FPDE presentation

Fractional Calculus
Fractional Partial Differential Equations
Finite Difference Approximation
Finite Difference Method
for Two Sided Space Fractional
Partial Differential Equations
Divyansh Verma - SAU/AM(M)/2014/14
Ajay Gupta - SAU/AM(M)/2014/20
South Asian University
Supervisor : Prof. Siraj-ul-Islam
November 24, 2015
Divyansh Verma | Ajay Gupta FPDE

Fractional Calculus
Overview
1 Fractional Calculus
History of Frational Calculus
Applications of Fractional PDE
Objective
2 Fractional Partial Differential Equations
Riemann-Liouville Fractional Derivative
Grünwald Definition for Fractional Derivative
Shifted Grünwald Formula/Estimate
3 Finite Difference Approximation
Approximating one-sided Fractional PDE
Stability Analysis
Approximating two-sided Fractional PDE
Stability Analysis

Fractional Calculus
Mathematics is the art of giving things misleading names. The
beautiful and at first look mysterious name the ”Fractional
Calculus” is just one of those misnomers which are the essence
of mathematics.
It does not mean the calculus of fractions, neither does it mean
a fraction of any calculus - differential, integral or calculus of
variations.
The ”Fractional Calculus” is a name for the theory of
integrals and derivatives of arbitrary order, which unify and
generalize the notion of integer-order differentiation and n-fold
integration.

Fractional Calculus
Objective
History of Fractional Calculus
The origin of fractional calculus dates back to the same time as
the invention of classical calculus. Fractional calculus
generalises the concept of classical caluculus a step furthermore
by allowing non-integer order.
The idea was ﬁrst raised by Leibniz in 1695 when he wrote a
letter to L’Hospital where he said: ‘Can the meaning of
derivatives with integer order to be generalized to derivatives
with non-integer orders?’
To this L’Hospital replied with a question of his own:‘What if
the order will be 1
2?’
To this, Leibniz said:‘It will lead to a paradox, from which one
day useful consequences will be drawn.’

Fractional Calculus
Objective
Fractional PDE models are widely used in :
Image Processing (eg. reconstructing a degraded image)
Financial Modelling (eg. for solving fractional equations
such as Fractional Black-Scholes equations arising in
financial markets)
Fluid Flow (eg. for solving fractional model of Navier
Stokes equation arising in unsteady flow of a viscous fluid)
Mathematical/Computational Biology (eg. for
solving time-fractional biological population models)

Fractional Calculus
Objective
Objective
To find a convergent numerical scheme using Finite
Difference Method for solving a two sided Fractional
Partial Differential Equation numerically.
To check the stability of numerical scheme using Matrix
Analysis Method.
Conclude the important results.

Fractional Calculus
We consider Fractional Partial Differential Equation (FPDE) of
the form :-
∂u(x, t)
∂t
= c+(x, t)
∂αu(x, t)
∂+xα
+ c−(x, t)
∂αu(x, t)
∂−xα
+ s(x, t) (1)
on finite domain L < x < R , 0 ≤ t ≤ T .
Initial Condition : u(x, t = 0) = F(x), L < x < R
Boundary Condition : u(L, t = 0) = u(R, t = 0) = 0
We consider the case 1 ≤ α ≤ 2 , where parameter α is the
fractional order of the spatial derivative. The s(x, t) is the
source term. The function c+(x, t) ≥ 0 and c−(x, t) ≥ 0 may be
interpreted as transport related coefficients.

Fractional Calculus
Riemann-Liouville Fractional Derivatives
The left-handed (+) fractional derivative in (1) is deﬁned by
Dα
L+f (x) =
dαf(x)
d+xα
=
1
Γ(n − α)
dα
dxα
x
L
f(ξ)
(x − ξ)α+1−n
dξ (2)
The right-handed (−) fractional derivative in (1) is deﬁned by
Dα
R−f (x) =
dαf(x)
d−xα
=
(−1)n
Γ(n − α)
dα
dxα
R
x
f(ξ)
(ξ − x)α+1−n
dξ
(3)
Dα
L+f (x) and Dα
R−f (x) are Riemann-Liouville fractional
derivatives of order α where n is an integer such that
n − 1 < α ≤ n

Fractional Calculus
if α = m, where m is an integer, then by above deﬁnition
Dα
L+f (x) =
dmf(x)
dxm
(4)
Dα
R−f (x) = (−1)m dmf(x)
dxm
(5)
gives the standard integer derivative.

Fractional Calculus
When α = 2 and setting c(x, t) = c+(x, t) + c−(x, t), equation
(1) becomes the following classical parabolic PDE
∂u(x, t)
∂t
= c(x, t)
∂2u(x, t)
∂x2
+ s(x, t) (6)
When α = 1 and setting c(x, t) = c+(x, t) + c−(x, t), equation
(1) becomes the following classical hyperbolic PDE
∂u(x, t)
∂t
= c(x, t)
∂u(x, t)
∂x
+ s(x, t) (7)
The case 1 < α < 2 represents the super diﬀusive process where
particles diﬀuse faster than the classical model (6) predicts.

Fractional Calculus
Grünwald Discretization for Fractional Derivative
The Grünwald discretization for right-handed and left-handed
fractional derivative are respectively given as
dαf(x)
d+xα
= lim
M+→∞
1
hα
M+
k=0
gk.f(x − kh) (8)
dαf(x)
d−xα
= lim
M−→∞
1
hα
M−
k=0
gk.f(x + kh) (9)
where M+,M− are positive integers, h+ = (x−L)
M+
, h− = (R−x)
M−
grünwald weights defined by g0 = 1 and gk = Γ(k−α)
Γ(−α)Γ(k+1), where
k = 1, 2, 3...

Fractional Calculus
We define shifted Grünwald formula as
dαf(x)
d+xα
= lim
M+→∞
1
hα
M+
k=0
gk.f[x − (k − 1)h] (10)
dαf(x)
d−xα
= lim
M−→∞
1
hα
M−
k=0
gk.f[x + (k − 1)h] (11)
which defines the following shifted Grünwald estimates resp.
dαf(x)
d+xα
=
1
hα
M+
k=0
gk.f[x − (k − 1)h] + O(h) (12)
dαf(x)
d−xα
=
1
hα
M−
k=0
gk.f[x + (k − 1)h] + O(h) (13)

Fractional Calculus
Stability Analysis
Stability Analysis
Approximating Left-handed Fractional PDE
If equation (1) only contains left-handed fractional derivative,
we omit the directional sign notation and write the fractional
PDE in the following form
∂u(x, t)
∂t
= c(x, t)
∂αu(x, t)
∂xα
+ s(x, t) (14)
we assume c(x, t) ≥ 0 over domain L ≤ x ≤ R , 0 ≤ t ≤ T.
Time grid : tn = n∆t, 0 ≤ tn ≤ T
Spatial grid : ∆x = h > 0, where h = R−L
K , x = L + ih for
i = 0, ..., K, L ≤ x ≤ R.
Deﬁne un
i be the numerical approximation for u(xi, tn) and
cn
i = c(xi, tn), sn
i = s(xi, tn).

Fractional Calculus
Stability Analysis
Stability Analysis
If the above equation (14) is discretized when 1 ≤ α ≤ 2 in time
by using an explicit (Euler) scheme,
u(x, tn+1 − u(x, tn))
∆t
= c(x, tn)
∂αu(x, tn)
∂xα
+ s(x, tn) (15)
and then in space with shifted Gr¨unwald estimate the equation
(14) takes the form
un+1
i − un
i
∆t
=
cn
i
hα
i+1
k=0
gkun
i−k+1 + sn
i (16)
for i = 1, 2, ...K − 1.

Fractional Calculus
Stability Analysis
Stability Analysis
the equation can be explicitly solved for un+1
i to give
⇒ un+1
i = un
i + ∆t
cn
i
hα
i+1
k=0
gk un
i−k+1 + sn
i ∆t (17)
⇒ un+1
i = un
i + β cn
i
i+1
k=0
gk un
i−k+1 + sn
i ∆t (18)
where β = ∆t
hα
⇒ un+1
i = βcn
i g0un
i+1 + (1 + βcn
i g1)un
i + βcn
i
i+1
k=2
gk un
i−k+1 + sn
i ∆t
(19)

Fractional Calculus
Stability Analysis
Stability Analysis
Stability Analysis
Result
The explicit Euler method (19) is stable if
∆t
hα
≤
1
αcmax
where cmax is the maximum value of c(x, t) over the region
L ≤ x ≤ R, 0 ≤ t ≤ T.
We will apply a matrix stability analysis to the linear system of
equations arising from the finite difference equations defined by
(19) and will use the Greschgorin Theorem to determine a
stability condition.

Fractional Calculus
Stability Analysis
Stability Analysis
Stability Analysis
The difference equations defined by (19), together with the
Dirichlet boundary conditions, result in a linear system of
equations of the form
Un+1
= A Un
+ ∆t Sn
(20)
where Un
= [un
0 , un
1 , un
2 , ..., un
K]T , Sn
= [0, sn
0 , sn
1 , sn
2 , ..., sn
K−1, 0]T
and A is the matrix of coefficients, and is the sum of a lower
triangular matrix and a superdiagonal matrix. The matrix
entries Ai,j for i = 1, ..., K − 1 and j = 1, ..., K − 1
Ai,j = 0 , when j ≥ i + 2
= 1 + g1 β cn
i , when j = i
= gi−j+1 β cn
i , when otherwise

Fractional Calculus
Stability Analysis
Stability Analysis
Stability Analysis
Note that g1 = −α and for 1 ≤ α ≤ 2 and i = 1 we have gi ≥ 0.
Also since ∞
k=0 gi = 0, this implies that −gi ≥ k=N
k=0,k=1 gi .
According to Greschgorin Theorem, the eigenvalues of the
matrix A lie in the union of the circles centered at Ai,i with
radius ri = K
k=0,k=i Ai,k.
Here we have Ai,i = 1 + g1cn
i β = 1 − αcn
i and
ri =
K
k=0,k=i
Ai,k =
i+1
k=0,k=i
Ai,k = cn
i β
i+1
k=0,k=i
gi ≤ α cn
i β (21)
⇒ Ai,i + ri ≤ 1

Fractional Calculus
Stability Analysis
Stability Analysis
Stability Analysis
⇒ Ai,i − ri ≥ 1 − 2 α cn
i β ≥ 1 − 2 α cmax β
Therefore for the spectral radius of the matrix A to be at most
one , it suffices to have (1 − 2 α cmax β) ≥ −1, which yields the
following condition on β
β =
∆t
hα
≤
1
αcmax
(22)
Under the condition on β defined by (22) the spectal radius of
matrix A is bounded by one. Therefore the Explicit Method
defined above is unconditionally stable.

Fractional Calculus
Stability Analysis
Stability Analysis
If the equation
∂u(x, t)
∂t
= c+(x, t)
∂αu(x, t)
∂+xα
+ c−(x, t)
∂αu(x, t)
∂−xα
+ s(x, t)
is discretized in time by using an implicit scheme, and in space
with shifted Gr¨unwald estimate. The equation takes the form
un+1
i − un
i
∆t
=
1
hα
cn+1
+,i
i+1
k=0
gkun+1
i−k+1 + cn+1
−,i
K−i+1
k=0
gkun+1
i+k−1 + sn+1
i
(23)
with h = (R − L)/K for i = 1, 2, ...K − 1.

Fractional Calculus
Stability Analysis
Stability Analysis
If the equation
∂u(x, t)
∂t
= c+(x, t)
∂αu(x, t)
∂+xα
+ c−(x, t)
∂αu(x, t)
∂−xα
+ s(x, t)
is discretized in time by using an explicit scheme, and in space
with shifted Gr¨unwald estimate. The equation takes the form
un+1
i − un
i
∆t
=
1
hα
cn
+,i
i+1
k=0
gkun
i−k+1 + cn
−,i
K−i+1
k=0
gkun
i+k−1 + sn
i
(24)
with h = (R − L)/K for i = 1, 2, ...K − 1.

Fractional Calculus
Stability Analysis
Stability Analysis
Stability Analysis
Result for Implicit Method
The implicit Euler method for two-sided Fractional PDE
deﬁned by (23) with 1 ≤ α ≤ 2 is unconditionally stable.
Result for Explicit Method
The explicit Euler method (24) is stable if
∆t
hα
≤
1
α (c+,max + c−,max)
where c+,max and c−,max are the maximum value of c(x, t) from
two sides over the region L ≤ x ≤ R, 0 ≤ t ≤ T.

Fractional Calculus
Stability Analysis
Stability Analysis
Conclusion
The explicit method using shifted Grünwald estimate for
approximating one-sided Fractional PDE is conditionally
stable, consistent and hence convergent with
O(∆t) + O(∆x).
The implicit method using shifted Grünwald estimate for
approximating two-sided Fractional PDE with 1 ≤ α ≤ 2 is
unconditionally stable, consistent and hence convergent
with O(∆t) + O(∆x).
The explicit method using shifted Grünwald estimate for
approximating two-sided Fractional PDE is conditionally
stable, consistent and hence convergent with
O(∆t) + O(∆x).

Fractional Calculus
Stability Analysis
Stability Analysis
References
M.M. Meerschaert and C. Tadjeran
Finite Difference Method for Two Sided
Igor Podlubny
Fractional Differential Equations
Academic Press, 1999
Kenneth S. Miller and Bertram Ross
An Introduction to the Fractional Calculus
and Fractional Differential Equations
John Wiley and Sons, 1993

Fractional Calculus
Stability Analysis
Stability Analysis
The End

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