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Fractional CalculusFractional Partial Differential Equations

Finite Difference Approximation

Finite Difference Methodfor Two Sided Space FractionalPartial Differential Equations

Divyansh Verma - SAU/AM(M)/2014/14Ajay Gupta - SAU/AM(M)/2014/20

South Asian University

Supervisor : Prof. Siraj-ul-Islam

November 24, 2015

Divyansh Verma | Ajay Gupta FPDE



Overview

1 Fractional CalculusHistory of Frational CalculusApplications of Fractional PDEObjective

2 Fractional Partial Differential EquationsFractional Partial Differential EquationsRiemann-Liouville Fractional DerivativeGrunwald Definition for Fractional DerivativeShifted Grunwald Formula/Estimate

3 Finite Difference ApproximationApproximating one-sided Fractional PDEStability AnalysisApproximating two-sided Fractional PDEStability Analysis




Mathematics is the art of giving things misleading names. Thebeautiful and at first look mysterious name the ”FractionalCalculus” is just one of those misnomers which are the essenceof mathematics.

It does not mean the calculus of fractions, neither does it meana fraction of any calculus - differential, integral or calculus ofvariations.

The ”Fractional Calculus” is a name for the theory ofintegrals and derivatives of arbitrary order, which unify andgeneralize the notion of integer-order differentiation and n-foldintegration.




History of Frational CalculusApplications of Fractional PDEObjective

History of Fractional Calculus

The origin of fractional calculus dates back to the same time asthe invention of classical calculus. Fractional calculusgeneralises the concept of classical caluculus a step furthermoreby allowing non-integer order.

The idea was first raised by Leibniz in 1695 when he wrote aletter to L’Hospital where he said: ‘Can the meaning ofderivatives with integer order to be generalized to derivativeswith non-integer orders?’

To this L’Hospital replied with a question of his own:‘What ifthe order will be 1

2?’

To this, Leibniz said:‘It will lead to a paradox, from which oneday useful consequences will be drawn.’





Applications of Fractional PDE

Fractional PDE models are widely used in :

Image Processing (eg. reconstructing a degraded image)

Financial Modelling (eg. for solving fractional equationssuch as Fractional Black-Scholes equations arising infinancial markets)

Fluid Flow (eg. for solving fractional model of NavierStokes equation arising in unsteady flow of a viscous fluid)

Mathematical/Computational Biology (eg. forsolving time-fractional biological population models)





Objective

To find a convergent numerical scheme using FiniteDifference Method for solving a two sided FractionalPartial Differential Equation numerically.

To check the stability of numerical scheme using MatrixAnalysis Method.

Conclude the important results.




Fractional Partial Differential EquationsRiemann-Liouville Fractional DerivativeGrunwald Definition for Fractional DerivativeShifted Grunwald Formula/Estimate

Fractional Partial Differential Equations

We consider Fractional Partial Differential Equation (FPDE) ofthe 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 < RBoundary Condition : u(L, t = 0) = u(R, t = 0) = 0

We consider the case 1 ≤ α ≤ 2 , where parameter α is thefractional order of the spatial derivative. The s(x, t) is thesource term. The function c+(x, t) ≥ 0 and c−(x, t) ≥ 0 may beinterpreted as transport related coefficients.





Riemann-Liouville Fractional Derivatives

The left-handed (+) fractional derivative in (1) is defined by

(DαL+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

(DαR−f

)(x) =

dαf(x)

d−xα=

(−1)n

Γ(n− α)

dα

dxα

∫ R

x

f(ξ)

(ξ − x)α+1−n dξ

(3)(DαL+f

)(x) and

(DαR−f

)(x) are Riemann-Liouville fractional

derivatives of order α where n is an integer such thatn− 1 < α ≤ n






if α = m, where m is an integer, then by above definition

(DαL+f

)(x) =

dmf(x)

dxm(4)

(DαR−f

)(x) = (−1)m

dmf(x)

dxm(5)

gives the standard integer derivative.






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 whereparticles diffuse faster than the classical model (6) predicts.





Grunwald Discretization for Fractional Derivative

The Grunwald discretization for right-handed and left-handedfractional 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−

grunwald weights defined by g0 = 1 and gk = Γ(k−α)Γ(−α)Γ(k+1) , where

k = 1, 2, 3...





Shifted Grunwald Formula/Estimate

We define shifted Grunwald 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 Grunwald 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)




Approximating one-sided Fractional PDEStability AnalysisApproximating two-sided Fractional PDEStability Analysis

Approximating Left-handed Fractional PDE

If equation (1) only contains left-handed fractional derivative,we omit the directional sign notation and write the fractionalPDE 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 ≤ TSpatial grid : ∆x = h > 0, where h = R−L

K , x = L+ ih fori = 0, ...,K, L ≤ x ≤ R.

Define uni be the numerical approximation for u(xi, tn) andcni = c(xi, tn), sni = s(xi, tn).






If the above equation (14) is discretized when 1 ≤ α ≤ 2 in timeby 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 Grunwald estimate the equation(14) takes the form

un+1i − uni

∆t=cnihα

i+1∑k=0

gkuni−k+1 + sni (16)

for i = 1, 2, ...K − 1.






the equation can be explicitly solved for un+1i to give

⇒ un+1i = uni + ∆t

cnihα

i+1∑k=0

gk uni−k+1 + sni ∆t (17)

⇒ un+1i = uni + β cni

i+1∑k=0

gk uni−k+1 + sni ∆t (18)

where β = ∆thα

⇒ un+1i = βcni g0u

ni+1 + (1 + βcni g1)uni + βcni

i+1∑k=2

gk uni−k+1 + sni ∆t

(19)





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 regionL ≤ x ≤ R, 0 ≤ t ≤ T .

We will apply a matrix stability analysis to the linear system ofequations arising from the finite difference equations defined by(19) and will use the Greschgorin Theorem to determine astability condition.





Stability Analysis

The difference equations defined by (19), together with theDirichlet boundary conditions, result in a linear system ofequations of the form

Un+1 = A Un + ∆t Sn (20)

where Un = [un0 , un1 , u

n2 , ..., u

nK ]T , Sn = [0, sn0 , s

n1 , s

n2 , ..., s

nK−1, 0]T

and A is the matrix of coefficients, and is the sum of a lowertriangular matrix and a superdiagonal matrix. The matrixentries Ai,j for i = 1, ...,K − 1 and j = 1, ...,K − 1

Ai,j = 0 , when j ≥ i+ 2

= 1 + g1 β cni , when j = i

= gi−j+1 β cni , when otherwise





Stability Analysis

Note that g1 = −α and for 1 ≤ α ≤ 2 and i 6= 1 we have gi ≥ 0.Also since

∑∞k=0 gi = 0, this implies that −gi ≥

∑k=Nk=0,k 6=1 gi .

According to Greschgorin Theorem, the eigenvalues of thematrix A lie in the union of the circles centered at Ai,i with

radius ri =∑K

k=0,k 6=iAi,k.

Here we have Ai,i = 1 + g1cni β = 1− αcni and

ri =

K∑k=0,k 6=i

Ai,k =

i+1∑k=0,k 6=i

Ai,k = cni βi+1∑

k=0,k 6=igi ≤ α cni β (21)

⇒ Ai,i + ri ≤ 1





Stability Analysis

⇒ Ai,i − ri ≥ 1− 2 α cni β ≥ 1− 2 α cmax β

Therefore for the spectral radius of the matrix A to be at mostone , it suffices to have (1− 2 α cmax β) ≥ −1, which yields thefollowing condition on β

β =∆t

hα≤ 1

αcmax(22)

Under the condition on β defined by (22) the spectal radius ofmatrix A is bounded by one. Therefore the Explicit Methoddefined above is unconditionally stable.





Approximating two-sided Fractional PDE

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 spacewith shifted Grunwald estimate. The equation takes the form

un+1i − uni

∆t=

1

hα

[cn+1

+,i

i+1∑k=0

gkun+1i−k+1 + cn+1

−,i

K−i+1∑k=0

gkun+1i+k−1

]+ sn+1

i

(23)

with h = (R− L)/K for i = 1, 2, ...K − 1.





Approximating two-sided Fractional PDE

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 spacewith shifted Grunwald estimate. The equation takes the form

un+1i − uni

∆t=

1

hα

[cn+,i

i+1∑k=0

gkuni−k+1 + cn−,i

K−i+1∑k=0

gkuni+k−1

]+ sni

(24)

with h = (R− L)/K for i = 1, 2, ...K − 1.





Stability Analysis

Result for Implicit Method

The implicit Euler method for two-sided Fractional PDEdefined 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) fromtwo sides over the region L ≤ x ≤ R, 0 ≤ t ≤ T .





Conclusion

The explicit method using shifted Grunwald estimate forapproximating one-sided Fractional PDE is conditionallystable, consistent and hence convergent withO(∆t) +O(∆x).

The implicit method using shifted Grunwald estimate forapproximating two-sided Fractional PDE with 1 ≤ α ≤ 2 isunconditionally stable, consistent and hence convergentwith O(∆t) +O(∆x).

The explicit method using shifted Grunwald estimate forapproximating two-sided Fractional PDE is conditionallystable, consistent and hence convergent withO(∆t) +O(∆x).





References

M.M. Meerschaert and C. Tadjeran

Finite Difference Method for Two SidedFractional Partial Differential Equations

Igor Podlubny

Fractional Differential EquationsAcademic Press, 1999

Kenneth S. Miller and Bertram Ross

An Introduction to the Fractional Calculusand Fractional Differential EquationsJohn Wiley and Sons, 1993





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