
in the formal solution of algebraic systems of differential equation without using the
Sumudu transform and Trzaska’s method for solving the first-order fractional linear
differential system, we fractionalize the one dimensional integro-differential equation
and we try to convert it into a system of fractional differential equation (FDE). The
main characteristic of using this technique is that reduces this problems to those of
Applied Sciences, Vol.11, 2009, pp.
78-90.
c Balkan Society of Geometers, Geometry Balkan Press 2009.
°
The fractional transport equation
79
solving a system of algebraic equations, thus greatly simplifying the problem and
making it computational plausible on the other hand that permit us to solve some of
the particular cases and then we can check that the solution is close to the dynamics of
some anomalous processes, the other aspect of this technique, it allows us to establish
a fractional derivative which performs the same mapping of a given linear operator,
it becomes to use the Riemann-Liouville definition for fractional derivatives and considerate the ordinary model and look that in the limit of some situations where the
ordinary model do not work fine it is necessary to introduce such fractional operators
in the model so we solve the problem.
The paper has been organized as follows. Section 2 contains preliminaries, and
Section 3 describes how to convert a transport equation into FDE in Section 4 we
report a specification application of the method.
Let us consider the following mono-energetic 3 − D transport equation:
Z
1
(1.1)
Ω.∇(r, Ω) + σt Ψ(r, Ω) =
σs (Ω, Ω0 )Ψ(r, Ω0 )dΩ0 +
Q(r)
4π
4π
Ω = (η, ξ)
(1.2)
angular variable,
and
(1.3)
σs (µ0 ) =
∞
X
2k + 1
k=0
4π
σsk Pk (µ0 ) differential cattering cross section
with µ0 = Ω.Ω0 and Pk = the k th Legendre polynomial.
2
Preliminaries
We enlist some definitions and basic results [5, 6, 9]
Definition 2.1. A real function f (x), x > 0 is said to be in the space
Cα,α∈R if there exists a real number p(> α), such that f (x) = xp f1 (x)
where f1 (x) = C [0, ∞) . Clearly Cα ⊂ Cβ if β ≤ α.
Definition 2.2. A function f (x), x > 0 is said to be in space Cαm ,
m ∈ N ∪ {0} , if f(m) ∈ Cα .
Definition 2.3. The (left sided) Riemann-Liouville fractional integral of order
µ > 0, of a function f ∈ Cα , α ≥ 1 is defined as:
Z t
1
µ
(2.1)
I f (t) =
(t − τ )µ−1 f (τ )dτ, µ > 0, t > 0,
Γ(µ) 0
I 0 f (t) = f (t)
Definition 2.4. The (left sided) Riemann-Liouville fractional derivative
m
of f, f ∈ C−1
, m ∈ N ∪ {0} of order α > 0, is defined as:
(2.2)
Dµ f (t) =
dm m−µ
I
f (t), m − 1 < µ ≤ m, m ∈ N.
dtm
80
Abdelouahab Kadem
Definition 2.5. The (left sided) Caputo fractional derivative of f,
m
f ∈ C−1
, m ∈ N ∪ {0} of order α > 0, is defined as:
½ £ m−µ (m) ¤
I
f (t) , m − 1 < µ ≤ m, m ∈ N,
(2.3)
Dcµ f (t) =
dm
µ = m.
dtm f (t)
Note that
(i) I µ tγ =
Γ(γ+1) γ+µ
,
Γ(γ+µ+1) t
P
µ > 0, γ > −1, t > 0.
k
m−1 (k)
(ii) I Dcµ f (t) = f (t)
f (0+ ) tk! , m´−
³ − k=0
Pm−1
k
(iii) Dcµ f (t) = Dµ f (t) − k=0 f (k) (0+ ) tk! ,
 α−β
µ
1 < µ ≤ m, m ∈ N.
m − 1 < µ ≤ m, m ∈ N.
f (t) if α > β,
f (t) if α = β,
 β−α
D
f (t) if α < β,
(v) Dcα Dm f (t) = Dα+m f (t), m = 0, 1, 2, ..., n − 1 < α < n.
From here on, we will use Cγ ([a, b]) (γ ∈ R) to denote the Banach space
©
ª
(2.4)
Cγ ([a, b]) = g(x) ∈ C((a, b]) :k g kCγ =k (x − a)γ g(x) kC < ∞ .
(iv) Dβ I α f (t) =
 I
In particular, C0 ([a, b]) represents the space of continuous functions in
[a, b], that is, C([a, b]).
3
Planar Geometry
We consider a planar-geometry problem with spatial variation only in the x−direction:
(3.1)
Q(r) = q(x),
(3.2)
Ψ(r, Ω) =
1
Ψ(x, µ)
2π
equation (1) simplifies to
(3.3)
∂Ψ
(x, µ) + σt Ψ(x, µ) =
µ
∂x
Z
1
σs (µ, µ0 )Ψ(x, µ0 )dµ0 +
−1
q(x)
,
2
with
(3.4)
σs (µ, µ0 ) =
∞
X
2k + 1
k=0
2
σsk Pk (µ)Pk (µ0 ).
So we consider equation (3.3) with 0 ≤ x ≤ a and −1 ≤ µ ≤ 1, and subject to the
boundary conditions
(3.5)
Ψ(x = 0, −µ) = f (µ),
and
(3.6)
Ψ(x = a, µ) = 0,
The fractional transport equation
81
where f (µ) is the prescribed incident flux at x = 0; Ψ(x, µ) is the angular flux in the
µ direction; σt is the total cross section; σsl , with l = 0, 1, ..., L are the components
of the differential scattering cross section, and Pk (µ) are the Legendre polynomials of
degree k. Now we fractionalize the equation (3.3) with the same boundary conditions
(3.5) and (3.6)
(3.7)
µ
∂β Ψ
(x, µ) + σt Ψ(x, µ) =
∂xβ
Z
1
σs (µ, µ0 )Ψ(x, µ0 )dµ0 +
−1
q(x)
,
2
with 0 < β ≤ 1.
Theorem 3.1. Consider the integro−dif f erential equation (3.7) subject to the
boundary conditions (3.5) and (3.6), then the f unction Ψ(x, µ) satisf y the f ollow
f irst − order linear dif f erential equation system f or the spatial component Yn (x)
N
X
n=0
1
αn,m
Yn(β) (x) +
L
N
X
X
σt π
2l + 1
q(x)
2
3
σsl αm,l
αn,l
Yn (x) +
Ym (x) =
2 − δm,0
2
2
n=0
l=0
where
Z
1
αn,m
1
Tm (µ)
µTn (µ) p
dµ,
1 − µ2
−1
Z 1
:=
Tn (µ)Pl (µ)dµ,
:=
2
αn,l
Z
3
:=
αn,l
−1
1
−1
Tn (µ)Pl (µ)
p
dµ,
1 − µ2
and Ym (x) are the coef f icients of the expansion of the Ψ(x, µ).
To prepare for the proof of the theorem we need the following result
Proposition 3.2. Let
Tn+1 (x) − 2xTn (x) + Tn−1 (x) = 0
and
Pl+1 (x) = 2xPl (x) − Pl−1 (x) − [xPl (x) − Pl−1 (x)] /(l + 1)
the recurrence relations f or the Chebyshev and the Legendre polynomials respectively
we have f or l > 2 and k = 2, 3
k
αn,l+1
:=
¤
2l + 1 £ k
l
k
αk
αn+1,l + αn−1,l
−
2l + 2
l + 1 n,j−1
3
2
assume the
and αn,l
Hence, in particular f or l = 0 and 1 the coef f icients αn,l
values
½
0
if
n + l odd,
2
αn,l
=
2
if
n
+ l even,
2
2
(1+l) −n
and
3
αn,l
=
πδn,l
2 − δl,0
82
Abdelouahab Kadem
P roof . It easy to see that
1
αn,m
=
πδ|n−m|
2(2 − δn+m,1 )
For k = 2 by the multiplication of the Chebyshev and the Legendre recurrence formulas we have
l
2l + 1
[Pl (µ)Tn+1 (µ) + Pl (µ)Tn−1 (µ)] −
Pl−1 (µ) [Tn+1 (µ) + Tn−1 (µ)]
2l + 2
2µ (l + 1)
it is known that
Tn+1 (µ) + Tn−1 (µ) = 2µTn (µ)
after doing some algebraic manipulations and integrating over µ ∈ [−1, 1] on the
resulting equation we get
2
αn,l+1
=
¤
2l + 1 £ 2
l
2
αn+1,l + αn−1,l
−
α2
2l + 2
l + 1 n,j−1
The case k = 3 is treated similarly but in this case we multiply the resulting expression
by √ 1 2 and integrate over µ ∈ [−1, 1] we get the desired result.
1−µ
Now we give a proof of Theorem 3.1.
P roof . Expanding the angular flux in the µ variable in terms of the Chebyshev
PN
(x)Tn (µ)
polynomials leads to Ψ(x, µ) = n=0 Yn√
with
2
1−µ
N = 0, 2, 4, ..., where the expansions coefficients Yn (x) should be determined. Here
Tn (µ) are the well known Chebyshev polynomials of order
√ n which are orthogonal in
the interval [−1, 1] with respect to the weight w(t) = 1/ 1 − t2 . After replacing this
ansatz into equation (3.7) it turns out
N n
o T (µ)
X
n
µYn(β) (x) + σt Yn (x) p
=
1
− µ2
n=0
L
X
2l + 1
(3.8)
l=0
2
σsl Pl (µ)
N
X
Z
Yn (x)
n=0
1
Tn (µ0 )
q(x)
Pl (µ0 ) p
dµ0 +
02
2
1−µ
−1
using the orthogonality of the Chebyshev polynomials, multiply the equation (3.8) by
Tm (µ), considering m = 0, 1, ..., N, and integrated in the µ variable in the interval
[−1, 1] . Thus we get the following first-order linear differential equation system for
the spatial component Yn (x)
(3.9)
N
X
n=0
1
αn,m
Yn(β) (x) +
L
N
X
X
2l + 1
σt π
q(x)
2
3
Ym (x) =
σsl αm,l
αn,l
Yn (x) +
2 − δm,0
2
2
n=0
l=0
where
Z
(3.10)
1
αn,m
=
1
Tm (µ)
µTn (µ) p
dµ,
1 − µ2
−1
The fractional transport equation
83
Z
2
αn,l
(3.11)
Tn (µ)Pl (µ)dµ,
−1
Z
3
αn,l
=
(3.12)
1
=
1
−1
Tn (µ)Pl (µ)
p
dµ,
1 − µ2
2
3
with δn,m denoting the delta of Kronecker. Here the coefficients αn,l
and αn,l
are
evaluated by the multiplication of the Chebyshev and Legendre recurrence formulas
and integration of the resulting equation (See proposition 3.2).
In the next step we solve the β-order linear differential equation system (3.9), we
rewrite this equation in matrix form
A.Y (β) (x) + BY (x) = C(x)
(3.13)
where Y (β) is the Caputo fractional derivative of order β, with 0 < β ≤ 1,
Y (x) = Col. [Y0 (x), Y1 (x), ..., YN (x)] and A is a real square matrix of order N + 1,
that is A ∈ MN +1 (R), and B ∈ C1−β ((0, x]), with the components
1
,
(A)i,j = αi−1,j−1
(3.14)
(3.15)
(B)i,j =
N
L
X
X
πσt
2l + 1
3
2
αj−1,l
δi,j −
σsl αi−1,l
2 − δ1,j
2
n=0
l=0
where each component of B belongs to space C1−β ((0, x]), and
q(x)
= Col. [C0 (x), C1 (x), ..., CN (x)] .
2
We rewrite the expression (3.14) as
(3.16)
C(x) =
Y (β) (x) + A−1 BY (x) = A−1 C(x)
(3.17)
We notice that the general solution to (3.18) where A−1 B ∈ MN +1 (R), and
A C ∈ C1−β ((0, x]), is given by [1]
Z x
A−1 B(x−ξ) −1
A−1 Bx
(3.18)
Y (x) = eβ
H+
eβ
A C(ξ)dξ,
−1
0
where H is an arbitrary constant matrix, and the β-exponential function eA
β
defined by
eβA
(3.19)
−1
Bx
= xβ−1
∞
X
(A−1 B)k
k=0
−1
−1
Bx
is
xkβ
Γ [(k + 1)β]
Moreover, it is easy to see that the function eβA B satisfies the following properties
[1]:
(i) If k A−1 B k= maxi,j | Ai,j Bi,j |, where Ai,j and Bi,j are the components of
matrix A and B respectively then
P∞
(k+1)β−1
−1
B
k eA
k ≤ k=0 k A−1 B kk xΓ[(k+1)β] (x > 0),
β
−1
Bx Kx
eβ
−1
Bx
D β eA
β
(ii) eA
β
(A−1 B+K)x
6= eβ
−1
(β 6= 1),
Bx
(iii)
= (A−1 B)eA
,
β
where A, B, K ∈ Mn (R) and β ∈ (0, 1] .
84
Abdelouahab Kadem
4
Specific application of the method
Consider the two-dimensional linear steady state transport equation given by
µ
p
∂β
∂β
Ψ(x, µ, φ) + 1 − µ2 cos φ β Ψ(x, µ, φ) + σt Ψ(x, µ, φ)
β
∂x
∂y
Z
(4.1)
Z
1
2π
=
0
0
0
0
0
0
σs (µ , φ → µ, φ)Ψ(x, µ , φ )dφ dµ + S(x, µ, φ)
−1
0
in the rectangular domain Ω = {x := (x, y): − 1 ≤ x ≤ 1, −1 ≤ y ≤ 1} , 0 < β ≤
1, and the direction in
D = {(µ, θ) : −1 ≤ µ ≤ 1, 0 ≤ θ ≤ 2π}. Here Ψ(x, µ, φ) is the angular flux,
0
0
σt and σs denote the total and the differential cross section, respectively, σs (µ , φ →
0
0
µ, φ) describes the scattering from an assumed pre-collision angular coordinates (µ , θ )
to a post-collision coordinates (µ, θ) and S is the source term.
Given the functions f1 (y, µ, φ) and f2 (x, µ, φ), describing the incident flux, we seek
for a solution of (4.1) subject to the following boundary conditions:
For 0 ≤ θ ≤ 2π, let
(
f1 (y, µ, φ), x = −1, 0 < µ ≤ 1,
(4.2)
Ψ(x = ±1, y, µ, θ) =
0,
x = 1, −1 ≤ µ < 0.
For −1 < µ < 1, let
(
(4.3)
Ψ(x, y = ±1, µ, θ) =
f2 (y, µ, φ),
y = −1, 0 < cos θ ≤ 1,
0,
y = 1, −1 ≤ cos θ < 0.
Theorem 4.1. Consider the integro − dif f erential equation (4.1) under the
boundary conditions (4.2) and (4.3), then the f unction Ψ(x, µ, θ) satisf ies the
f ollowing f irst − order f ractional linear dif f erential equation f or the spatial
component Ψi (x)
∂β
µ β Ψk (x, µ, θ) + σt Ψk (x, µ, θ)
∂x
Z 1 Z 2π
0
0
0
0
0
0
σs (µ , φ → µ, φ)Ψk (x, µ , φ )dθ dµ + Gk (x, µ, θ)
−1
0
P roof . Expanding the angular flux Ψ(x, µ, θ) in terms of the Chebyshev polynomials
in the y variable, leads to
(4.4)
Ψ(x, µ, θ) =
I
X
Ψi (x, µ, θ)Ti (y).
i=0
Below we determine the first component, i.e., Ψ0 (x, µ, θ) explicitly, whereas the other
components, Ψi (x, µ, θ), i = 1, ...I, will appear as the unknowns in I one dimensional
The fractional transport equation
85
transport equations. We start to determine Ψ0 (x, µ, θ), by inserting (4.4) into the
boundary conditions (4.3) at y = ±1, to find that:
(4.5)
Ψ0 (x, µ, θ) = f2 (x, µ, φ) −
I
X
(−1)i Ψi (x, µ, θ), 0 < cos θ ≤ 1,
i=1
(4.6)
Ψ0 (x, µ, θ) = −
I
X
Ψi (x, µ, θ),
−1 ≤ cos θ < 0.
i=1
where −1 ≤ x ≤ 1, −1 < µ < 1, and we have used the fact that for the Chebyshev
polynomials T0 (x) ≡ 0, Ti (1) ≡ 1 and Ti (−1) ≡ (−1)i .
If we now insert Ψ from (4.4) into (4.1), multiply the resulting equation by √Tk (y)2 ,
1−y
k = 1, ..., I, and integrate over y we find that the components Ψk (x, µ, θ), k = 1, ..., I,
satisfy the following I one-dimensional equations:
µ
Z
Z
1
(4.7)
2π
∂β
Ψk (x, µ, θ) + σt Ψk (x, µ, θ)
∂xβ
0
0
0
0
0
0
σs (µ , φ → µ, φ)Ψk (x, µ , φ )dθ dµ + Gk (x, µ, θ)
−1
0
The same procedure with the boundary condition (4.2) at x = −1, and (4.4) yields
(4.8)
Ψ(−1, y, µ, θ) = f1 (y, µ, φ) =
I
X
Ψi (−1, µ, θ)Ti (y).
i=0
Now multiply (4.8) by √Tk (y)2 , k = 1, ..., I, and integrate over y we find that
1−y
(4.9)
Ψk (−1, µ, θ) =
2
π
Z
1
Tk (y)
dy.
f1 (y; µ, θ) p
1 − y2
−1
Similarly, (note the sign of µ below), the boundary condition at x = 1 is written as
(4.10)
I
X
Ψi (1, −µ, θ)Ti (y) = 0
0 < µ ≤ 1.
i=0
Multiplying (4.10) by √Tk (y)2 , k = 1, ..., I and integrating over y, we get
1−y
(4.11)
Ψk (1, −µ, θ) = 0, 0 < µ ≤ 1, 0 ≤ θ ≤ 2π.
We can easily check that Gk in (4.7) is written as
(4.12)
Gk (x, µ, θ) = Sk (x, µ, θ) −
p
1 − µ2 cos θ
I
X
i=k+1
Aki Ψk (x, µ, θ)
86
Abdelouahab Kadem
where
Aki
(4.13)
2
=
π
Z
1
d
Tk (y)
(Ti (y)) p
dy
dy
1 − y2
−1
and
(4.14)
2
Sk (x, µ, θ) =
π
Z
1
Tk (y)
S(x, y, µ, θ) p
dy.
1 − y2
−1
Note that the solutions to the one-dimensional problems given through the equation
(4.7)-(4.14) define the components Ψk (x, µ, θ), for k = 1, ..., I, in this decreasing order
to avoid the coupling of the equations. Once this is done, the angular
PIflux given by
(4.4) is completely determined. Here we have used the convention i=I+1 ... = 0.
Hence the starting GI (x, µ, θ) ≡ SI (x, µ, θ). Note also that although the solution,
developed in here, rely on specific boundary conditions the procedure is quite general
in the sense that the expression for the first component, Ψ0 (x, µ, θ), keeps the information from the boundary conditions in the y variable, while the other components
are derived based on the boundary conditions in x.
Now consider the corresponding discrete ordinates equation [3]
µm
(4.15)
∂ β Ψα
(x, µm , φm ) + σt Ψα (x, µm , φm ) =
∂β x
M
X
ωn Ψα (x, µm , φm ) + Gk (x; µm , φm )
n=1
Theorem 4.2. Let 0 < β ≤ 1, Consider the f ractional integro − dif f erential
equation (4.15), then the f unction Ψα (x, µm , φm ) satisf ies the f ollowing f irst −
order f ractional linear dif f erential equation f or the spatial component χk



I
J
M p
X
X
X
∂ β χm 
µm
Aα
Bjα sin φm  χm
1 − µ2m 
+ σt −
i cos φm −
∂xβ
m=0
i=α+1
j=α+1
π
=
2
Z
1
Sα (x, µm , φm )Tl (µm )dµm .
−1
Here χk are the coef f icients of the expansion of the Ψα (x, µm , φm ).
P roof . We expand Ψα (x, µm , φm ) in a truncated series of Chebyshev polynomials
i.e.
(4.16)
Ψα (x, µm , φm ) =
M
X
χk (x, φm )Tk (µm )
p
1 − µ2m
k=0
bringing the equation (4.16) in equation (4.15) to get
"M
#
"M
#
X χk (x, φm )Tk (µm )
∂ β X χk (x, φm )Tk (µm )
p
p
µm β
+ σt
=
∂x
1 − µ2m
1 − µ2m
k=0
k=0
The fractional transport equation
M
X
(4.17)
"
ωn
n=1
87
#
M
X
χk (x, φm )Tk (µm )
p
+ Gα (x; µm , ηm )
1 − µ2m
k=0
with
Gα (x; µm , ηm ) = Sα (x, µ, φ) −
(4.18)

p
1 − µ2

M
J
M
X
X
X
χk (x, φm )Tk (µm )
χk (x, φm )Tk (µm ) 
p
p
× cos φ
Aα
+ sin φ
,
Bjα
i
2
1
−
µ
1 − µ2m
m
α=i+1
α=j+1
k=0
k=0
I
X
multiply the equation (4.17) par Tl (µm ) and integrate over µm ∈ [−1, 1] we find
µm
Z 1
Z 1
M
M
X
∂β X
Tk (µm )Tl (µm )
Tk (µm )Tl (µm )
p
p
χ
(x,
φ
)
dµ
+σ
χ
(x,
φ
)
dµm =
k
m
m
t
k
m
2
∂xβ
1
−
µ
1 − µ2m
−1
−1
m
k=0
k=0
(4.19)
M
X
ωn
n=0
M
X
Z
1
Tk (µm )Tl (µm )
p
dµm +
1 − µ2m
χk (x, φm )
−1
k=0
Z
1
Gα (x; µm , ηm )Tl (µm )dµm
−1
with
Z
Z
1
1
Gα (x; µm , ηm )Tl (µm )dµm =
Sα (x, µm , φm )Tl (µm )dµm −cos φm
−1
−1
×
M
X
Z
1
Ck (x, φm )
−1
k=0
(4.20)
×
M
X
p
1 − µ2m
i=α+1
J
X
p
Tk (µm )Tl (µm )
p
Bjα
dµm + sin φm 1 − µ2m
1 − µ2m
j=α+1
Z
1
χk (x, φm )
−1
k=0
Tk (µm )Tl (µm )
p
dµm
1 − µ2m
where
Aα
i
(4.21)
2
=
π
Bjα =
(4.22)
2
π
Z
1
−1
Z
1
−1
Z
1
−1
Z
1
−1
Ti (y)
d
(Tα (y)) p
Tl (µm )dydµm
dy
1 − y2
Tj (z)
d
(Tα (z)) √
Tl (µm )dzdµm
dz
1 − z2
by using the properties of Chebyshev polynomials to equation (4.20) to get
Z 1
Gα (x; µm , ηm )Tl (µm )dµm =
−1
Z
1
Sα (x, µm , φm )Tl (µm )dµm −
−1
I
X
hπp
2
i
1 − µ2m χk (x, φm )
Aα
i
88
Abdelouahab Kadem

×
(4.23)
I
X
J
X
Aα
i cos φm +
i=α+1

Bjα sin φm 
j=α+1
then the equation (4.15) becomes



M
I
J
X
X
X
p
∂ β χm 
µm
+ σt −
1 − µ2m 
Aα
Bjα sin φm  χm
i cos φm −
∂xβ
m=0
i=α+1
j=α+1
(4.24)
=
π
2
Z
1
Sα (x, µm , φm )Tl (µm )dµm .
−1
after written in vector and matrix notation and regrouping the coefficients χm together
in equation (4.24), we can derive the following differential equation
∂ β χm
+ DCm = Em
∂xβ
(4.25)
where Dm =
1
µm Bm
(4.26)
and Em =
Am :=
π
2
1
µm Am
Z
with
1
Sα (x, µm , φm )Tl (µm )dµm
−1


J
I
M
X
X
X
p
Bjα sin φm 
Aα
:= σt −
1 − µ2m 
i cos φm −

(4.27)
Bm
m=0
i=α+1
j=α+1
the solution of differential equation for the vector χm is thus constructed as follows
[1]
Z x
−(x−ξ)D
(4.28)
χm (x) = e−Dx
χ
(0)
+
eβ
Em (ξ)dξ
m
β
0
where
(4.29)
β−1
eDx
β =x
∞
X
Dk
k=0
xkβ
Γ [(k + 1)β]
equation (4.28) depend on vector χm (0). Having established an analytical formulation
for the exponential appearing in equation (4.29), the unknown components of vector
χm (0) for the boundary problem (4.1) can be readily obtained applying the boundary
conditions (4.2), (4.3) and (4.4).
5
Study of the spectral approximation
Now we expand
(5.1)
Ψα,N (x, φm ) =
(N )
X
m=0
)
χ(N
m cos(mφm )
The fractional transport equation
89
(N )
where χm is the approximation to the coefficient χm by the consideration of the
truncated series Ψα,N . From spectral analysis, we know that when a function is infinitely smooth and all its derivatives exist, then the coefficients appearing in its
sine or cosine series go to zero faster than 1/n. Moreover, if the function and all
its derivatives are periodic, then the decay is faster than any power of 1/n. However, as indicated by Canuto. (1988) [2], in practice this decay cannot be observed
before enough coefficients that represent the essential structures of the function are
considered.
In the calculation, one can test the convergence of the cosine truncated series
defined in equation (5.1) by evaluating
·
¸
| ΨN +1 (k) − ΨN (k) |
(5.2)
sup
≤²
ΨN (k)
k
where ² is the required precision. In general, the few first coefficients of the series are
enough to generate the angular flux.
If N is the chosen value, we can write
)
χ(N
m =0
(5.3)
for all
n > N,
Combining therefore equations (5.3) and (4.28) we shall now describe the necessary
(N )
algorithm to obtain all the cosine coefficients χm
Step 0: N = 0; for n = N = 0
Z x
(0)
−D(x−s)
−Dx (0)
(5.4)
χ0 (x) = eβ χ0 (0) −
eβ
A0 (x)dx,
0
with
Z
(5.5)
1
A0 := π
Sα (x, µ0 , φ0 )Tl (µ0 )dµ0
−1
(0)
which is well known, and thus χ0 (x) is completely determined. To finish the step,
we apply equation (5.1) to obtain the first approximation to the angular flux, i.e., Ψ0 .
Step 1: N = 1; for n = 0,
Z x
(1)
(1)
(0)
−
(5.6)
χ0 (x) = e−Dx
C
e−D(x−s) A1 (x)dx,
0
β
0
with
(5.7)
A1 :=
π
2
Z
1
Sα (x, µ1 , φ1 )Tl (µ1 )dµ1
−1
Step 2: for n = 1
Z
(5.8)
(1)
(1)
χ1 (x) = e−Dx
χ1 (0) −
β
0
x
D(x−s)
eβ
A1 (x)dx,
with
(5.9)
A1 :=
π
2
Z
1
Sα (x, µ1 , φ1 )Tl (µ1 )dµ1
−1
90
Abdelouahab Kadem
(0)
Bringing the approximated solution for χ0 obtained at step 0 inside equation (5.8)
and iterating with equation (5.4), we obtain immediately the approximated coeffi(1)
(1)
cients χ0 and χ1 . To finish the step, we evaluate through equation (5.1) the new
approximation Ψ1 and perform the precision condition defined in equation (5.2). If
equation is verified, the calculation is stopped; if not, we go to step 2 and to likewise
until the convergence condition in equation (5.2) is fulfilled.
6
Conclusions and future work
In this paper, we have shown that we can reduce a fractional transport equation. It is
interesting to mention that we can extend this procedure for higher spatial dimensions.
These will be our future investigations.
References
[1] B. Bonilla, M. Rivero, and J. J. Trujillo, On systems of linear fractional differential
equations with constant coefficients, Applied Mathematics and Computation 187, 1
(2007), 68-87.
[2] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid
Mechanics, New York: Springer, 1988.
[3] S. Chandrasekhar, Radiative Transfer, Dover, New York, 1960.
[4] A. Kadem, Solving the one-dimensional neutron transport equation using Chebyshev
polynomials and the Sumudu transform, Anal. Univ. Oradea, fasc. Math. 12 (2005),
153-171.
[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional
Differential Equations, North-Holland Mathematics Studies 204, 2006.
[6] Y. Luchko, and R. Gorenflo, An operational method for solving fractional differential
equation with the Caputo derivatives, Acta Math. Vietnamica 24, 2 (1999), 207-233.
[7] K. Oldham, and J. K. Spanier, The Fractional Calculus, Academic Press, New York.
1974.
[8] F. W. J. Olver, Asymptotic and Special Functions, Academic Press, New York, 1974.
[9] G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives:
Theory and Applications, Gordon and Breach, Yverdon, 1993.
[10] Z. Trzaska, An efficient algorithm for partial expansion of the linear matrix pencil inverse, J. of the Franklin Institute 324 (1987), 465-477.
[11] G. K. Watugala, Sumudu transform - a new integral transform to solve differential
equations and control engineering problems, Math. Engrg. Indust. 6, 4 (1988), 319-329.
[12] J. Zabadal, M. T. Vilhena, and P. Livotto, Simulation of chemical reaction using fractional derivatives, II Nuovo Cimento 116-B, 3 (2001a).
Author’s address:
Abdelouahab Kadem
L.M.F.N Mathematics Department,
University of Setif, Algeria.
E-mail: abdelouahabk@yahoo.fr