discrete-space time-fractional processes

$: Discrete-space time-fractional processes$
RESEARCH PAPER

DISCRETE-SPACE TIME-FRACTIONAL PROCESSES

Colin Atkinson ∗ and Adel Osseiran ∗∗

Abstract

A time-fractional diffusion process defined in a discrete probability set-ting is studied. Working in continuous time, the infinitesimal generators ofrandom processes are discretized and the diffusion equation generalized byallowing the time derivative to be fractional, i.e. of non-integer order. Theproperties of the resulting distributions are studied in terms of the Mittag-Leffler function. We discuss the computation of these distribution functionsby deriving new global rational approximations for the Mittag-Leffler func-tion that account for both its initial Taylor series and asymptotic power-lawtail behaviours. Furthermore, we derive integral representations for boththe continuous and the discrete time-fractional distributions and use theseto prove a convergence theorem.

MSC 2010 : 26A33, 33E12, 35R11, 60G22Key Words and Phrases: fractional calculus, time-fractional process,

discrete-space process, Mittag-Leffler function

1. Introduction

Working in continuous time, random processes can be studied on bothcontinuous and discrete spaces. Markov processes have been researched indepth, in both these settings; having the Markov property means that, giventhe present state of the process, future states are independent of the paststates. Markov processes have been studied in numerous books including

c© 2011 Diogenes Co., Sofiapp. 201–232 , DOI: 10.2478/s13540-011-0013-9

202 C. Atkinson, A. Osseiran

those by Feller [14], and Chung [8] provides a thorough account of thegeneral theory for Markov chains. The theory of continuous-time Markovchains is presented in Grimmett and Stirzaker [19], and also in Anderson [2]with focus on their applications. These are typically defined by specifying agenerator from which the transition semigroup can be defined. One can alsostart with a time-parametrized random variable that maps the sample spaceof a probability space to some countable set. With the resulting discreteprobabilities one can interpret the coordinate elements of the generator asthe infinitesimal transition probabilities.

Approximating diffusion processes with continuous-time Markov chainshas also been studied in, for example, Platen [38]. One way to do this is todiscretize the generator of a diffusion, define a discrete-space process withthe discretized generator and work with the diffusion equation in discretespace. The convergence of such discrete processes to their continuous coun-terparts has been studied by Kushner and Dupuis [26] and Albanese andMijatovic [1].

One can construct random processes with different properties from thestandard Markov and Gaussian ones by employing fractional derivativesin the diffusion equation. Fractional calculus involves the generalizationof differentiation and integration to non-integer orders; for example, canone rigorously define the square root of the operator d

dx? The study offractional calculus has a long history dating back to Leibniz and Euler inthe 17th and 18th centuries, to a series of papers by Liouville 1832-1937 andan 1847 paper by Riemann defining the fractional integral. Books coveringthe topic of fractional calculus began to emerge in the 20th century insuch works as Oldham and Spanier [37], McBride and Roach [33], Samkoet al. [44], Kiryakova [23], Rubin [43] and Kilbas et. al. [22]. Integraltransforms play an important role in the study of fractional calculus andwe refer to McBride [32] for a treatment of fractional calculus and integraltransforms of generalized functions, and to Mainardi [29] for a study of theroles played by integral transforms and their importance in the study offractional diffusion processes. For the history of fractional calculus, onecan see also in the recent surveys by Tenreiro Machado, Kiryakova andMainardi [49], [50], [51].

While the fractional derivative known as the Riemann-Liouville deriv-ative had existed for many years, Caputo [5] introduced a slightly differentdefinition, which ultimately came to be named after him as the Caputofractional derivative. The key difference between the two is that the Ca-puto derivative of a function incorporates the initial values of the functionand of its integer derivatives of lower order, compared to the Riemann-Liouville derivative which incorporates initial values of the function and of

DISCRETE-SPACE TIME-FRACTIONAL PROCESSES 203

its fractional derivatives. When dealing with physical applications in whichsolutions to boundary value problems are required, especially those whichlend themselves to solution via Laplace transform techniques, the Caputoderivative is the more natural of the two when it comes to interpretingthe boundary conditions with respect to the fractional derivative. Specifi-cally, this is because the initial conditions are typically expressed in termsof the values of the field variables at the boundary and its derivatives ofinteger-order.

A fractional differential equation is specified by generalizing the stan-dard integer-order derivatives to fractional ones of arbitrary order. Suchequations differ from standard ones in that the nature of the fractionalderivative introduces a memory effect, thus increasing its modeling ability.Time-fractional diffusion equations generalize the standard diffusion equa-tions, and the interpretation of the fractional derivative is that it representsa degree of memory in the diffusing material. Research into such equationshas been the subject of increased interest in science and engineering ap-plications where dynamical systems can be described by fractional orderequations, and these are the subject of recent books as those by Miller andRoss [34], Podlubny [39], Diethelm [9], and many papers such as Kochubei[25].

In the context of probability theory one typically applies fractional cal-culus to partial differential equations such as the diffusion equation, andthe resulting processes are the subject of recent interest, e.g. Gorenflo andMainardi [18] [31]. When the time derivative is taken to be fractional inthe diffusion equation, the “time-fractional process” is that whose spatialprobability density function evolves in time according to the equation. Suchprocesses are characterized by variances given by

σ2(t) =2tα

Γ(1 + α),

when α, the real number that replaces the order one of the time derivative,is in the range 0 < α ≤ 1. The case of α = 1 is non-fractional, and thevariance is given by 2t as expected; for α < 1 these processes are said to besub-diffusive. Furthermore, these processes exhibit initial exponential-likedecay and asymptotic hyperbolic decay, with the cross-over between thesetwo behaviours occurring at smaller values the further α is from one.

The Mittag-Leffler functions arise naturally as solutions of fractionaldifferential and integral equations. These entire functions generalize theexponential function and are characterized by a parameter α related tothat of the order of fractional differentiation. The Mittag-Leffler functionis discussed in, amongst other places, Erdelyi [12] in the context of it be-ing a special function in its own right, and its behaviour in the complex


plane studied by Cartwright [6]. This behaviour is studied further in suchworks as Gorenflo et. al. [17] in which its computation is studied for var-ious arguments z in the complex plane. Like many special functions, itsproperties must be explored differently in different regions of the complexplane. Additionally, early insights into the complete monotonicity of theMittag-Leffler function date back to Pollard [42] and Feller [13]. For multi-index analogues of the Mittag-Leffler function, related to many importantcases of special functions of fractional calculus, one can see recent papersby Kiryakova, among them [24].

Solutions to time-fractional equations have been investigated by severalauthors, other than the books on fractional differential equations like thoseby Miller and Ross [34], Podlubny [39], Kilbas, Srivastava and Trujillo [22],Diethelm [9], etc. For example, Wyss [56] derives a closed-form solutionfor the time-fractional diffusion equation in terms of the Fox functions,and Wyss and Schneider [46] the fractional diffusion and wave equations.Mainardi et. al. [30] solve the space-time fractional diffusion equationproviding a Mellin-Barnes integral representation of the Green functions inthe complex plane. See also our recent paper, Atkinson and Osseiran [3].

Another representation of the fractional derivative was introduced byGrunwald in 1867 and Letnikov in 1868. By starting with the limit defini-tion of a function f at a point x, the authors generalize the limit definitionof the nth order derivative to one or non-integer order q. This definitionlends itself naturally to the study of discrete fractional calculus, and is uti-lized by Podlubny [40] and Podlubny et. al. [41] with a matrix approach.Gorenflo and Abdel-Rehim [16] study the convergence of the Grunwald-Letnikov scheme for the time-fractional diffusion. Other works on discretefractional calculus include that of Lubich [28] in which the author discretizesthe fractional integral. Shen et. al. [47] provide a detailed analysis of aconservative difference approximation, and Lin and Xu [27] apply finite dif-ference and spectral approximations, both for the time-fractional diffusionequation. Other works on numerical solutions to fractional equations arediscussed by, amongst others, Boyadjiev et. al. [4], Diethelm and Ford [10]and Diethelm and Luchko [11].

While discretizations of the time-fractional diffusion equation have beenstudied, these have been done from a numerical perspective, and the discrete-space time-fractional process has itself not been studied in its own right.

Our goal is to work in continuous-time and define and study a discrete-space time-fractional process. To this end, in Section 2 we describe a nat-ural discrete probability setting, starting with the generator of a randomprocess. This is generalized to obtain a time-fractional diffusion equation,taking the time derivative to be Caputo’s fractional derivative of order


0 < α ≤ 1. The time-fractional case leads us to study Mittag-Lefflerfunctions of generator matrices, the computation of which is discussed inSection 3. In Section 4, the derivation of integral representations for thetime-fractional process’s probability functions allows us to prove a con-vergence result. Specifically, the convergence of the time-evolution of theprobability mass function in space of the discrete space process to that ofthe probability density function of the continuous one as the spacing in thediscrete space goes to zero.

2. Fractional calculus and time-fractional processes

A fractional differential equation is specified by the generalization ofthe standard integer-order derivatives to fractional ones of arbitrary order.In an ordinary differential equation, when the values of the derivatives inthe problem are known at a particular point, then the value of the solutionof the problem can be found. In a fractional differential equation, however,the previous values of the derivatives and of the solution are required tofind the solution. That is to say, these equations differ from standard onesin that the nature of the fractional derivative introduces a memory effect.This is due to the convolution effect appearing in the fractional case, whichwe see in the above definition of the fractional derivative.

Consider a random process whose spatial probability density evolves ac-cording to a time-fractional diffusion equation in which the time derivativeis generalized to fractional order

∂α

∂tαP (x; t) =

12

∂2

∂x2P (x; t), (2.1)

where ∂α

∂tα denotes a fractional derivative of order α, taken to be the Caputofractional derivative of order α, where 0 < α ≤ 1. The operator on the righthand side is a linear operator which represents the generator of Brownianmotion. In the case where 0 < α < 1, the Caputo derivative of a suitablefunction f(t) is given by

∂α

∂tαf(t) =

1Γ(1 − α)

∫ t

0

∂f(s)∂s

ds

(t − s)α. (2.2)

When α = 1 the fractional derivative reduces to the standard derivative∂∂tf(t) and equation (2.1) describes the time-evolution of the probabilitydensity function in space of Brownian motion. In the discrete case theoperator on the right hand side of equation (2.1) is given by a matrix,specifically, half the discrete Laplace operator in this case.

A process whose density satisfies this equation is referred to as a time-fractional process; equation (2.1) defines the time-evolution of its probabil-ity density function in space. Owing to the memory introduced by using


the fractional derivative a process whose space PDF time-evolution satisfiesthis equation is now non-Markovian, in fact it has long-term memory.

2.1. Continuous-time discrete-space processes

With the goal of defining a discrete-space time-fractional equation, webegin this section with a discussion of generators and diffusion equationsbefore discretizing space and generalizing to the time-fractional case. LetXt be an Ito diffusion satisfying the stochastic differential equation

dXt = μ(Xt)dt + σ(Xt)dBt, (2.3)

where Bt is a 1-dimensional Brownian motion, and μ and σ are the driftand diffusion coefficients. It is a well-known fact that the infinitesimalgenerator A of Xt, as defined by the limit

Af(x) = limt↓0

E[f(Xt)] − f(x)t

, (2.4)

can be calculated explicitly in terms of the coefficients of the above SDE,[36]. The real value x is the initial point of the process Xt, i.e. X0 = x.The infinitesimal generator A of the process Xt of equation (2.3), is theoperator defined to act on functions f in C2

0 (R) as

Af(x) = μ(x)df(x)dx

+σ2(x)

2d2f(x)

dx2. (2.5)

The partial differential equation∂

∂tpt(x) = Apt(x), (2.6)

subject to an initial condition p0(x), describes the time evolution of prob-ability density function of Xt.

To obtain a time-fractional diffusion, we again generalize the time deriv-ative of order one in the above PDE with a fractional derivative of arbitraryorder 0 < α ≤ 1 and specify the equation

∂α

∂tαpt(x) = Apt(x) (2.7)

where A is the generator given in equation (2.5), acting on the space coor-dinate. The processes whose probability density function is defined as thesolution to this equation, owing to the memory introduced by using thefractional derivative, is now non-Markovian.

We note that the generator of Brownian motion Bt is given by 12Δ =

12

∂2

∂x2 , half the Laplace operator. When this is used as the generator inequation (2.6) we obtain the standard diffusion equation, the underlyingprocess of which is both Gaussian and Markov. When used in equation (2.7)the result is the time-fractional diffusion equation studied in the literature.


In order to work with a more general process, i.e. not be constrained tothe Laplace operator and the related processes, but to be able to use anyrequired generator, we describe and work with a discrete setting.

2.2. Discrete space

Our goal is to discretize the state-space of the time-fractional diffusion,retaining the linear non-fractional order of the space derivatives, but takingtime to be fractional. Specifically, we look for a discretization of the partialdifferential operators that form the generator of equation (2.5). Naturaldiscretizations of these operators are the discrete Laplacian Δh and discretegradient ∇h operators given by

∇hf(x) =f(x+h) − f(x)

hand Δhf(x) =

f(x+h) + f(x−h) − 2f(x)h2

,

where h, a small positive real number, defines the spacing between thestates of the discrete space Ω, and f is a function defined on Ω. In the limitas h → 0, we pass to the continuum, and these finite difference operatorsapproach the first and second order derivatives respectively. Written incoordinate terms x and y in Ω, these operators are

∇h(x, y)=1h

⎧⎨⎩−1 y = x,

1 y = x + h,0 otherwise,

and Δh(x, y)=1h2

⎧⎨⎩−2 y = x,

1 y = x + h, x − h,0 otherwise.

(2.8)The discrete state space Ω can be taken to be hZ, viewed as a discretesubset of R. This state space can be truncated at points on the far left andright so that it becomes a finite state space. These can be chosen at a stateto which the transition is of a very low order of probability; in this case thegenerator is a matrix. We return to this in more detail in Section 3. Wenow define G to be the discrete operator version of the above generator inequation (2.5). Acting on a suitable function f(x), the tri-diagonal matrixG is given by

Gf(x) = μ(x)∇hf(x) +12σ2(x)Δhf(x). (2.9)

The discrete space time-fractional diffusion equation is the analogue ofequation (2.7) using G, defined as

∂α

∂tαPα

t = GPαt , (2.10)

subject to the condition limt→0+ Pαt (x, y) = δx,y, the Kronecker delta.

When α = 1, the Caputo derivative reduces to the standard derivative∂∂t , and the solution is given by the matrix exponential etG ; in this case theprocess in question is a continuous-time Markov chain. The solution for


general α involves the Mittag-Leffler matrix function which is the subjectof the next section.

The coordinate values of the generator G of a process taking values ina countable state space Ω can be intuitively thought of as the first orderchange in the transition probabilities between states x and y in Ω duringthe infinitesimal time-interval dt. As equations, we have

P[Xt+dt = y |Xt = x] = G(x, y)dt , x �= y,

andP[Xt+dt = x |Xt = x] = 1 + G(x, x)dt.

We note that by its definition, the operator G allows the process to tran-sit from its current state to the neighbouring states only. Furthermore itsatisfies the properties of generating positive probabilities and conservingprobability in that G(x, y) ≥ 0 for all x �= y, and

∑y G(x, y) = 0 for all x,

where x and y are in Ω.

2.3. The Mittag-Leffler matrix function

We now solve the discrete-space time-fractional diffusion equation toobtain the Mittag-Leffler function of the generator matrix as the solutionPα

t . Applying the Laplace transform to the time-fractional equation, wehave

L{

∂α

∂tαPα

t ; s, t}

= L{GPαt ; s, t}

⇒ sαF (s) − sα−1Pα0 = GF (s),

where F (s) is the Laplace transform of the function Pαt (acting on the time

parameter). The equation is subject to the condition Pα0 = I, and we solve

the above equation to obtain F (s):

F (s) = sα−1(sαI − G)−1 =1s

(I − G/sα)−1 . (2.11)

From here, F (s) can be expanded in powers of s and the inverse Laplacetransform computed term by term to yield the Mittag-Leffler matrix func-tion

Pαt = Eα(tαG) =

∞∑k=0

tαkGk

Γ(1 + αk). (2.12)

In the literature, Mittag-Leffler matrix functions appear in [21] as the so-lution of systems of time-fractional differential equations, and in [7] in agame-theory context.


We note the case where α = 1 the Gamma function in the denominatorbecomes Γ(1 + k) = k!, because k is an integer, and the Mittag-Lefflerreduces to the matrix exponential.

E1(tG) =∞∑

k=0

tkGk

k!= etG . (2.13)

We justify the existence of the Laplace transform in the matrix case inSection 2.5 below by assuming a spectral decomposition for G.

2.3.1. Convergence of Mittag-Leffler matrix series. To establish conver-gence of the Mittag-Leffler matrix series, for an arbitrary N × N squarematrix G and 0 < α ≤ 1, we write

Eα (G) =∞∑

n=0

Gn

Γ(1 + αn)= I +

∞∑n=1

Gn

Γ(1 + αn). (2.14)

Letδ = max

1≤x,y≤N|G(x, y)|, and Gn(x, y) =

(G(n)(x, y)

),

then |G2(x, y)| ≤ N × δ2, and, by induction, |Gn(x, y)| ≤ Nn−1 × δn, 1 ≤x, y ≤ N . The series is thus bounded by

∞∑n=1

|Gn(x, y)|Γ(1 + αn)

≤∞∑

n=1

Nn−1δn

Γ(1 + αn)=

1N

∞∑n=1

(Nδ)n

Γ(1 + αn)(2.15)

and it remains to verify that the series on the right hand side, which is theMittag-Leffler function of Nδ is convergent ∗ . The limit of the ratio ofthe (n + 1)th and nth coefficients, limn→∞

Γ(1+αn)Γ(1+α(n+1)) goes to zero, for all

0 < α ≤ 1, and the series thus converges via the ratio test. The limit canbe seen by taking Stirling’s approximation of the Gamma function.

2.4. Infinitesimal probability conservation

Here we discuss the probability conservation condition of the generator,in its specification, specifically equation (2.9). Firstly, to see the behaviourfor small t, we let δt be a small time; then, using the Mittag-Leffler matrixseries of equation (2.12) we have

Pαδt = Eα (G(δt)α) ∼ I +

GΓ(1 + α)

(δt)α = e1

Γ(1+α)G(δt)α

as δt → 0.

(2.16)

∗ The term |Gn(x, y)| denotes the absolute value of the x, y entry of the matrix Gn.


That is, the Mittag-Leffler function starts at t = 0 as a stretched exponen-tial, and in terms of coordinates x and y is

Pα(δt)α (x, y) = Eα (G(δt)α) (x, y) = δxy+

G(x, y)Γ(1 + α)

(δt)α+O((δt)2α

), (2.17)

which when x = y is

Pα(δt)α (x, x) = 1 +

G(x, x)Γ(1 + α)

(δt)α + O((δt)2α

).

For x �= y the probability of moving away from x in time dt is the sum overall possible states y (y �= x) given by

∑y �=x

Eα (G(δt)α) (x, y) =∑y �=x

(δxy +

G(x, y)Γ(1 + α)

(δt)α)

=1

Γ(1 + α)

∑y �=x

G(x, y)(δt)α (2.18)

because δxy = 0 for x �= y. For each of these to be an infinitesimal proba-bility, they must all be positive:

G(x, y) ≥ 0 ∀ y �= x,

and probability conservation must also be satisfied, i.e.∑y

Pαδt(x, y) =

∑y

Eα (G(δt)α) (x, y) = 1.

So

1 +G(x, x)

Γ(1 + α)(δt)α +

1Γ(1 + α)

∑y �=x

G(x, y)(δt)α = 1,

then

G(x, x) = −∑y �=x

G(x, y). (2.19)

From the analysis here we can see that these conditions are actually thesame as those of the well known continuous time Markov chain: despite thefact that the time-fractional process is not Markovian, the constraints onthe discrete space specification are the same. However, in equation (2.17)the error term, which is of order (δt)2α, must be smaller than δt for theseconditions to be consistent with the Markov case; this would require α > 1

2 .


2.5. Spectral properties of generator matrices

A square N ×N matrix G is diagonalizable if there exists an invertiblematrix U such that U−1GU is a diagonal matrix, which we denote D, andthe matrix G has the spectral decomposition given by G = UDU−1. Thediagonal matrix D contains the N eigenvalues dj , j = 1, 2, ..., N along thediagonal.

The eigenvalues dj can be complex, but for a matrix G satisfying thegenerator properties above, the eigenvalues lie along the negative real axisor are complex with a negative real part. When the matrix is symmetric,the eigenvalues are real. In general, when complex, the eigenvalues appearin conjugate pairs. A matrix is said to be stable if and only if all itseigenvalues have negative real part, and a matrix G can be shown to bestable if all the solutions of x′(t) = Gx(t) converge to zero as t goes toinfinity. The matrices G obtained from the above discretization of secondorder differential operators are all stable as they satisfy this property. Forexample, the generator of discrete Brownian motion, 1

2Δh has eigenvaluesdj = 2

h [cos(πjh) − 1] , j = 1, 2, ..., N . In the general case, we write the jthcomplex eigenvalue as dj = Rje

iθj .The spectral properties of the generator matrix G become important:

to have a meaningful Laplace transform of equation (2.10), specifically, amatrix argument, leading to equation (2.11), and a meaningful inverse, thefunction F must be well defined along the real line. If we think aboutthe matrix function in terms of the diagonalized form, we immediately seethat given these elements have negative real parts, the specification of theLaplace transform and its inverse are both meaningful.

Furthermore, given a generator G with the spectral properties describedin this section, one can apply asymptotic formulae for the Mittag-Lefflerfunction to the case of large t. Assuming G is an invertible matrix, then

Eα(tαG) ∼ − G−1

tαΓ(1 − α), t → ∞. (2.20)

This can also be derived using the well defined Laplace transform pair

L{Eα (tαG) ; t, s} = sα−1 (sα − G)−1

noting that under certain conditions satisfied here, the limit of the Laplacetransform F of a function f satisfies

lims→0

sF (s) = limt→∞ f(t),

by the final value theorem.


3. Computing the Mittag-Leffler Function

In this section we discuss the computation of the Mittag-Leffler func-tion, specifically that of matrix argument. To highlight the complexity ofbeing in a non-Markov environment, we contrast the Markov case of α = 1and the matrix exponential with the Mittag-Leffler case.

3.1. The Matrix Exponential

When α = 1, the Mittag-Leffler function reduces to the exponentialfunction, and likewise, as discussed in the matrix case, the Mittag-Lefflermatrix function reduces to the matrix exponential. Amongst other uses,the matrix exponential can be utilized to solve systems of linear ordinarydifferential equations, and exponentiating the generators of Markov chainsyields their transition probability matrices. Accordingly, many methods forcomputing the matrix exponential have been developed and refined; theseare surveyed in Moler and Van Loan [35].

The ”scaling and squaring” method, which has recently been refinedby Higham [20], takes advantage of the fact that it is possible to multiplymatrices with a large number of elements efficiently. More importantly,it uses the property of the exponential eA =

(e

An

)n. We describe the

algorithm in the context of computing the transition probability matrix ofa continuous-time Markov chain: Obtaining the matrix of transitions fromtime 0 to time t involves exponentiating the generator matrix G, i.e. findingetG . For a small enough time scale δt, we can approximate e(δt)G with atruncated Taylor series I +(δt)G + · · · , or via a Pade approximation. Then

the transition matrix can be computed as etG =(e(δt)G) t

δt . If we select δt

so that t/δt is a power of 2, say 2j , then this involves just squaring thematrix j times. A common criteria for the selection of δt is that t/δt is thesmallest power of 2 for which ∗ δt‖G‖ ≤ 1.

We refer to [20] and [35] for more details, but note that this property ofthe probabilities as exponentiated generators is essentially the Chapman-Kolmogorov equation, written in matrix format. In the context of Mittag-Leffler functions, as discussed earlier, the memory introduced by the frac-tional derivative means the processes that are governed by the time-fractionaldiffusion equations will be non-Markovian, and will thus not satisfy theChapman-Kolmogorov equation. The ”scaling and squaring” algorithm,and any other algorithm that benefits from such properties of the exponen-tial function, will fail in the Mittag-Leffler case. We note that while the

∗ The 2-norm defined for a matrix A by ‖x‖ =[∑N

i=1 |xi|2]1/2

, ‖A‖ =

max||x||=1 ‖Ax‖


addition of memory makes the processes interesting, it also makes their usemore difficult.

3.2. The Mittag-Leffler function

In the non-matrix case, the computation of the Mittag-Leffler functionis studied in [17]. The authors study the function of a complex variablefor different positions of argument z in the complex plane. The Taylorseries definition and asymptotic series ∗ serve the cases where |z| is smalland large respectively. Integral representations are applied for intermediatevalues of |z|.

In the context of computing the Mittag-Leffler matrix function itself,Kaczorek [21] solves a system of time-fractional linear ODEs in terms ofa Mittag-Leffler matrix and then extends the classical Cayley-Hamiltontheorem to fractional systems. This method is discussed under polynomialmethods in the computation of the matrix exponential in the survey [35].While it is interesting that this method works for the Mittag-Leffler matrixfunction in the same way it does for the matrix exponential, the inefficiencyof the method carries over. While possible for systems involving only ahandful of equations, i.e. a matrix with few elements, for larger sizedmatrices it is not viable.

Weideman and Trefethen [52] study various contours in the search foran effective method for the numerical inversion of the Laplace transform,based on the Bromwich contour. Specifically, they obtain the function f(t)

f(t) =1

2πi

∫ σ+i∞

σ−i∞eztF (z)dz , σ > σ0,

where σ0 is the convergence abscissa and σ is chosen as such so that thesingularities of F (z) all lie in the open half plane Re z < σ. The two defor-mations of the Bromwich contour are the parabola and a hyperbola, bothbeginning and ending in the left hand plane, winding around the negativereal axis. The goal behind this is to increase the decay in the integrandsthus speeding up convergence and making the use of a quadrature viable.

∗ The Mittag-Leffler function admits the following series; these appear in Erdelyi [12]but the later of the two dates back to Wiman [53], [54]:

Eα(z) =1

αez1/α −

N∑k=1

z−k

Γ(1 − αk)+ O

(|z|−1−N

)(3.1)

for | arg z| ≤ 34απ, |z| → ∞, and

Eα(z) = −N∑

k=1

z−k

Γ(1 − αk)+ O

(|z|−1−N

)(3.2)

for 34απ ≤ | arg z| ≤ 2π − 3

4απ, |z| → ∞.


As an application the authors apply these to computing the solution of thetime-fractional diffusion equation ∂α

∂tα ut = uxx by changing the problem toone on a discrete space and working with matrices. Specifically the use ofthe discrete Laplace operator whose eigenvalues lie along the negative realaxis and are thus inside the selected Bromwich contours.

Firstly, starting with the series definition of the Mittag-Leffler func-tion, if tα‖G‖ is sufficiently small, then the Mittag-Leffler matrix functionEα(tαG) can be computed with a truncated Taylor series, the error fromwhich can be easily approximated. The size of tα‖G‖ determines how manyterms one will ultimately have to compute to achieve a specific degree ofaccuracy. On the other hand, if tα‖G‖ is large, then one can apply theasymptotic series of the function, again truncating with a number of terms.

3.3. Global rational approximations

Pade approximations for the Mittag-Leffler function are discussed inStarovoitov and Starovoitova [48], where the authors show that the ap-proximants serve uniformly on the compact set {|z| ≤ 1}. While the Padeapproximation is better than a truncated Taylor series, it is not necessarilycompatible with the asymptotic behaviour for large arguments. These arenaturally extendable to matrices. However, when tα‖G‖ 1 both Taylorseries and Pade approximations require many terms, making them in effi-cient, and these methods also suffer from roundoff errors, which make themunreliable.

In a detailed study of fractional-order viscoelasticity, Freed et. al. [15]provide a table of coefficients of rational approximants to the functionEα(−xα), x ≥ 0, α ∈ {0.01, 0.02, 0.03, . . . , 0.98, 0.99}, noting the need fora method that covers intermediate values of x, where neither the Taylorseries nor asymptotic series of the function will suffice. The authors beginwith an integral representation of the function that can be used to calculatevalues to arbitrary accuracy in a finite element setting, then using these ex-act values of the function for a set of points x in [0.1, 15] the coefficients ofthe approximant are solved for. These results are superior to those of [48]because it is the exact values of the function at intermediate values beingapproximated with a rational function, compared to just the Taylor series.Nonetheless, we find it interesting to derive rational approximations in adifferent manner.

Winitzki [55] discusses a global rational approximation, that providesuniform approximations for transcendental functions. These allow one touse both Taylor and asymptotic series that the author uses to find goodapproximations of several functions including the error function, the inverse


tangent and some Bessel functions. We apply this method to the Mittag-Leffler function.

The function Eα(−x) is finite everywhere on the interval (0,∞) andthe function admits the two series:

Eα(−x) =m−1∑k=0

(−x)k

Γ(1 + αk)+ O (xm) ≡ a(x) + O (xm) , x → 0,

Eα(−x) = −n−1∑k=1

(−x)−k

Γ(1 − αk)+ O

(x−n

) ≡ b(x−1

)+ O

(x−n

), x → ∞.

To get the first coefficient of the asymptotic series to be 1, we work insteadwith the function Γ(1 − α)xEα(−x), which admits the two series above,multiplied by Γ(1 − α)x. We now look for a rational approximation of theform

Γ(1 − α)xEα(−x) ≈ p(x)q(x)

≡ p0 + p1x + · · · pvxv

q0 + q1x + · · · qvxv, (3.3)

for some integer v to be optimally chosen. The problem is to find thecoefficients pi and qi such that equation (3.3) has the correct expansions atx = 0 and x = ∞. Since the leading term of equation (3.3) at x = ∞ ispv/qv, we can set pv = qv = 1. This formulation is similar to the problemof Hermite-Pade interpolation with two anchor points, except that one ofthe points is at infinity, where we use an expansion in x−1. The unknowncoefficients pi and qi are found from the system of linear equations

p(x) − q(x)a(x) = O(xm) at x = 0, (3.4)p(x)xv

− q(x)xv

b(x−1) = O(x−n) at x = ∞. (3.5)

After the choice pv = qv = 1, these two equations form an inhomogeneouslinear system of (m + n − 1) equations for 2v unknowns pi, qi, 0 ≤ i < v.Therefore when a solution exists it is unique if m + n is odd.

As an example, we take v = 2 and are thus searching for an approxi-mation for the function Γ(1 − α)xEα(−x) of the form

Γ(1 − α)xEα(−x) ≈ p(x)q(x)

=p0 + p1x + x2

q0 + q1x + x2,

where the two series for Γ(1 − α)xEα(−x) are truncated to orders m = 3and n = 2, yielding the functions

a(x) = Γ(1 − α)x − Γ(1 − α)Γ(1 + α)

x2


and

b(x−1

)= 1 − Γ(1 − α)

Γ(1 − 2α)x.

Making the necessary substitutions and collecting powers of x, we find thefollowing values:

p0 = 0,

p1 =−Γ(1−α)2Γ(1+α)

Γ(1−2α) + Γ(1 + α)

(Γ(1 + α)Γ(1 − α) − 1),

q0 =−Γ(1−α)Γ(1+α)

Γ(1−2α) + Γ(1+α)Γ(1−α)

(Γ(1 + α)Γ(1 − α) − 1),

q1 =Γ(1 + α) − Γ(1−α)

Γ(1−2α)

(Γ(1 + α)Γ(1 − α) − 1). (3.6)

Separating the function Eα(−x), its global rational approximation is given,in terms of these parameters as

Eα(−x) ≈ 1Γ(1 − α)

p1 + x

q0 + q1x + x2(3.7)

for α in (0, 1) and x ≥ 0. As an example, for α = 1/2, using the valuesΓ(1/2) =

√π and Γ(3/2) = 1

2

√π, we have

E1/2(−x) ≈1 + π−2√

πx

1 +√

πx + (π − 2)x2. (3.8)

What is interesting about approximations such as this is that it agreeswith the behaviour of Eα(−x) very well for small x and large x whileproviding a good approximation over intermediate values, uniformly. Forexample, the maximum error over the negative real axis for the v = 2global rational approximation in the case of α = 1/2 is less than 0.8%(this, as expected occurs at an intermediate value of x). Higher-orderapproximations, i.e. values of v greater than 2, can be computed to gaingreater accuracy, and these require nothing more than solving a system ofequations ∗ . Heuristically the approximation is best when m and n areclose.

In terms of matrices, this rational approximation can be applied tocompute the function Eα(tαG). If the matrix admits a known real spec-trum, such as the discrete Laplacian as discussed in Section 2.5, then therational approximation which has a unique solution for specified powers,

∗ We note that these systems of equations can even be solved symbolically in termsof the Gamma functions via any number of packages. This yields the coefficients pi andqi to any degree of accuracy.


can be applied to approximate each of the terms Eα(tαdi) in one matrixcomputation, and thus approximate Eα(tαG). Alternatively this can beapplied directly to the matrix, but the terms on the denominator will needto be inverted after being summed ∗ .

Figure 1. The central row of the Mittag-Leffler matrixfunction Eα(tαG) where G is half the discrete Laplace oper-ator, α = 0.75, h = 0.5 and variance σ2 = 15%, plotted fordifferent times t.

We end this section by returning to the earlier discussion of the dimen-sion of the state space. The space hZ is countable, but to work with finitematrices one must truncate this. Two points can be selected, one on eitherside of the initial state of the process, so that the probability of reachingthese states is negligibly small. One can continue to expand the state spaceuntil the end points have such probability.

It remains to specify the behaviour of the process at the boundary ofthe domain, i.e. at these two points. The choice of absorbing boundaryconditions works well as long as the discrete state-space is large enough sothat the process doesn’t reach the boundary. In terms of the generator,these conditions are imposed by simply setting (all elements of) the firstand the last row to zero identically; and this does not violate the perviousconditions on the generator.

We also note that this discrete space setup also lends itself to boundaryvalue problems. If one wishes to impose an absorbing condition at a pointL in Ω, then one assigns row L of the matrix, and all rows beyond it (be

∗ A matrix inversion is less computationally intense than a full matrix diagonalization,assuming both exist.


it above or below the starting point) to be zero. The same holds for twoabsorbing states.

4. Time-fractional convergence

In this section we first derive integral representations for the evolutionof both the continuous-space time-fractional process and a discrete-spacetime-fractional process which we describe below. These integral representa-tions play a key role in our convergence theorem. We denote, with pα

t (x, y),the solution of the time-fractional diffusion equation:

∂α

∂tαpα

t =12

∂2

∂x2pα

t (4.1)

subject to the condition limt→0 pαt (x, y) = δ(x − y), the initial condition

given by the Dirac delta function concentrated at y. The function pαt (x, y),

viewed as a function of the starting point x, as the solution of this equation,defines the time evolution of the probability density function in space ofthe time-fractional diffusion. Solving fractional equations such as that ofequation (4.1) lead us to solutions in terms of Mittag-Leffler functions.

4.1. Continuous-space time-fractional PDF

To solve equation (4.1) in terms of a Mittag-Leffler function we firstobtain a spectral representation of the space operator, half the Laplace op-erator, on the right hand side of equation (4.1) using the Fourier transform∗ . It is well known that

F(

dn

dxnf(x)

)(p) = (ip)nF(f)(p).

Thus, F(Δf)(p) = −p2F(f)(p) and so the spectral representation of ourspace operator is

F 12ΔF−1(g)(p) = −1

2p2g(p). (4.3)

Noting that by applying Laplace transforms to the equation dα

dtα f(t) =−λf(t) one obtains the solution f(t) = Eα(−λtα), we combine this with

∗ We take the definition of the Fourier transform of an integrable function f as

F(f)(p) :=1√2π

∫ ∞

−∞f(x)e−ipxdx (4.2)

and the inverse Fourier transform

F(g)(x) :=1√2π

∫ ∞

−∞g(p)eipxdp.


the above Fourier transform to find the following representation:

pαt (x, y) =

12π

∫R

Eα

(tα(−p2

2

))eip(x−y)dp. (4.4)

This reduces to the non-fractional case when α = 1, and the Mittag-Lefflerfunction reduces to the exponential function. In this case one obtainspt(x, y) = 1

2π

∫R

e−tp2/2eip(x−y)dp = 1√2πt

exp(−(x − y)2/2t

)that we rec-

ognize as the transition probability density function of a Gaussian process.We note that the integral representation of equation (4.4) is consistent withthose derived in the literature, for example in [45].

4.2. Discrete-space time-fractional PMF

In this section we again search for an integral representation in this caseof the time-evolution of the probability mass function of the discrete statespace process. This function, which we denote as P h,α

t (x, y), is the solutionof the equation defined as follows:

Definition 4.1 (Discrete-space Time-fractional Diffusion Equation).The discrete-space time-fractional diffusion equation is the time-fractionalgeneralization of the discrete-space diffusion equation given by

∂α

∂tαP h,α

t =12ΔhP h,α

t (4.5)

subject to the condition limt→0 P h,αt (x, y) = δxy, the Kronecker delta-

function.

In this case we have defined the operator Δh as in Section 2.2 and weassume the countable state space hZ, viewed as a subset of R. To pro-ceed to the integral representation of P h,α

t (x, y) we must obtain a spectralrepresentation for the discrete Laplace operator Δh. Before attemptinga spectral decomposition we know that as a real symmetric matrix, Δh

will have a real spectrum. To find this spectrum we apply a semi-discreteFourier transform, essentially the analogue of the fourier transform above,defined by:

Fh(f)(p) :=∑n∈Z

f(hn)

√h

2πe−inhp


for any point p in the range[−π

h , πh

]. The operator 1

2Δh then has thespectral representation∗

Fh12Δh F−1

h (φ)(p) =(

cos(hp) − 1h2

)φ(p),

where φ is any element of L2([−π

h , πh

]). The representation of the proba-

bility mass function P h,αt now follows, using the spectral decomposition of

equation (4.7), we have

P h,αt (x, y) =

h

2π

∫ πh

−πh

Eα

(tα(

cos(hp) − 1h2

))eip(x−y)dp. (4.7)

This function gives the time evolution of the probability mass functionin discrete space. We note that the integral representations obtained sofar have the distinct property that the time and space parameters appearindependently; the integrand is a product of two functions, one dependingon t, the other on the difference x − y. This will ultimately allow us tobound the difference between these two in the limit as h goes to zero andestablish convergence of the discretization in the time-fractional case.

4.3. The convergence

We have established representations for both the time evolution of thecontinuous-space time-fractional probability density function, and the timeevolution of the discrete-space probability mass function, both as integrals,given by equations (4.4) and (4.7) respectively. In this section we stateand prove a theorem about the convergence of the discrete-space time-fractional process PMF to the continuous-space time-fractional PDF asthe spacing h goes to zero. In the time-fractional context the numericalschemes in the difference approximations, in space and in time, in bothShen et. al. [47] and Lin and Xu [27] are shown to be convergent. Resultsfor the Markov case can be found in Kushner and Dupuis [26] in which

∗ Operating with Fh on Δh(f) for suitable functions f gives

FhΔh(f)(p) =

√h

2π

∑n∈Z

f(h(n + 1)) + f(h(n − 1)) − 2f(hn)

h2e−inhp

=1

h2

(eiph

∑n∈Z

f(hn)

√h

2πe−inhp + e−iph

∑n∈Z

f(hn)

√h

2πe−inhp

−2∑n∈Z

f(hn)

√h

2πe−inhp

)

=2 (cos(hp) − 1)

h2Fh(f)(p). (4.6)


the authors discretize both space and time, and analyse the convergence indistribution of the approximating Markov chains to the diffusion process.Also, Albanese and Mijatovic [1] establish the optimal convergence rate forthe probability density function of the discretized process as the distanceh between the nodes of the state-space of the Markov chain goes to zero.We can now state and prove the convergence theorem.

Theorem 4.1. For t > 0 and α ∈ (0, 1), denote by pαt (x, y) and

P h,αt (x, y) the time evolutions of the probability densities of the continuous-

space (R) and discrete-space (hZ) time-fractional processes, with the inte-gral representations in equations (4.4) and (4.7) respectively. Then thereexist positive constants C and δ such that the inequality

∣∣∣∣pαt (x, y) − 1

hP h,α

t (x, y)∣∣∣∣ ≤ Ch

holds for all h < δ, independently of x and y.

P r o o f. To prove this result, we need to estimate the difference whichwe denote as

D(h) :=∣∣∣∣pα

t (x, y) − 1h

P h,αt (x, y)

∣∣∣∣ .Since Pt(x, y) is defined on the discrete space hZ, then for a fixed y thisshould be compared with the probability that the continuous process liesin the interval [y, y + h), that is

∫ y+hy pα

t (x, z)dz. Thus, 1hP h,α

t (x, y) shouldconverge to pα

t (x, y) which as h goes to zero is the limit of the average ofpα

t (x, y) over the interval [y, y + h).

We define a(h) to be the function

a(h) = 5√

log(1/h), (4.8)

so that the function a(h) goes to infinity as h → 0 but is bounded aboveby the function π/h as h → 0. We also note that a(h) satisfies the limitha(h)5 → 0 as h → 0.

We now split the domains of integration into 3 the regions: (−∞,−π/h]∪[π/h,∞), [−π/h,−a(h)]∪[a(h), π/h] and [−a(h), a(h)], over each of whichthe integrands behave differently. Writing the difference D(h) over these


new domains, we obtain the following inequality:

2πD(h) =

∣∣∣∣∣∫ π

h

−πh

Eα

(tασ2 cos(hp) − 1

h2

)eip(x−y)dp

−∫

R

Eα

(−tασ2 p2

2

)eip(x−y)dp

∣∣∣∣ (4.9)

≤ 2∫ a(h)

0

∣∣∣∣Eα


h2

)− Eα

(−tασ2 p2

2

)∣∣∣∣ dp

+ 2∫ π/h

a(h)

∣∣∣∣Eα


h2

)− Eα

(−tασ2 p2

2

)∣∣∣∣ dp

+ 2∫ ∞

π/hEα

(−tασ2 p2

2

)dp. (4.10)

The modulus is taken inside yielding this inequality, and the terms eip(x−y)

vanish as they are of modulus one. The factors of 2, appearing in frontof each term, are the result of omitting the integrals over the symmetricregions and noting that the integrands are even functions. The removal ofthe modulus from the integrand of the last term is justified by the positivityof the Eα(−x), x ≥ 0 as discussed above. In the remainder of the proof, wehandle each of these integrals separately, labeling them by I1(h), I2(h) andI3(h) respective to the order they appear in equation (4.10). The combinedresults for all three integrals yield the required result of the theorem.

Bounds for the integral I1(h): The integral I1(h) is given by

I1(h) =∫ a(h)

0

∣∣∣∣Eα

(tασ2 cos(hq) − 1

h2

)− Eα

(−tασ2 q2

2

)∣∣∣∣ dq

=

√2

tασ2

∫ √tασ2

2a(h)

0

∣∣∣∣∣Eα

(tασ2

h2cos

(hp

√2

tασ2

)−1

)− Eα

(−p2)∣∣∣∣∣ dp.

Over the domain of integration, noting that a(h) is bounded above by π/hfor small h, the family of functions

p → tασ2

h2

(cos

(hp

√2

tασ2

)− 1

)

converges uniformly to the function p → −p2. This follows from the Taylorseries expansion of the cosine, from which we also deduce that:

tασ2

h2

(cos

(hp

√2

tασ2

)− 1

)≤ −p2 + h2 p4

4!4

tασ2, (4.11)


because the error from truncating an alternating series with monotonicallydecreasing terms is bounded above by the absolute value of the first termafter the truncation, here being h2 p4

4!4

tασ2 . Our strategy is to approximatethe difference in I1(h) by exploiting these properties along with the mono-tonicity of the Mittag-Leffler function, specifically Eα(−x) being monotonedecreasing for all x ≥ 0, 0 < α < 1, that gives us the inequalities

0 ≤ Eα

(tασ2

h2cos

(hp

√2

tασ2

)− 1

)− Eα

(−p2)

(4.12)

≤ Eα

(−p2 + h2 p4

3!tασ2

)− Eα

(−p2). (4.13)

The term Eα

(−p2 + h2 p4

3!tασ2

)we now expand as follows:

Eα

(−p2 + h2 p4

3!tασ2

)= Eα

(−p2)

+ h2 p4

3!tασ2

∂

∂p2Eα

(−p2)

(4.14)

+ h4 12!

(p4

3!tασ2

)2(∂

∂p2

)2

Eα

(−p2)

+ · · ·

≤ Eα

(−p2)− h2 p4

3!tασ2

∂

∂p2Eα

(−p2), (4.15)

where the series appearing after the equality is alternating, by virtue ofthe completely monotonic nature of the function, and in which each of the

derivatives of the function Eα

(−p2)

are bounded for p ∈[0,√

tασ2

2 a(h)].

The inequality follows again from the bound on a truncated alternatingseries, and the fact that h is small. To conclude our bound on the integralI1(h) we note that ∂

∂p2 Eα

(−p2)

is an increasing function on the domainof integration, again in reference the complete monotonicity which tells usthat its derivative, the second derivative of Eα

(−p2)

with respect to p2, ispositive. Using these, we now find

I1(h) ≤√

2tασ2

∫ √tασ2

2a(h)

0h2 p4

3!tασ2

[− ∂

∂p2Eα

(−p2)]

dp (4.16)

≤√

2tασ2

h2

3!tασ2max

0≤p≤√

tασ2

2a(h)

[− ∂

∂p2Eα

(−p2)]∫ √

tασ2

2a(h)

0p4dp (4.17)

=

√2

tασ2

h2

3!tασ2

15

(√tασ2

2a(h)

)5 [− ∂

∂p2Eα

(−tασ2 p2

2

)∣∣∣∣p=0

](4.18)


=tασ2

2h2

3!15a(h)5

1Γ(1 + α)

(4.19)

= C1h2a(h)5. (4.20)

The first inequality follows from equation (4.15). The second uses the factthat the term inside the brackets of equation (4.16) is both positive and

bounded on the interval[0,√

tασ2

2 a(h)], and the integral of equation (4.16)

is thus bounded by that in equation (4.17) multiplied by the maximum pre-ceding it. It is also the case that owing to the monotonic decreasing natureof this term, the point p = 0 appearing in equation (4.18) is where themaximum is attained over the domain of integration. The Gamma func-tion appearing in equation (4.19), the value of the derivative at zero, is seenby conducting a term by term differentiation of the Mittag-Leffler series andevaluating at the point zero. Equation (4.20) concludes by expressing thebound on the first of our integrals in terms of powers of h and a(h), and apositive constant C1 independent of h.


I2(h) =∫ π/h

a(h)

∣∣∣∣Eα


h2

)− Eα

(−tασ2 p2

2

)∣∣∣∣ dp. (4.21)

The strategy here is to obtain the asymptotic behaviour of this integralas h → 0, making use of the asymptotic power series for Mittag-Lefflerfunction given in equation (3.2). Knowing that the arguments of bothMittag-Leffler functions in the integrand of I2(h) lie along the negative realaxis, we have the following asymptotic power series as x → ∞:

Eα (−x) = −N−1∑k=1

(−x)−k

Γ(1 − αk)+ O(|x|−N ), (4.22)

which holds for all integers N ≥ 2. Starting with the Nth order seriesof the form in equation (4.22) for the Mittag-Leffler functions in equation(4.21), we apply linearity of the asymptotic series to obtain an asymptoticseries, to N terms, for their difference:

Eα


h2

)− Eα

(−tασ2 p2

2

)

=N∑

k=1

(tασ2

)−k

Γ(1 − αk)

[(−p2

2

)−k

−(

cos(hp) − 1h2

)−k]

(4.23)


as h → 0. By definition the function a(h) goes to infinity in this limit andthus both these series are justified on the domain of integration [a(h), π/h],independently and then combined.

We now apply the following property of asymptotic power series: thedifference between a function f(x) and the first two terms b0 and b1

x of itsN term asymptotic power series, when it exists, is integrable. The seriesappearing here involves only even powers of p, so b1 is zero, as is the termb0, implying that the right hand side of the above equation is integrable.We have:

I2(h) =∫ π/h

a(h)

N∑k=1

(tασ2

)−k

Γ(1 − αk)

[(−p2

2

)−k

−(

cos(hp) − 1h2

)−k]

dp

=N∑

k=1

(−tασ2)−k

Γ(1 − αk)

∫ π/h

a(h)

[(p2

2

)−k

−(

1 − cos(hp)h2

)−k]

dp. (4.24)

The difference inside the brackets is negative for all k ≥ 1, p ∈ [a(h), π/h]and the whole of the right hand side of equation (4.24) positive for allN ≥ 1. The interchange of the sum and integral justified by the MonotoneConvergence Theorem. The difference inside the brackets is an integrablefunction on [a(h), π/h] for all k ≥ 1 and making use of the integral ∗ :

∫1

(1 − cos(x))ndx = − 1

2n−1

n−1∑j=0

(n − 1

j

)cot2j+1

[x2

]2j + 1

, (4.25)

we now argue as follows that the integral of the terms inside the the bracketof equation (4.24), is of leading term hk. Noting the Laurent series for cot(z)about the origin,

cot(z) =1z− 1

3z − 1

45z3 − · · · ,

the first of these integrals is, when k = 1,

∫ π/h

a(h)

[2p2

− h2

1 − cos(hp)

]dp =

[−2

p+ h cot

(hp

2

)]π/h

a

(4.26)

∗ This integral can be proven by an inductive argument, taking the derivative on thenth step to verify the hypothesis.


= −2hπ

+2a− h cot

(ha

2

)

= −2hπ

+2a− h

(2ha

− 13

ha

2− 1

45

(ha

2

)3

− · · ·)

= −2hπ

+h2a

6+

145

h4a3

23+ · · · . (4.27)

We note that the terms on the right hand side of the last equality, startingwith that of order h2, form a convergence series ∗ . The next integral,corresponding to k = 2, is∫ π/h

a

[4p4

− h4

(1−cos(hp))2

]dp=

1h

∫ π

ah

[4h4

x4− h4

(1−cos(x))2

]dx (4.28)

= h3

[−4

31x3

+12

cot[x2

]+

16

cot3[x2

]]π

ah

= h3

[−4

31π3

+43

1(ah)3

− 12

cot[ah

2

]− 1

6cot3

[ah

2

]].

For small h we again use the Laurent series for cot(z) about the origin toget

−16

cot3[ah

2

]= −1

6

(2ah

− 16ah − 1

45(ah)3 − · · ·

)3

= −43

1(ah)3

+1

6ah+· · ·

and

−12

cot[ah

2

]= − 1

ah+

112

ah +12

145

(ah)3 − · · · ,

where the omitted terms in the first series are of order h and higher. Wefind ∫ π/h

a

[4p4

− h4

(1 − cos(hp))2

]dp

= h3

[−4

31π3

+1

5ah+

112

ah +12

145

(ah)3 − · · ·]

. (4.29)

It is clear from this equation that the term of interest arises from the secondterm in the sum, multiplied by h3 to give h2

5a which, using the definitionof a = a(h), goes to zero as h goes to zero at a rate faster than h2. Therest of the terms, including the first one, go to zero at a faster rate thanthis. For k ≥ 3, values of the integral of equation (4.24) go at the rateclaimed, using formula (4.25) and the Laurent series for cot(z), noting that

∗ Its convergence can be seen by looking at the terms in the series of the functioncot(z)− 1

z, which involve Bernoulli numbers, and can be shown to be convergent by any

one of a number of standard tests.


via by definition of the function a(h) we know that limh→0 ha(h) = 0. Weconclude that

I2(h) ∼N∑

k=1

hkck

(−tασ2)−k

Γ(1 − αk)(4.30)

as h → 0, for arbitrary N . The coefficients ck are a set of decreasing termsin k which we pick up through the various integral evaluations, the firstof which, c1, is observed in and only in equation (4.27), and is c1 = − 2

π .The first term of the series of equation (4.30) is thus positive, noting thatΓ(1−α) > 0 when 0 < α < 1, and this concludes our analysis of the integralI2(h).


I3(h) =∫ ∞

π/hEα

(−tασ2 p2

2

)dp (4.31)

which we treat using the following limit:

limx→∞

∫∞x Eα

(−λp2)dp

xEα (−λx2)= 1 , for any λ ∈ (0,∞). (4.32)

That is to say,∫∞x Eα

(−λp2)dp and xEα

(−λx2)

are asymptotically equiv-alent as h → 0.

To establish this limit we shall need to know the asymptotic behaviourof the derivative of the Mittag-Leffler function Eα(−x2) with respect tox. Firstly, the asymptotic power series for Eα(−x2), can be obtained fromequation (4.22). Noting that the derivative of the Mittag-Leffler functionEα(−x) is a monotone increasing on [0,∞) and is a bounded function, itadmits an asymptotic series for x → ∞. The existence of the asymptoticseries for the derivative, and knowledge of the series for the function itselfmean we can legitimately differentiate the latter, term by term, to obtainthat of the former:

∂

∂xEα

(−x2)

=∂

∂x

[−

N−1∑k=1

(−x2)−k

Γ(1 − αk)+ O

(x−2N

)]

= 2N−1∑k=1

(−1)k+1x−2k−1

Γ(1 − αk)+ O

(x−2N

)(4.33)

the first term of which is 2Γ(1−α)x

−3.


For the application of L’Hospital’s rule to be valid we note the followinglimits for the two separate parts of the quotient in equation (4.32)

limx→∞

[∫ ∞

xEα

(−λp2)dp

]= 0 and lim

x→∞[xEα

(−λx2)]

= 0,

where for the second limit we have used the asymptotic power decay of theMittag-Leffler function. Applying L’Hospital’s rule to the limit in equation(4.32) we have

limx→∞

∫∞x Eα

(−λp2)dp

xEα (−λx2)= lim

x→∞Eα

(−λx2)

Eα (−λx2)−2λxE′α (−λx2)

(4.34)

= limx→∞

1λΓ(1−α)x2 + O(x−4)

1λΓ(1−α)x2 + O(x−4) − 4

λΓ(1−α)x3 + O(x−5)(4.35)

= 1, (4.36)

and thus, in the limit as h → 0, the integral I3(h) grows like[xEα

(−λx2)]

x=π/hand we note that the function xEα

(−λx2)

has an as-

ymptotic form of 1xΓ(1−α) +O(x−3) for large x. This completes the analysis

of I3.

Combining the results obtained for I1(h) in equation (4.20), I2(h) inequation (4.30) and that of I3(h) above, with the inequality of equation(4.10), concludes the proof of Theorem (4.1). �

5. Conclusion

In this paper we defined and studied in continuous-time a discrete-spacetime-fractional process. This involved the application of a natural dis-cretization scheme to the generator of a diffusion and generalizing the timederivatives to one of arbitrary order. Using fractional calculus we solvedthese equations and studied the solutions given in terms of Mittag-Lefflermatrix functions. The computation of such matrix functions was studiedand new global rational approximations derived. The derivation of inte-gral representations for the time-fractional process’s probability functionsallows us to prove the convergence of the time-evolution of the probabilitymass function in space of the discrete space process to that of the proba-bility density function of the continuous one as the spacing in the discretespace goes to zero.

References

[1] C. Albanese and A. Mijatovic, Convergence rates for diffusions oncontinuous-time lattices. Working paper, 2006.


[2] W.J. Anderson, Continous-time Markov Chains. Springer, 1991.[3] C. Atkinson and A. Osseiran, Rational solutions for the time-fractional

diffusion equation. SIAM J. Appl. Math., 71 (2011), 92–106.[4] L. Boyadjiev, S.L. Kalla, and H.G. Khajah. Analytic and numerical

treatment of a fractional integro-differential equation of Volterra type,Math. Comput. Model., 25 (1997), 1–9.

[5] M. Caputo, Linear models of dissipation whose Q is almost frequencyindependent, Part II. Geophys. Journal Royal Astronomic Soc., 13(1967), 529–539 (Reprinted in: Fract. Calc. Appl. Anal., 11, No 1(2008), 3–14).

[6] M.C. Cartwright, Integral Functions. Cambridge Tracts in Mathematicsand Mathematical Physics, No 44, Cambridge Univ. Press, 1956.

[7] A.A. Chikrii and S.D. Eidel’man, Generalized Mittag-Leffler matrixfunctions in game problems for evolutionary equations of fractionalorder. J. of Cybernetics and Systems Analysis, 36, No 3 (2000), 315–338.

[8] K.L. Chung, Markov Chains with Stationary Transition Probabilities.Springer-Verlag, Berlin, Second edition, 1967.

[9] K. Diethelm, The Analysis of Fractional Differential Equations.Springer-Verlag, Berlin-Heidelberg, 2010.

[10] K. Diethelm and N.J. Ford, Numerical solution methods for distributedorder differential equations. Fract. Calc. Appl. Anal., 4 (2001), 531–542.

[11] K. Diethelm and Yu. Luchko. Numerical solution of linear multiterminitial value problems of fractional order. Comput. Methods of Numer-ical Analysis, 6 (2004), 243–263.

[12] A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi. HigherTranscendental Functions, Bateman Project, Volume 3. McGraw-Hill,New York, 1955.

[13] W. Feller. Fluctuation theory of recurrent events, Trans. Amer. Math.Soc., 67 (1949), 98–119.

[14] W. Feller. An Introduction to Probability Theory and Its Applications,Volume 2. Wiley, New York, Second Edition, 1971.

[15] A. Freed, K. Diethelm, and Yu. Luchko. Fractional-order viscoelasticity(FOV): Constitutive development using the fractional calculus. FirstAnnual Report, NASA/TM-2002-211914, Gleen Research Center.

[16] R. Gorenflo and E.A. Abdel-Rehim. Convergence of the Grunwald-Letnikov scheme for time-fractional diffusion. Journal of Comput. andAppl. Mathematics, 205, No 2 (2007), 871–881.

[17] R. Gorenflo, J. Loutchko, and Yu. Luchko. Computation of the Mittag-Leffler function Eα,β(z) and its derivatives. Fract. Calc. Appl. Anal., 5,No 4 (2002), 491–518.


[18] R. Gorenflo and F. Mainardi. Some recent advances in theory andsimulation of fractional diffusion processes. Journal of Comput. andAppl. Mathematics, 229, No 2 (2009), 400–415.

[19] G.R. Grimmett and D. Stirzaker. Probability and Random Processes.Oxford University Press, Third edition, 2001.

[20] N.J. Higham. The scaling and squaring method for the matrix expo-nential revisited. SIAM Journal on Matrix Anal. and Appl., 26, No 4(2005), 1179–1193.

[21] T. Kaczorek. Fractional positive continuous-time linear systems andtheir reachability. Int. J. Appl. Math. Comput. Sci., 18, No 2 (2008),223–228.

[22] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo. Theory and Applica-tions of Fractional Differential Equations. Elsevier, Amsterdam, 2006.

[23] V. Kiryakova. Generalized Fractional Calculus and Applications. Long-man Sc. and Tech. & J. Wiley, Harlow - N. York, 1994.

[24] V. Kiryakova. The multi-index Mittag-Leffler functions as im-portant class of special functions of fractional calculus. Comput-ers and Math. with with Appl., 59, No 5 (2010), 1885–1895,doi:10.1016/j.camwa.2009.08.025.

[25] A.N. Kochubei. A Cauchy problem for evolution equations of fractionalorder. Differential Equations, 25(1989), 967-974.

[26] H. Kushner and P. Dupuis. Numerical Methods for Stochastic ControlProblems in Continuous Time. Springer, 2001.

[27] Yumin Lin and Chuanju Xu. Finite difference/spectral approximationsfor the time-fractional diffusion equation. Journal of ComputationalPhysics, 225 (2007), 1533–1552.

[28] Ch. Lubich. Discretized fractional calculus. SIAM Journal on Math.Anal., 17, No 3 (1986), 704–719.

[29] F. Mainardi. Applications of integral transforms in fractional diffusionprocesses. Integr. Transf. Spec. Funct., 15, No 6 (2004), 477–484.

[30] F. Mainardi, Yu. Luchko, and G. Pagnini. The fundamental solution ofthe space-time fractional diffusion equation. Fract. Calc. Appl. Anal.,4, No 2 (2001), 153–192.

[31] F. Mainardi and R. Gorenflo. Time-fractional derivatives in relaxationprocesses: A tutorial survey. Frac. Calc. Appl. Anal., 10, No 3 (2007),269–308.

[32] A.C. McBride. Fractional Calculus and Integral Transforms of Gener-alized Functions. Pitman Press, San Francisco, 1979.

[33] A.C. McBride and G. Roach. Fractional Calculus. Research Notes inMathematics, Vol. 138, Pitman, Boston-London-Melbourne, 1985.


[34] K.S. Miller and B. Ross. An Introduction to the Fractional Calculusand Fractional Differential Equations. John Wiley, New York, 1993.

[35] C. Moler and C. Van Loan. Nineteen dubious ways to compute theexponential of a matrix, twenty-five years later. SIAM Review, 45, No1 (2003), 3–30.

[36] B. Øksendal. Stochastic Differential Equations. Springer, Sixth edition,2003.

[37] K.B. Oldham and J. Spanier. The Fractional Calculus. Academic Press,New York, 1974.

[38] E. Platen. Higher order weak approximation of Ito diffusions byMarkov chains. Probability in the Engineering and Information Sci.,6 (1992), 391–408.

[39] I. Podlubny. Fractional Differential Equations. Academic Press, SanDiego, 1999.

[40] I. Podlubny. Matrix approach to discrete fractional calculus. Frac.Calc. Appl. Anal., 3, No 4 (2000), 359–386.

[41] I. Podlubny, A. Chechkin, T. Skovranek, YangQuan Chen, andB. M. Vinagre Jara. Matrix approach to discrete fractional calculus II:Partial fractional differential equations. J. of Computational Physics,228, No 8 (2009), 3137–3153.

[42] H. Pollard. The complete monotonic charachter of the Mittag-Lefflerfunction Eα(−x). Bull. Amer. Math. Soc., 54 (1948), 1115–1116.

[43] B. Rubin. Fractional Integrals and Potentials. Pitman Monographs andSurveys in Pure and Appl. Math., Vol. 82, Longman, Harlow, 1996.

[44] S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals andDerivatives, Theory and Applications. Gordon and Breach, Amsterdam,1993.

[45] E. Scalas, R. Gorenflo, F. Mainardi, and M. Raberto. Revisiting thederivation of the fractional diffusion equation. In: Scaling and Disor-dered Systems (Eds.: F. Family et. al.), World Scientific, 2002, 281–289.

[46] W.R. Schneider and W. Wyss. Fractional diffusion and wave equations.J. Math. Phys., 30 (1989), 134–144.

[47] S. Shen, F. Liu, V. Anh, and I. Turner. Detailed analysis of a conserva-tive difference approximation for the time-fractional diffusion equation.J. Appl. Math. Comput., 22, No 3 (2006), 1–19.

[48] A.P. Starovoitov and N.A. Starovoitova. Pade approximants of theMittag-Leffler functions. Sbornik: Mathematics, 198, No 7 (2007),1011–1023.


[49] J. Tenreiro Machado, V. Kiryakova, F. Mainardi. Recent his-tory of fractional calculus. Communications in Nonlinear Sci-ence and Numerical Simulations, 16, No 3 (2011), 1140–1153,doi:10.1016/j.cnsns.2010.05.027.

[50] J. Tenreiro Machado, V. Kiryakova, F. Mainardi. A note and posteron the recent history of fractional calculus. Fract. Calc. Appl. Anal.,13, No 3 (2010), 329–334, http://www.math.bas.bg/∼fcaa.

[51] J. Tenreiro Machado, V. Kiryakova, F. Mainardi. A poster about theold history of fractional calculus. Fract. Calc. Appl. Anal., 13, No 4(2010), 447–454, http://www.math.bas.bg/∼fcaa.

[52] J.A.C. Weideman and L.N. Trefethen. Parabolic and hyperbolic con-tours for computing the Bromwich integral, Working paper. Dec. 2005.

[53] A. Wiman. Uber den Fundamentalsatz in der Theorie der FunktionenActa Math., 29 (1905), 191–201.

[54] A. Wiman. Uber die Nullstellen der Funktionen Acta Math., 29 (1905),217–234.

[55] S. Winitzki. Uniform approximations for transcendental functions. Lec-ture Notes in Computer Science, Springer-Verlag, Volume 2667, 2003.

[56] W. Wyss. The fractional diffusion equation. J. Math. Phys., 27 (1986),2782–2785.

∗,∗∗ Department of MathematicsImperial College180 Queen’s Gate, London SW7 2BZ, UK

∗ Corresponding author, e-mail: [email protected]∗∗ e-mail: [email protected] Received: June 15, 2010

http://www.math.bas.bg/~fcaa

http://www.math.bas.bg/~fcaa

discrete-space time-fractional processes

Documents