an overview of numerical solution techniques for fractional … · 2017. 11. 28. · in this...

An Overview of Numerical Solution Techniques forFractional Order Problems

B. A. Jacobs

School of Computer Science and Applied Mathematics, University of the Witswatersrand,Johannesburg, Private Bag 3, Wits 2050, South Africa

DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS)

B. A. Jacobs (Wits) Numerics for Fractional Problems 1 / 55

Introduction

Acknowledgement

Special thanks to Luca Gerardo-Giorda for the invitation as well asGianni Pagnini and the workshop organisers.

I would like to acknowledge my collaborators, B.I. Henry, C.Angstmann, A.V. McGann, C. Harley, E. Momoniat, and their effortsthat made this work possible.


Introduction

Introduction

In this presentation I aim to give an overview of some (lesser-known)numerical techniques for fractional differential equations.

This work is primarily limited to ‘parabolic’ type equations:sub-diffusion, time-fractional Fokker-Planck, although we do deviatefrom this path slightly.

References are provided for examples, rigorous numerical analysis andin depth explanation/derivation of each method.


Finite Difference Type Discretizations

Grunwald-Letnikov

Natural extension of Taylor series, useful tool for discretization offractional derivatives.

GLDα0,tx(t) =

m−1∑k=0

x (k)(0)t−α+k

Γ(−α + k + 1)+

1

Γ(m − α)

∫ t

0(t − τ)m−α−1x (m)(τ)dτ

(1)or

GLDα0,tx(t) = lim

∆t→0,n∆t→t∆t−α

n∑k=0

(−1)k(α

k

)x(t − k∆t) (2)

Gives rise to “finite-difference” approximation to fractional derivatives.



Grunwald-Letnikov

An Explicit Finite Difference Method and a New von Neumann-TypeStability Analysis for Fractional Diffusion Equations [1].

A Second-Order Accurate Numerical Approximation for the FractionalDiffusion Equation [2]. (Space-fractional)

Pros

Simple extension of traditional finite-difference type methods.

Numerical analysis (stability, convergence, consistency) is similar totraditional approaches.

Cons

Explicit schemes can be computationally intractable for small α dueto stability conditions.

Obtaining higher-order accurate (in time) schemes can bechallenging, typically schemes are O(∆tα).



Transform Methods

Primary benefit of treating fractional derivative algebraically,

L{CD

α0,tx(t)

}= sαX (s)−

m−1∑k=0

sα−k−1x (k)(0). (3)

Transforms can be coupled with other transforms or discretizationtechniques [3, 4].

Laplace transform is a natural choice for time-fractional problems ontemporal domains t ∈ [0,∞)



Transform Methods

Pros

Circumvent time marching.

Can be coupled with high-order spatial discretization(Pseudo-Spectral, Compact Finite Difference, etc).

Cons

Inversion of the transform is non-trivial but can be done numericallywith geometric convergence O(e−1/δt).

Restricted to linear problems.



Quasi-Linearization

One may use a quasi-linearization approach to render non-lineartime-fractional PDEs tractable for transform methods.

Application of Hybrid Laplace Transform Method to Non-linearTime-Fractional Partial Differential Equations via Quasi-Linearization,Submitted to International Journal of Computer Mathematics.

In essence this method is an extension of a Netwon-Raphson iterativemethod in a functional space, where a series of linear functionalapproximations to the non-linear equation is constructed whichconverge to the non-linear dymamics quadratically.



Integrablization

Integrablization - the process of making something integrable.

Discretization of Fractional Differential Equations by a PiecewiseConstant Approximation to appear in Mathematical Modelling ofNatural Phenomena.

Integrablization of Time Fractional PDEs [5]



Integrablization of Fractional ODEs

Consider a general αth order fractional differential equation,

CDα0,tx(t) = f (x(t)), x(0) = x0. (4)

We construct an integrable form of the equation by assuming the solutioncan be approximated by a series of piecewise constant functions,

CDα0,tx(t) ≈ f

(x(δt⌊ t

δt

⌋)), (5)

where

limδt→0

f(x(δt⌊ t

δt

⌋))= f (x(t)). (6)

This yields an integrable form of the equation,

CDα0,tx(t) ≈ f (x0) +

∞∑m=1

(f (x(mδt))− f (x((m − 1))δt))u(t −mδt)) (7)




Applying the Laplace Transform to Equation (7) gives

sαL{x(t)} − sα−1x0 = s−1f (x0) (8)

+ s−1∞∑

m=1

(f (x(mδt))− f (x((m − 1)δt)))e−smδt .

After rearranging, inverting the Laplace transform and replacing t = nδtwe obtain the following numerical scheme.

x(nδt) = x0 +(nδt)α

Γ(1 + α)f (x0) (9)

+n−1∑m=1

((n −m)δt)α

Γ(1 + α)(f (x(mδt))− f (x((m − 1)δt)))




Considering a simple linear example

CDα0,tx(t) = −cx(t), x(0) = x0 (10)

with exact solutionx(t) = x0Eα(−ctα) (11)




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Figure : The Grunwald-Letnikov discretization (Orange Circles) and the piecewiseconstant integrablization on the time interval [0, 3] (Blue Squares, fixed timestep, Yellow Diamonds, random time steps, and Purple Triangles, non-uniformspaced time steps) with α = 0.5, c = 1, x0 = 1, and δt = 0.25. The left panelshows the solutions, with the exact solution given as a solid black line, and theright panel shows the difference between the exact solution and the discritisation.



Integrablization of Time-Fractional PDEs

Following the work for ODEs, a similar approach can be taken fortime-fractional PDEs,

CDα0,tw(x , t) = ρ

(x ,(δt⌊ t

δt

⌋),w(x , δt

⌊ t

δt

⌋)). (12)

Resulting in

CDα0,tw(x , t) = ρ (x , 0,w(x , 0)) (13)

+∞∑

m=1

(ρ (x , (mδt),w(x ,mδt))

−ρ (x , (m − 1)δt,w(x , (m − 1)δt))) u(t −mδt).




Again solving Equation (13) using the Laplace transform, we obtain (indiscrete form),

w(x , lδt) =n−1∑k=0

w (0,k)(x , 0)(lδt)k

k!+ ρ (x , 0,w(x , 0))

(lδt)α

Γ(α + 1)

+l−1∑m=1

(ρ (x , (mδt),w(x ,mδt)) (14)

−ρ (x , (m − 1)δt,w(x , (m − 1)δt)))((l −m)δt)α

Γ(α + 1),

where n − 1 ≤ α < n and n ∈ Z.




We consider now the time-fractional diffusion equation

CDα0,tw(x , t) =

∂2

∂x2w(x , t), (15)

with 0 < α ≤ 1. After ’integrabilizing’ the above, we obtain

w(x , lδt) = w(x , 0) + w (2,0)(x , 0)(lδt)α

Γ(α + 1)(16)

+l−1∑m=1

(w (2,0)(x ,mδt)− w (2,0)(x , (m − 1)δt)

) ((l −m)δt)α

Γ(α + 1).




0.001 0.005 0.010 0.050 0.100

0.5

1

5

10

δt

E(δt,2

)

0.005 0.010 0.050 0.100 0.500 1

0.001

0.010

0.100

1

10

t

E(δt,t)

0.001 0.0050.010 0.0500.100 0.500 1

10-4

0.001

0.010

0.100

1

10

t

E(δt,t)

0.001 0.0050.010 0.0500.100 0.500 1

10-5

10-4

0.001

0.010

t

R(δt,t)



The error in the integrablization of the fractional heat equation giveninitial conditions given in by a Taylor series of exp(x − x2) about x = 0 oforder p. Top left: The absolute error as a function of δt for p = 6evaluated at t = 2. The dashed line is a linear best fit in δt. Top right:The evolution of the absolute error as a function of time. The arrowindicates decreasing δt. Bottom left: The time evolution of the absoluteerror for p = 4, 5, 6, 7, 8, 9, 10, at fixed δt = 1

1000 . The arrow indicatesdecreasing p. Bottom right: The time evolution of the relative error forp = 4, 5, 6, 7, 8, 9, 10, at fixed δt = 1

1000 . The arrow indicates decreasing p.



Integrablization

Pros

Straightforward to implement.

Broad applicability: FDEs, FPDEs, any choice of α, nonlinearproblems. See [5] for diffusion-wave equation, Burgers’ equation.

Can be coupled with traditional spatial discretization schemes.

Cons

Only accurate to O(δt).

Reduces to Euler scheme for α = 1.

Related to (fractional) Adam-Bashforth methods.



Discrete Time Random Walk

The Discrete Time Random Walk (DTRW) framework:

is derived from well-posed phenomenological stochastic processes,

provides numerical properties (boundedness, stability, convergenceand consistency) ‘for free’,

allows fractional derivatives to arise naturally from physicallymeaningful processes,

is able to deal with non-linearities in simply,

results in an explicit time-stepping numerical method.

The primary caveat to this, is that a given fPDE cannot be ‘discretized’ inthe traditional sense: stochastic processes are defined and the modelequation is then ‘discovered’ in the continuum limit.



Discrete Time Random Walk

To date the DTRW approach has been successfully applied to

SIR type models, with fractional order recovery and infectivity, as wellas generalised compartment models with fractional dynamics (underreview) [6, 7],

anomalous diffusion, with and without reactions [8, 9],

the general form of fractional Fokker-Planck equations [10],

time-fractional pure advection equation [11, 12],

as well as integer-order models of the above.



Discrete Time Random Walks for Time-Fractional PDEs

We consider the random walk of a particle on a lattice,

{L1, . . . , i − 1, i , i + 1, . . . , L2},

then Ψ(j , n|i ,m) is a transition function which dictates the probability of aparticle jumping to site j at time n given that it was at site i at time m.We assume that this transition function is constituted of an independentjump probability distribution λ(j |i) and a sufficiently stationary waitingtime distribution ψ(n −m), hence

Ψ(i , n|j ,m) = λ(i |j)ψ(n −m), (17)

with m < n.




In this work we restrict a particle’s motion to nearest neighbours for anygiven jump. The jump probability distribution is illustrated below.

prpl

ps

i - 2 i - 1 i i + 1 i + 2

Where

pr = λ(i + 1|i), (18)

pl = λ(i − 1|i),ps = λ(i |i).




The choice of waiting time distribution recovers temporal derivativesoperators in the continuum limit.

A Markovian/Exponential waiting time distribution recovers thetemporal derivative to order 1.

Any heavy-tailed distribution will recover a fractional derivative in thelimit, in this work we use the Sibuya distribution because the inverseZ-transform exists in closed form.

Heavy-tailed waiting time distributions capture the memory effectsassociated with fractional derivatives.




In order to capture reaction dynamics we introduce the process ofannihilation of a particle. We denote the annihilation of a particle at pointi at time n by A(i , n). Then the probability of surviving the annihilationprocess at site i from time m to n is given by

Θ(i , n,m) =n−1∏l=m

(1− A(i , l)). (19)

The probability of a particle at a site not jumping by time step n, giventhe particle arrived at the earlier time m is given by

Φ(n −m) = 1−n−m∑k=0

ψ(k). (20)




The probability of a particle arriving at site i on the nth time step, giventhat it is created and begins walking at site i0 on the nth0 time step, isrecursively defined by the probability flux;

Q(i , n|i0, n0) = δi ,i0δn,n0 +

L2∑j=L1

n−1∑m=0

λ(i |j)ψ(n −m) Θ(j , n,m)Q(j ,m|i0, n0)

(21)where Q(i ,m|i0, n0) = 0 for all m < n0. This equation expresses that theflux into site i at time step n is the sum of all the fluxes into site j at theearlier time step m that survived until time step n when they transition tosite i .




The probability of the particle being at site i on the nth time step can thenbe written

X (i , n|i0, n0) =n∑

m=0

Φ(n −m) Θ(i , n,m)Q(i ,m|i0, n0). (22)

The right hand side is the sum over all possibilities of the particle arrivingat an earlier time step, m, and not being annihilated or jumping before thenth time step.



Omitting some of the more technical points we obtain the generalisedmaster equation for a single species discrete general reaction diffusionprocess.

U(i , n)− U(i , n − 1) =

L2∑j=L1

λ(i |j)n−1∑m=0

K (n −m) Θ(j , n,m)U(j ,m)

−n−1∑m=0

K (n −m) Θ(i , n,m)U(i ,m)− A(i , n − 1)U(i , n − 1) + C (i , n).

(23)

If the jump probability mass function is defined by

λ(i |j) =r

2δi+1,j +

r

2δi−1,j + (1− r)δi ,j , (24)

where r is the probability of a self jump, then the pmf is symmetric andthe spatial process is symmetric and we have no advective terms arising inthe diffusion limit of the master equation.



Waiting Time Distributions

By selecting a Markovian/Exponential waiting time we obtain the memorykernel

K (n) = ωδ1,n, (25)

where ω is the probability of the particle jumping to a new site on anygiven time step, and taking the diffusion limit of the GME we obtain

∂u(x , t)

∂t= ωD

∂2u(x , t)

∂x2− a(x , t)u(x , t) + c(x , t), (26)

where

D = lim∆x→0,∆t→0

r∆x2

2∆t. (27)



Sibuya Waiting Time

The kernel obtained from a Sibuya waiting time may be written as,

K (n) =

(1− αn

)(−1)n − δ0,n + δ1,n, (28)

and the kernel can be obtained from a recursion relation [13]. It isinteresting to note that the terms in the memory kernel can be related tothe Grunwald-Letnikov fractional derivative via

D1−α(f (x)) = limh→0

n∑k=0

(K (n) + δ0,n − δ1,n)f (x − kh)

h1−α . (29)

Moreover the diffusion limit in this case results in

∂u(x , t)

∂t= Dα

∂2

∂x2

[θ(x , t, 0) 0D

1−αt

(u(x , t)

θ(x , t, 0)

)]−a(x , t)u(x , t)+c(x , t) .

(30)with

Dα = lim∆x→0,∆t→0

r∆x2

2∆tα(31)



Comments

This recovers the fractional reaction-diffusion equation derived from thediffusion limit of continuous time random walks in [14] where 0D

1−αt is

the Riemann-Liouville fractional derivative, formally equivalent to theGrunwald-Letnikov fractional derivative in this setting. We note that Eq.(30) is potentially a non-linear equation, as the annihilation and creationrates, a(x , t) and c(x , t) respectively, are defined by the reaction kinetics,which may be dependent on the concentration u(x , t).



Nonsymmetric Jump Distributions

If we now define a generalized master equation as

U(i , n) = pr (i−1, n−1)U(i−1, n−1)+pl(i +1, n−1)U(i +1, n−1) (32)

where the jump probabilities are defined by Boltzmann weights

pr (i , n) =exp(−βV (xi + ∆x , tn))

exp(−βV (xi −∆x , tn)) + exp(−βV (xi + ∆x , tn)), (33)

andpl(i , n) = 1− pr (i , n), (34)

where

F (x , t) = −∂V (x , t)

∂x. (35)



Taking the diffusion limit of this scheme we obtain

∂u(x , t)

∂t= D

∂2u(x , t)

∂x2− 2βD

∂F (x , t)u(x , t)

∂x+O(∆t) +O(∆x4), (36)

with D = ∆x2/2∆t. Finally noting that O(∆t) = O(∆x2) we see thatthe scheme will be convergent with order ∆x2.This scheme is robust for F (x , t) an arbitrary function of x and t as wellas being a function of u(x , t). If F (x , t) is a known function of x and tthen the potential can be computed exactly but if it is a function ofu(x , t) then the potential can be approximated using standard quadrature.



The Viscous Burgers’ Equation

Taking F (x , t) = u(x , t), D = ν, β = 1/ν we obtain the viscous Burgers’equations

∂u(x , t)

∂t= ν

∂2u(x , t)

∂x2− u(x , t)

∂u(x , t)

∂x. (37)

Interestingly, if the CFL condition is violated the scheme remains stable,but is unable to accurately capture the propagation speed of a wavefront.The numerical scheme obtained when using a standard trapezoidalquadrature is

U(i , n) =U(i − 1, n − 1)

1 + exp(−∆x8ν (U(i − 2, n − 1) + 2U(i − 1, n − 1) + U(i , n − 1)))

+U(i + 1, n − 1)

1 + exp( ∆x8ν (U(i + 2, n − 1) + 2U(i + 1, n − 1) + U(i , n − 1)))

.

(38)




Burgers Equation with time dependent Dirichlet boundary conditions.(Top Left) Error as a function of time. The arrow indicates direction ofincreasing ∆x . Red points indicate that the CFL condition is broken. (TopRight) Error as a function of ∆x at t = 6250

81 . (Bottom Left) Plot of thesolutions at t = 6250

81 , the arrow indicates the direction of increasing ∆x .The exact solution is shown as a solid black line. (Bottom Right)Difference between the exact solution and the numerical approximation att = 6250

81 .




Turning our attention now to a purely advective equation, we seek toapproximate the dynamics of

∂u

∂t= −D1−α

0,t

∂u

∂x(39)

We achieve this by only allowing particles to jump in one direction, defining

λ(i |j) = pr (j , n)δj ,i−1 + ps(j , n)δj ,i . (40)

We take

pr (i , n) =1

Cα(41)

where

Cα = lim∆x ,∆t→0

∆x

∆tα(42)




-1.0 -0.5 0.0 0.5 1.0

0.0

0.5

1.0

1.5

x

u(x,t)

-1.0 -0.5 0.0 0.5 1.0

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

x

Difference

Figure : Left, Plot of the exact solution, together with the one-sided jumpdistribution DTRW (red circles) and the two-sided jump distribution DTRW (bluesquares). Right, the difference between the exact solution and the one-sided jumpdistribution DTRW (red circles) and the two-sided jump distribution DTRW (bluesquares). Here α = 1

2 , Cα = 2, and ∆x = 263 . The solutions are evaluated at

t = 2003√

7.




5.×10-4 0.001 0.005 0.010 0.050 0.1000.03

0.04

0.05

0.06

Time

L ∞

0.02 0.05 0.10 0.20

0.02

0.05

0.10

0.20

Δx

L ∞

Figure : Left, L∞ scaling time for the the one-sided jump distribution DTRW (redcircles) and the two-sided jump distribution DTRW (blue squares) with ∆x = 2

63 .Right, L∞ vs ∆x for the one-sided jump distribution DTRW (red circles) and thetwo-sided jump distribution DTRW (blue squares) at t = 1

16 . Here α = 12 and

Cα = 2.



Discrete Time Random Walks for Fractional ODEs

The DTRW framework can also be used to solve fractional orderODEs, and models arising from compartmental models.

Fractional order recovery and infectivity for SIR type models havealready been considered.

The work Discrete Time Random Walk Scheme for the NumericalSimulation of Fractional Order Compartment Models, currently underreview, aims to provide a general framework for practitioners outsidethe field of applied mathematics to numerically solve theircompartmental models.

The examples presented below are taken from the above mentionedwork.




1 2λ(t)β1

β2

μ

τ1,α1

τ2,α2

Figure : Box diagram for the case considered in the implementation section. Thesolid arrows indicate a Markovian transition from one compartment to anotherwith a rate parameter indicated above the arrow. Arrows originating orterminating on no compartment indicate transitions in and out of the systemrespectively. Dashed arrow indicate a non-Markovian transition with the firstparameter above the arrow being a time scale and the second parameter beingthe exponent.




The numerical scheme describing the dynamics of the above system aregiven by,

X1(n) = X1(n − 1) + Q2,1(n) + Q3,1(n)− Q1,2(n), (43)

X2(n) = X2(n − 1) + Q1,2(n) + Q3,2(n)− Q2,1(n)− Q2,3(n), (44)

where the fluxes are computed as before based on survival probabilitiesdrawn from an inhomogeneous Poisson process.



Chromium Poisoning Model

We consider a one-compartment model describing the clearance ofchromium in mice first considered in [15]. This compartment modelcomprises of a single compartment that has an initial dose of chromium,c0, that exits out of the compartment via a non-Markovian process, seeFigure 5.

c τ,α

Figure : A one compartment model for the clearance of chromium. There is asingle non-Markovian transition out of the compartment with time scale τ andexponent α.




The amount of chromium in the compartment, c(t), changes according tothe fractional order DE,

dc

dt= −τα 0D

1−αt c(t), (45)

where τ is a time scale and α the exponent, subject to the initial conditionc(0) = c0. The exact solution to Eq. (45) is given by,

c(t) = c0Eα,1 (− (τ t)α) . (46)




The only flux into the compartment, Q(k), is the initial dosage and hencewe may write,

Q2,1(n) = c0δn,0, (47)

where δn,0 is the Kronecker delta function and c0 is the initial dosageconcentration. The non-Markovian survival function is given by

Φ1,2(n) = Eα,1 (− (τn∆t)α) , (48)

and there are no Markovian processes. Hence,

Q1,2(n) = c0(Φ1,2(n − 1)− Φ1,2(n)), (49)

for n ≥ 1.




Finally our approximation for the solution is,

c∆t(n∆t) = X1(n) = X1(n − 1)− c0(Φ1,2(n − 1)− Φ1,2(n)), (50)

for n ≥ 1 and X (0) = c0. This recursive relation can be solved to give,

c∆t(n∆t) = X1(n) = c0Eα,1 (− (τn∆t)α) . (51)

Hence in this case our numerical approximation gives the exact solution atdiscrete points, t = n∆t.



HIV Model

V

I δI ,α

λ δV-1

Figure : Diagram for the HIV model. Here the Infected cells die with anon-Markovian transition. The virions grow proportionally to the death of theInfected cells, such that λ = NδαI 0D1−α

t I , and themselves are cleared via aMarkov transition.



HIV Model

We consider a simplistic two-compartment model describing the dynamicsof HIV and the CD4+ T cells, targeted by the virus, in the case of acombined antiretroviral therapy with 100% efficacy, leading to noreplenishment of the infected T-cells from uninfected cells [15]. The modelis governed by

dI

dt= −δαI 0D1−α

t I , (52)

anddV

dt= NδαI 0D1−α

t I − δVV , (53)

where I denotes the number of infected CD4+ T cells and V the numberof HIV virons, subject to the initial conditions,

I (0) = I0, (54)

V (0) = V0. (55)



HIV Model

The solution of these equations is given by [15],

I (t) = I0Eα,1 (−(δI t)α) . (56)

V (t) =e−δV t I0N

([1− eδV tEα,1 (−(δI t)α)

]+ δV

∫ t

0eδV sEα,1 (−(δI s)α) ds

)+ V0e

−δV t

(57)

In representing this as a compartment model, we consider threecompartments, with I is taken to be the first compartment, V to be thesecond, and a third dummy compartment for outflows an initial inflows.



HIV Model

Defining the fluxes through survival functions we have,

Q3,1(n) = I0δn,0, (58)

Q3,2(n) = NQ1,3(n) + V0δn,0. (59)

Q1,3(n) = I0(Φ1,3(n − 1)− Φ1,3(n)), (60)

withΦ1,3(n) = Eα,1 (− (δIn∆t)α) . (61)

Then the flux from the V compartment to the dummy compartment is,

Q2,3(n) =n−1∑k=0

(Θ2,3(n − 1, k)−Θ2,3(n, k))Q3,2(k), (62)

with,

Θ2,3(n,m) = exp

(−∫ n∆t

m∆tδV dτ

). (63)



HIV Model

Our approximation can then be found from,

I∆t(n∆t) = X1(n) = X1(n − 1) + Q3,1(n)− Q1,3(n), (64)

V∆t(n∆t) = X2(n) = X2(n − 1) + Q3,2(n)− Q2,3(n). (65)



HIV Model

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■

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0 1 2 3 40.0

0.2

0.4

0.6

0.8

1.0

t

I(t),V

(t)

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■

■■ ■ ■

■■■■■■■■■■■■■■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

0 1 2 3 40.000

0.005

0.010

0.015

0.020

0.025

t

ϵ Δt(t)

Figure : Left: Comparison of approximate DTRW solution for I compartment (◦)and V compartment (�) against exact solutions (solid line). Right: L1 error inapproximate DTRW solution for I compartment (◦) and V compartment (�).Plots are given on the domain t ∈ [0, 4], with α = 0.7 and ∆t = 1/8.



HIV Model

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●

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●

●

●

●

●

●

■

■

■

■

■

■

■

■

■

■

◆

◆

◆

◆

◆

◆

◆

◆

◆

◆

0.001 0.005 0.010 0.050 0.100

1.×10-4

5.×10-40.001

0.005

0.010

0.050

0.100

Δt

ϵ Δt(0.4)

Figure : Plot of the convergence of the L1 error for the numerical approximationof the concentration in the V compartment at t = 0.4 over ∆t for α = 0.1 (◦),α = 0.5 (�) and α = 0.9 (�).



Conclusions and Open Questions

Recent work for CTRW and DTRW has been focused on sub-diffusionprocesses on arbitrarily growing domains [16, 17].

The DTRW framework is still being developed for odd-orderderivatives (KdV Type equations).

Future research aims to consider non-local jump distributions forspace-fractional problems as well as techniques for obtaininghigher-accuracy for the derived schemes.



Thank you

Thank you.


References

S. B. Yuste, L. Acedo, An explicit finite difference method and a newvon Neumann-type stability analysis for fractional diffusion equations,SIAM J. Numer. Anal. 42 (5) (2005) 1862–1874.

C. Tadjeran, M. M. Meerschaert, H.-P. Scheffler, A second-orderaccurate numerical approximation for the fractional diffusion equation,J. Comp. Phys. 213 (1) (2006) 205–213.

B. Jacobs, C. Harley, Two hybrid methods for solving two-dimensionallinear time-fractional partial differential equations, in: Abstract andApplied Analysis, Vol. 2014, Hindawi Publishing Corporation, 2014.

B. A. Jacobs, High-order compact finite difference and laplacetransform method for the solution of time-fractional heat equationswith dirchlet and neumann boundary conditions, Numerical Methodsfor Partial Differential Equations 32 (4) (2016) 1184–1199.

C. N. Angstmann, B. I. Henry, B. Jacobs, A. V. McGann,Integrablization of time fractional pdes, Computers & Mathematicswith Applications 73 (6) (2017) 1053–1062.


References

C. Angstmann, B. Henry, A. McGann, A fractional order recovery sirmodel from a stochastic process, Bulletin of mathematical biology78 (3) (2016) 468–499.

C. N. Angstmann, B. I. Henry, A. V. McGann, A fractional-orderinfectivity sir model, Physica A: Statistical Mechanics and itsApplications 452 (2016) 86–93.

C. Angstmann, I. C. Donnelly, B. I. Henry, J. A. Nichols, A discretetime random walk model for anomalous diffusion, Journal ofComputational Physics 293 (2015) 53–69.

C. N. Angstmann, I. C. Donnelly, B. I. Henry, B. Jacobs, T. A.Langlands, J. A. Nichols, From stochastic processes to numericalmethods: A new scheme for solving reaction subdiffusion fractionalpartial differential equations, Journal of Computational Physics 307(2016) 508–534.

C. Angstmann, I. C. Donnelly, B. I. Henry, T. Langlands, P. Straka,Generalized continuous time random walks, master equations, andB. A. Jacobs (Wits) Numerics for Fractional Problems 55 / 55

References

fractional fokker–planck equations, SIAM Journal on AppliedMathematics 75 (4) (2015) 1445–1468.

C. Angstmann, B. I. Henry, B. A. Jacobs, A. V. McGann, et al.,Numeric solution of advection-diffusion equations by a discrete timerandom walk scheme, arXiv preprint arXiv:1610.05417.

C. Angstmann, B. Henry, B. Jacobs, A. McGann, A time-fractionalgeneralised advection equation from a stochastic process, Chaos,Solitons & Fractals.

C. N. Angstmann, I. C. Donnelly, B. I. Henry, J. A. Nichols, A discretetime random walk model for anomalous diffusion, J. Comp. Phys. 293(2015) 53–69.

C. N. Angstmann, I. C. Donnelly, B. I. Henry, Continuous timerandom walks with reactions, forcing, and trapping, Math. Model.Nat. Phenom. 8 (2) (2013) 17–27.


References

C. N. Angstmann, A. M. Erickson, B. I. Henry, A. V. McGann, J. M.Murray, J. A. Nichols, Fractional order compartment models, SIAMJournal on Applied Mathematics 77 (2) (2017) 430–446.

F. Le Vot, E. Abad, S. Yuste, Continuous-time random-walk model foranomalous diffusion in expanding media, Physical Review E 96 (3)(2017) 032117.

C. Angstmann, B. Henry, A. McGann, Generalized fractional diffusionequations for subdiffusion in arbitrarily growing domains, PhysicalReview E 96 (4) (2017) 042153.


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