numerical schemes for integro differential equations with erd lyi kober fractional operator

Keywords Erdelyi-Kober operator· Fractional calculus · Finite difference ·Integro-differential equation 1 Introduction Fractional calculus constitutes a very vast area in which many inte

DOI 10.1007/s11075-016-0247-z

ORIGINAL PAPER

Numerical schemes for integro-differential equations

with Erd´elyi-Kober fractional operator

Łukasz Płociniczak 1 · Szymon Sobieszek 1

Received: 29 May 2016 / Accepted: 30 November 2016

© The Author(s) 2016 This article is published with open access at Springerlink.com

Abstract This work investigates several discretizations of the Erd´elyi-Kober

frac-tional operator and their use in integro-differential equations We propose twomethods of discretizing E-K operator and prove their errors asymptotic behaviour forseveral different variants of each discretization We also determine the exact form oferror constants Next, we construct a finite-difference scheme based on a trapezoidalrule to solve a general first order integro-differential equation As is known fromthe theory of Abel integral equations, the rate of convergence of any finite-differentmethod depends on the severity of kernel’s singularity We confirm these results inthe E-K case and illustrate our considerations with numerical examples

Keywords Erdelyi-Kober operator· Fractional calculus · Finite difference ·Integro-differential equation

1 Introduction

Fractional calculus constitutes a very vast area in which many interesting cal and physical objects reside From the point of view of the latter, fractional modelsmany times happen to describe natural phenomena with incredible accuracy proba-bly thanks to its intrinsic nonlocal properties [21,38] These, in turn, can be used tomodel history of the considered process and a variety of memory effects [36,46].There are many examples of applications of fractional models [5,21] One of the

mathemati- Łukasz Płociniczak

lukasz.plociniczak@pwr.edu.pl

1 Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wyb Wyspia´nskiego 27, 50370 Wrocław, Poland

most successful is anomalous diffusion [34,36,37], which can be observed in ety of situations such as moisture percolation in porous media [39], protein randomwalks in cells [53], telomere motion [7,23] and diffusion of cosmic rays across themagnetic fields [11] When considering self-similar solutions to a sub-diffusive evo-lution equation [19,47], the fractional derivative operator (either Riemann-Liouville

vari-or Caputo) becomes the so-called Erd´elyi-Kober (E-K) fractional integral [15,27]which possesses many interesting mathematical and physical features [20,41,50]

The E-K operator, which we will denote by I a,b,c, is weakly-singular and can beviewed as one of Volterra (or Abel) type Its precise definition will be given in Section

2but note that the general theory can be applied to it However, utilizing specificfeatures of E-K operator leads to many interesting results A thorough expositionconcerning the theory of E-K fractional integral is presented in the book [25] In [1,

26], a number of solutions to the E-K integral equations have been obtained (butsee also [33]) while in [24] further results for hyper-Bessel operator were given.Moreover, some questions about existence and uniqueness were answered in [22,52].There is a very abundant literature about numerical methods for both fractionaldifferential equations and integral equations To state only a few, we start from men-tioning two classic monographs concerning numerical methods for Volterra (andAbel) integral equations [8,31] Also, the reader will find there a thorough treat-ment of integro-differential equations with Volterra operators A more modern reviewpaper [4] summarizes recent results on a variety of numerical ways of solving con-sidered equations As being inegro-differential operators, fractional derivatives can

be treated similarly to the more general cases However, certain advantages can begained from exploiting particular structure of these operators A review of numericalmethods (as well as analytical results) for ordinary fractional differential equationshas been given in [6], where a modern overview and practical algorithms are given.The book contains a number of interesting references to which interested reader

is referred to Also, the paper [16] discusses some general methods for ordinaryfractional differential equations while [29] gives a analysis of a non-uniform gridapproximation Lastly, we would like to mention several papers discussing numericalapproaches to time-fractional diffusion A very thorough treatment has been given

in [30] where a combined space-time spectral method was used A similar setting offinite differences was also applied in [31] Some recent works on the nonlinear caseinclude papers on finite difference schemes for inverse problem [13] and single-phaseflow in porous media [3]

The motivation behind this paper is a self-similar solution of time-fractionalporous medium equation (see [40,51]) As we noted before, E-K operator appears insuch a situation very naturally as a part of ordinary integro-differential equation mod-elling moisture distribution in a variety of building materials [14,28] (also see somenew experimental results [55]) In our preceding works [42–45], we have devised asystematic way of approximating the solution of that equation by a simple, analyticalformulas It was then compared with the numerical solution to verify its applicabil-ity and accuracy We noticed that the finite difference scheme for the time-fractionalpartial differential equation was very demanding on the computer power and obtain-ing an array of solutions for different values of was not practical Nonlocality andnonlinearity of the investigated equation is the obvious reason for such a case This

paper is a first step in deriving a more optimal numerical method which is structed for the self-similar ordinary rather than original partial differential equation.

con-In what follows, we introduce two types of discretization of the E-K operator, findtheirs truncation errors with exact error constants and apply those results to construct

a second-order finite difference scheme which approximates the solution of the first

order integro-differential equation with E-K operator I a,b,c, namely

The objective for future work will be to extend these results to the self-similarnonlinear time-fractional diffusion

2 Discretization of the Erd´elyi-Kober operator

Let us define the Erd´elyi-Kober (E-K) fractional integral operator by the formula

I a,b,c y(x):= 1

(b)

 1 0

(1− s) b−1s a

y

s 1/c x

where y is at least locally integrable The above definition is one of the few equivalent

ones found in the literature Others can be obtained by a change of the variable [26].The definition that will be particularly useful for numerical calculations arises from

the transformation t = s 1/c x made in the (2) This leads to the Volterra operatorrepresentation

I a,b,c y(x)= cx −c(a+b)

We will take the above assumption to be valid for the rest of our work unless

differ-ently stated This specific choice of domains for a and b is required for the integral

(2) to be convergent However, by the analytic continuation, a and b can be assumed

to lie within the domain of Beta function but we will not pursue this route here (butsee [43]) Note also, that in some important applications, we have a= 0 Apart fromthat, as can be seen from the self-similar analysis of the time-anomalous diffusionequation (see [9,12,42]), the particular version of the E-K operator that arises there

requires c < 0 However, we defer the analysis of such case to our future work and

in the present paper we assume that c > 0 Additional results concerning self-similar

solutions of the fractional differential equations and E-K operators can be found forexample in [10,17,18,48]

The main idea behind discretization of E-K operator is to apply a quadrature rule

for approximating only the function y and not the rest of the integrand This will

allow us to conduct a part of calculations analytically minimizing the discretizationerror The type of quadrature can be chosen according to be suited for a particularapplication (or preference) and here we consider rectangular, mid-point and trape-

zoidal quadratures This overall procedure, throughout the literature, is called product integration method (see [32])

First, fix x and consider the representation (2) Introduce a grid of the [0, 1]

interval

0= s0< s1< s2< · · · < s i < · · · < s n = 1, (6)where maxi (s i+1− s i ) → 0 as the grid is refined, i.e n → ∞ At this point, it is not necessary to discretize the x variable Now, we have

y

s 1/c x

and we consider several ways of approximating y on a subinterval [s i , s i+1) More

specifically, we apply a chosen quadrature to the function Y (s) := y(s 1/c x)for fixed

x and c.

• Rectangular rule Here, on each subinterval we build an approximating

rectan-gle with its height equal to Y (s i ) By L r a,b,cdenote the operator which gives the

discretization of I a,b,c It has the form

v i r (a, b):= 1

(b)

 s i+1

s i (1− s) b−1s a

case, a= 0, can be evaluated explicitly

v r i ( 0, b)=(1− s i ) b − (1 − s i+1) b

• Mid-point rule Here, the height of the approximating rectangle is Y (s i +1/2 ),

where s i +1/2 := (s i+1+ s i )/2 The mid-point rule discretization is as follows

• Trapezoidal rule In the trapezoidal rule, we approximate the function Y by the

line segments, i.e Y (s)≈(Y (s i+1) − Y (s i )) / (s i+1− s i )

Note that all of the above discretizations need to evaluate y at a point s i 1/c x, which

depends on the parameter c and the [0, 1] grid This is a drawback of the method since given the x-grid it would require additional approximation by interpolating values of

y : s i 1/c x does not have to belong to the x-grid.

As it will become clear, more sensible in most situations is to consider the Volterrarepresentation of the E-K operator (3) Here, we fix x and define the grid

0= t0< t1< t2< · · · < t i < · · · < t n = x. (16)Similarly as above we expand the E-K integral (3)

I a,b,c y(x)=cx −c(a+b)

) b−1t c(a +1)−1 y(t)dt, (17)

and apply the standard interpolations to the function y This has the advantage that y

will be calculated on the grid points

• Rectangular rule We approximate y on [y i , y i+1) by y(t i )and obtain

K a,b,c r y(x):= cx −c(a+b)

where the weights

w i r (a, b, c):= B((t i+1/x) c ; a + 1, b) − B((t i /x) c ; a + 1, b)

differ from (9) only by points at which the Incomplete Beta Function is evaluated

The dependence on c has moved from the argument of y into the weight The special case a= 0 is

where the weights are the same as in the rectangular rule

• Trapezoidal rule We use the first-order Lagrange polynomial to approximate

y(x)for each interval[t i , t i+1], i.e y(t) ≈ (y(t i+1) − y(t i ))/(t i+1− t i )

Lemma 1 For a, b ∈ R and c > 0 we have

a−1

1−



i n

cb−1

=

 1 0

s a−1(1− s c ) b−1ds. (28)

After substitution t = s c the last integral defines Beta function c−1B(ac−1, b).

Now, assume that a < 0 and b > a The sum (27) can then be rewritten as

n −a n−1

i=1



1−i n

cb−1

where we have moved the largest power of n in front of the series We have to show that the sum above converges to ζ (1 − a) First, when b ≥ 1 the sequence (1 − (i/n) c ) b−1is nondecreasing for any fixed i and thus bounded from above by 1 and

by 1− (b − 1)(i/n) cfrom below Hence, we can write

Notice that the majorizing series from above converges to ζ (1 − a) The estimate

from below has exactly the same limit and in order to see that we have to consider the

magnitude of c > 0 More specifically, we have an elementary result which follows

from the asymptotics of partial sums of the Riemann Zeta function (which can beshown by the Euler-Maclaurin formula)

Assume now that a < b < 1 Notice that the function f n (x) := (1−(x/n) c ) b−1x a−1

for 0≤ x ≤ n has a minimum at x min = n(1 + c(b − 1)/(a − 1)) −1/c≥ 0 Define

i min := [x min ] (where [x] is the integral part of x) and decompose

We will show that the last integral converges to 0 as n → ∞ Set  := 1/n and use

the L’Hospital’s Rule

cb−1

Combining (38) and (36) with (33) proves the case of a < b < 1.

The inverted dependence b < a follows the same reasoning with slight tions For example, when b < 0 and a ≥ 1, the series (27) by a change of summation

modifica-variable i → n − i can be written as

series by integral and apply Lebesgue’s Theorem)

Assume now that a = 0 and b > 0 (the case with b < 0 is proved) Other case, i.e b = 0 and a > 0 can be demonstrated in a similar way The instance with b ≥ 1

is a consequence of the Lebesgue’s Theorem just as above (or elementary estimates)

For 0 < b < 1, we anticipate logarithmic asymptotic behaviour and to prove it use

the bounds by appropriate integrals Start with the estimate from below

cb−1

i n

A change of variable x = ny and x + 1 = ny in each integral respectively lets us

refine the estimate into

(45)

We thus can see that the second integral goes to a positive constant while the last term

becomes zero as n→ ∞ It suffices to show that the first term above has logarithmicasymptotics To this end use the L’Hospital’s rule and obtain (we implicitly substitute

 = 1/n to conduct the differentiation)

cb−1

i n

−1

We are left with proving the case a = b < 0 For equal parameters, the function

f n (x) defined above has its minimum at x min = n(1 + c) −1/c Setting i min = [x min]and decomposing our sum yields

c1−a

i1−a (48)

We change the summation index in the last term to n − i to obtain

1−a . (49)

Observe that nc(1 +c) −1/c ≤ i min and nc(1 +c) −1/c < n −i min −1 ≤ n/2−1 hence

by the same argument as before we can use the Lebesue Dominated ConvergenceTheorem and (40) to finally get

1−a

i1−a = (1 + c a−1)ζ (1− a), (50)

for a = b < 0 This concludes the proof.

For further reference, it will be useful to define the following function

γ a,b,c (s) := (1 − s c ) b−1s a , s ∈ (0, 1), a, b ∈ R, c > 0. (51)

The next lemma is an auxiliary result giving another property of γ a,b,c

Lemma 2 Fix m ≥ 1 If −1 < a < m − 1 or 0 < b < m, we have

Proof Notice that the mth derivative of γ a,b,c will contain terms of the form (1−

σ i c ) α s β , where by inspection the lowest exponents will be α := b − m − 1 or β :=

a − m (lowest exponents dominate the asymptotics) Each differentiation will bring one c − 1 term into the β exponent but overall it will be larger than a − m (since

c >0) Hence, it suffices to consider only two cases

If a > m − 1 and b > m, then α > 0 and β > −1 and thus by Lemma 1 (and the fact i/n < σ i < (i + 1)/n) it follows that T n = O(n −(k+m−1) ) , which for m > 1 is

of smaller order than n −(k−1).

Now, assume that a < m − 1 and b > m Again, by Lemma 1 the dominant term is O(n m −δ+1 ) , hence T

n = O(n −(k+δ−1) as n → ∞ Because δ > 0, we have

k + δ − 1 > k − 1 therefore T n = o(n −(k−1) ) Logarithmic behaviour is dealt with

the same way

We can now find the orders of discretization errors of the approximation operators

L a,b,c and K a,b,c We will limit our reasoning to the case of uniform grids, i.e s i =

i/n and t i = x i/n, where n is the number of grid divisions Note that s i+1−s i = 1/n and t i+1− t i = x/n.

Theorem 1 (Discretization errors) Fix a, b, c > 0 an assume that y ∈ C2( 0, X) Then, for a fixed x ∈ (0, X) the discretization errors corresponding to the operator

I a,b,c have the following asymptotic behaviour as n → ∞.

(53)

 1

c− 1y(σ1

c x) 12(b)

⎧

⎪

⎪

n−2B 1

O(n−2), c(a + 1) ≥ 1 and b ≥ 1;

O(n −(1+min{c(a+1),b}) ), 0 < c(a + 1) < 1 or0 < b < 1.

(56)

Here, σ ∈ (0, 1) and τ ∈ (0, x) depend on parameters a, b, c, function y and can be different for each discretization.

Remark 1 Note that it is possible that some of the above formulas can indicate that

the difference between discretization and the E-K operator is asymptotic to zero Thissimply means that the error is of higher order than stated In other words, asymp-totic relations above give the lowest order of discretization error Finding the wholeasymptotic expansion of these quantities is one of the objectives of our future work

Proof Theorem 1 Let us start with the simplest case of the rectangular rule First, consider the discretization of the first type, i.e operator L r a,b,c defined in (8)

Expanding in the Taylor series we have for s ∈ [s i , s i+1)and someσ ∈ (s i , s i+1)

I a,b,c y(x) = L r

a,b,c y(x) + R r , (58)where

(1− s) b−1s a

Then, use the Mean Value Theorems, in turn for integrals and sums (notice that (s−

s i ) does not change its sign for s ∈ (s i , s i+1)), to obtain

inte-of discretization error To proceed, we expand F in the Taylor series noting that

in the above formula is O(n−2)(by Lemma 1) and we have to show that the second

is always of higher order When we expand the derivative we obtain

Lemma 2 immediately states that the right-hand side is o(n−1) (take k = 2 with

m = 2 and m = 1 for the first and second sum respectively) From this estimate on,

the remainder in (62) we can conclude

R r n→∞∼ x

c

y(σ1

c x) 2(b)

where Lemma 1 has once again been used in determining the asymptotic form of the

series (the i= 0 term in (61) vanishes due to the convergence of integral)

The discretization error for the second method (19) can be obtained in a similar

way In the second form of the operator I a,b,c(17) expand y at t = t iinto the Taylorseries to obtain