# Computing Hadamard type operators of variable fractional order111this is a preprint of a paper whose final and definite form will be published in applied mathematics and computation, issn: 0096-3003. submitted 10/june/2014; revised 03/dec/2014; accepted 16/dec/2014.

###### Abstract.

We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard–Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound formula for the error is provided. We exemplify our method by applying the proposed numerical procedure to the solution of a fractional differential equation and a fractional variational problem with dependence on the Hadamard–Marchaud fractional derivative.

###### Key words and phrases:
Fractional calculus, variable fractional order, numerical methods, fractional differential equations, fractional calculus of variations
###### 2010 Mathematics Subject Classification:
Primary: 26A33, 33F05; Secondary: 34A08, 49M99.

Ricardo Almeida and Delfim F. M. Torres

Center for Research and Development in Mathematics and Applications (CIDMA)

Department of Mathematics, University of Aveiro

3810–193 Aveiro, Portugal

## 1. Introduction

Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation and integration operators [26, 29]. It has been called “The calculus of the XXI century” (K. Nishimoto, 1989) and claimed that “Nature works with fractional time derivatives” (S. Westerlund, 1991) . Several definitions for fractional derivatives and fractional integrals are found in the literature. Although the most common ones seem to be the Riemann–Liouville and Caputo fractional operators, recently there has been an increasing interest in the development of Hadamard’s XIX century fractional calculus : see [1, 4, 5, 6, 7, 12, 13, 14, 15, 16] and references therein. This calculus is due to the French mathematician Jacques Hadamard (1865–1963), where instead of power functions, as in Riemann–Liouville and Caputo fractional calculi, one has logarithm functions. The left and right Hadamard fractional integrals of order are defined by

and

respectively, while the left and right Hadamard fractional derivatives of order are given by

and

 tDαbx(t)=−tΓ(1−α)ddt∫bt(lnτt)−αx(τ)τdτ,

respectively. Uniqueness and continuous dependence of solutions for nonlinear fractional differential systems with Hadamard derivatives is discussed in . The main purpose of this paper is to extend the previous definitions to the case where the order of the integrals and of the derivatives is not a constant, but a function that depends on time. Such time-dependence of has already been considered for Riemann–Liouville and Caputo fractional operators, and has proven to describe better certain phenomena (see, e.g., [8, 9, 10, 17, 21, 24, 25, 27]). To the best of our knowledge, an extension to Hadamard fractional operators is new and no work has been carried out so far in this direction. This is due to practical difficulties in computing such fractional derivatives and integrals of variable order. For this reason, here we propose a simple but effective numerical method that allows to deal with variable fractional order operators of Hadamard type.

The organization of the paper is the following. In Section 2 we extend known definitions of Hadamard fractional operators by considering the order to be a function, and present a new definition of derivative, the Hadamard–Marchaud fractional derivative, which is an intrinsic variable order operator. In Section 3 we prove expansion formulas for the given fractional operators, using only integer-order derivatives. Finally, in Section 4 we give some concrete examples of the usefulness of the proposed method, including the application to the solution of a fractional differential system of variable order (Section 4.1) and to the solution of a fractional variational problem of variable order (Section 4.2).

## 2. Hadamard operators of variable fractional order

Along the text, the order of fractional operators is given by a function , and the space of functions is such that each of the following integrals are well-defined, where are two reals with .

###### Definition 2.1 (Hadamard integrals of variable fractional order).

The left and right Hadamard fractional integrals of order are defined by

 aIα(t)tx(t)=1Γ(α(t))∫ta(lntτ)α(t)−1x(τ)τdτ

and

 tIα(t)bx(t)=1Γ(α(t))∫bt(lnτt)α(t)−1x(τ)τdτ,

respectively.

###### Definition 2.2 (Hadamard derivatives of variable fractional order).

The left and right Hadamard fractional derivatives of order are defined by

and

 tDα(t)bx(t)=−tΓ(1−α(t))ddt∫bt(lnτt)−α(t)x(τ)τdτ,

respectively.

The two Definitions 2.1 and 2.2 coincide with the classical definitions of Hadamard when the order is a constant function. Besides these definitions, we introduce a different one inspired on Hadamard and Marchaud fractional derivatives .

###### Definition 2.3 (Hadamard–Marchaud derivatives of variable fractional order).

The left and right Hadamard–Marchaud fractional derivatives of order are defined by

and

 (3) tDα(t)bx(t)=x(t)Γ(1−α(t))(lnbt)−α(t)+α(t)Γ(1−α(t))∫btx(t)−x(τ)τ(lnτt)−α(t)−1dτ,

respectively.

###### Remark 1.

Splitting the integrals in Definition 2.3 into two, and integrating by parts, we get that (2) is equivalent to

while (3) is equivalent to

 tDα(t)bx(t)=x(b)Γ(1−α(t))(lnbt)−α(t)−1Γ(1−α(t))∫bt(lnτt)−α(t)x′(τ)dτ.

Other Hadamard notions of variable fractional order are possible. For example, motivated by the Caputo fractional derivative, we can set

Integrating by parts, we then obtain that

Similar calculations can be done for the right Hadamard–Caputo derivative of variable fractional order, .

###### Definition 2.4 (Left Hadamard–Caputo derivative of variable fractional order).

The left Hadamard–Caputo fractional derivatives of order is defined by

###### Remark 2.

If we consider the particular case , a constant, then Definition 2.4 simplifies to the Hadamard–Caputo fractional derivative studied in :

###### Lemma 2.5.

Let . If

 x(t)=(lnta)β,

then the left Hadamard fractional integral of order is given by

 aIα(t)tx(t)=Γ(β+1)Γ(β+α(t)+1)(lnta)β+α(t),

the left Hadamard fractional derivative of order is given by

where is the Psi function, that is, is the derivative of the logarithm of the Gamma function,

 ψ(t)=ddtln(Γ(t))=Γ′(t)Γ(t),

and the left Hadamard–Marchaud fractional derivative of order is given by

###### Proof.

Starting with Definition 2.1, we arrive to

 aIα(t)tx(t)=1Γ(α(t))(lnta)α(t)−1∫ta(1−lnτalnta)α(t)−1(lnτa)βdττ.

Performing the change of variable

 lnτa=slnta,

it follows that

 aIα(t)tx(t)=1Γ(α(t))(lnta)β+α(t)∫10(1−s)α(t)−1sβds=1Γ(α(t))(lnta)β+α(t)B(α(t),β+1)=1Γ(α(t))(lnta)β+α(t)Γ(α(t))Γ(β+1)Γ(β+α(t)+1)=Γ(β+1)Γ(β+α(t)+1)(lnta)β+α(t),

where is the Beta function, that is,

 B(λ,μ)=∫10tλ−1(1−t)μ−1dt,λ,μ>0.

The formula for is obtained in a similar way, using (4):

To prove the formula for the left Hadamard fractional derivative of order , we start with the same change of variables as before, to get

The intended formula follows directly by computing the derivative in (6). ∎

As a consequence of Lemma 2.5, we have that . Next, we establish a relation between these two types of differential operators.

###### Theorem 2.6.

The following relation between the left Hadamard and the left Hadamard–Marchaud fractional derivatives of order holds:

###### Proof.

Starting with the definition, and differentiating the integral, we obtain that

Integrating,

 ∫te−ϵa(lntτ)−α(t)−1x(t)τdτ=x(t)α(t)(ϵ−α(t)−(lnta)−α(t)).

Also, since , then

 limϵ→0x(te−ϵ)−x(t)ϵα(t)=0.

Using these two relations, we prove that

The next corollary is a trivial consequence of Theorem 2.6. It asserts that it only makes sense to distinguish between Hadamard and Hadamard–Marchaud fractional derivatives in the variable-order case: for the classical situation of constant order , the fractional derivatives coincide.

###### Corollary 1.

If , then both left Hadamard and left Hadamard–Marchaud fractional derivatives coincide.

## 3. Approximations

In this section we exhibit several results about approximations for the Hadamard fractional operators, which are expressed by only using integer-order derivatives of the function. With this in hand, given any problem that depends on these fractional operators, we are able to rewrite it by eliminating all the fractional operators, and by doing so obtaining a classical problem, with dependence on integer-order derivatives only. Then one can apply any known technique from the literature. We mention , where an analogous idea was carried out for the Riemann–Liouville fractional derivative. In the following, given , we define the sequences and recursively by the formulas

 x0,0(t)=x(t),xk+1,0(t)=tddtxk,0(t), for k∈N∪{0},

and

 x0,1(t)=x′(t),xk+1,1(t)=ddt(txk,1(t)), % for k∈N∪{0}.

The following definition is useful to describe our approximations.

###### Definition 3.1 (Left and right moment of a function).

The left moment of of order is given by

 Vk(t)=(k−n)∫ta(lnτa)k−n−1x(τ)τdτ,

and the right moment of of order by

 Wk(t)=(k−n)∫bt(lnbτ)k−n−1x(τ)τdτ.
###### Theorem 3.2.

Fix and , and let , where . Then,

 (7)

with

 A(k)=1Γ(α(t)+k+1)[1+N∑p=n−k+1Γ(p−α(t)−n)Γ(−α(t)−k)(p−n+k)!],k=0,…,n,B(k)=Γ(k−α(t)−n)Γ(α(t))Γ(1−α(t))(k−n)!,k=n+1,…,N.

Relation (7) gives an approximation for the left Hadamard fractional integral of order with error bounded by

###### Proof.

Similar to the one given in , replacing by . ∎

###### Theorem 3.3.

Fix and , and let , where . Then,

with

 (8) S1(t)=n∑k=0A(k)(lnta)k−α(t)xk,0(t)+N∑k=n+1B(k)(lnta)n−α(t)−kVk(t),

where

 A(k)=1Γ(k+1−α(t))[1+N∑p=n−k+1Γ(p+α(t)−n)Γ(α(t)−k)(p−n+k)!],k=0,…,n,B(k)=Γ(k+α(t)−n)Γ(−α(t))Γ(1+α(t))(k−n)!,k=n+1,…,N,

and

 (9) S2(t)=tx(t)α′(t)Γ(1−α(t))(lnta)1−α(t)⎡⎢ ⎢⎣ln(lnta)1−α(t)−1(1−α(t))2−ln(lnta)N∑k=0(−α(t)k)(−1)kk+1+N∑k=0(−α(t)k)(−1)kN∑p=11p(k+p+1)]+tα′(t)Γ(1−α(t))(lnta)1−α(t)×[ln(lnta)N+n+1∑k=n+1(−α(t)k−n−1)(−1)k−n−1k−n(lnta)n−kVk(t)−N+n+1∑k=n+1(−α(t)k−n−1)(−1)k−n−1N∑p=11p(k+p−n)(lnta)n−k−pVk+p(t)].

The error of approximating the left Hadamard fractional derivative by is bounded with

 (10) |E1,N(t)|≤maxτ∈[a,t]|xn,1(τ)|exp((n−α(t))2+n−α(t))Γ(n+1−α(t))(n−α(t))Nn−α(t)(lnta)n−α(t)(t−a)

and

 (11) |E2,N(t)|≤maxτ∈[a,t]|x′(τ)|∣∣∣t(2t−a)α′(t)(lnta)2−α(t)∣∣∣exp(α2(t)−α(t))Γ(2−α(t))N1−α(t)[∣∣∣ln(lnta)∣∣∣+1N].
###### Proof.

Doing the change of variables

 tτ=ua,

and some calculations needed, we deduce that

Define

 ¯¯¯¯S1(t)=x(a)Γ(1−α(t))(lnta)−α(t)+1Γ(1−α(t))∫ta(lntτ)−α(t)x′(τ)dτ

and

 ¯¯¯¯S2(t)=tα′(t)Γ(1−α(t))∫ta(lntτ)−α(t)ln(lntτ)x(τ)τdτ.

When , i.e., when is constant, formula is equivalent to the left Hadamard fractional derivative (1) (see ), and following the same techniques as the ones given in , we obtain formula as in (8) and the upper bound formula for as in (10). About the sum , starting with the relation

 ¯¯¯¯S2(t)=tα′(t)Γ(1−α(t))∫tax(τ)⋅[ddτ∫τa(lntu)−α(t)ln(lntu)duu]dτ,

and performing integration by parts, we get

 ¯¯¯¯S2(t)=tα′(t)Γ(1−α(t))[x(t)∫ta(lntu)−α(t)ln(lntu)duu−∫tax′(τ)[∫τa(lntu)−α(t)ln(lntu)duu]dτ].

From simple computations, we have

 ∫ta(lntu)−α(t)ln(lntu)duu=(lnta)1−α(t)⎡⎢ ⎢⎣ln(lnta)1−α(t)−1(1−α(t))2⎤⎥ ⎥⎦.

Also, by Taylor’s theorem, we have the two following formulas:

 (lntu)−α(t)=(lnta)−α(t)(1−lnualnta)−α(t)=(lnta)−α(t)N∑k=0(−α(t)k)(−1)k(lnua)k(lnta)k+E′N(t)

and

 ln(lntu)=ln(lnta)+ln(1−lnualnta)=ln(lnta)−N∑p=11p(lnua)p(lnta)p+E′′N(t).

Combining all the previous equalities, we get

 ¯¯¯¯S2(t)=tα′(t)Γ(1−α(t))⎡⎢ ⎢⎣x(t)(lnta)1−α(t)⎡⎢ ⎢⎣ln(lnta)1−α(t)−1(1−α(t))2⎤⎥ ⎥⎦−∫tax′(τ)(lnta)−α(t)ln(lnta)N∑k=0(−α(t)k)(−1)k(lnta)k(∫τa(lnua)kduu)dτ+∫tax′(τ)(lnta)−α(t)N∑k=0(−α(t)k)(−1)k(lnta)kN∑p=1∫τa(lnua)k+pduup(lnta)pdτ⎤⎥ ⎥ ⎥⎦+E2,N(t)=tα′(t)Γ(1−α(t))(lnta)−α(t)⎡⎢ ⎢⎣x(t)lnta⎡⎢ ⎢⎣ln(lnta)1−α(t)−1(1−α(t))2⎤⎥ ⎥⎦−ln(lnta)N∑k=0(−α(t)k)(−1)k(lnta)k(k+1)∫tax′(τ)(lnτa)k+1dτ+N∑k=0(−α(t)k)(−1)k(lnta)kN∑p=1∫tax′(τ)(lnτa)k+p+1dτp(lnta)p(k+p+1)⎤⎥ ⎥ ⎥⎦+E2,N(t).

Formula (9) is deduced using relations

 ∫tax′(τ)(lnτa)k+1dτ=x(t)(lnta)k+1−Vk+n+1(t)

and

 ∫tax′(τ)(lnτa)k+p+1dτ=x(t)(lnta)k+p+1−Vk+p+n+1(t).

The error that occurs on this approximation is bounded by

 |E2,N(t)|≤∣∣ ∣∣tα′(t)Γ(1−α(t))(lnta)−α(t)∣∣ ∣∣×∣∣ ∣ ∣∣−ln(lnta)∞∑k=N+1(−α(t)k)(−1)kk+1∫tax′(τ)(lnτa)k+1(lnta)kdτ+∞∑k=N+1(−α(t)k)(−1)k∞∑p=N+11p(k+p+1)∫tax′(τ)(lnτa)k+p+1(lnta)k+pdτ∣∣ ∣ ∣∣.

Since

 ∣∣(−α(t)k)∣∣≤exp(α2(t)−α(t))k1−α(t),

it follows that

 |E2,N(t)|≤maxτ∈[a,t]|x′(τ)|∣∣ ∣