Fractional variational calculus of variable order
ABSTRACT We study the fundamental problem of the calculus of variations with variable
order fractional operators. Fractional integrals are considered in the sense of
RiemannLiouville while derivatives are of Caputo type.

Conference Paper: Variable order fractional variational calculus for double integrals
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ABSTRACT: We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain a fractional EulerLagrange necessary optimality condition for variable order twodimensional fractional variational problems.Decision and Control (CDC), 2012 IEEE 51st Annual Conference on; 01/2012  SourceAvailable from: Delfim F. M. Torres[Show abstract] [Hide abstract]
ABSTRACT: We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free boundary value problems. The fractional actionlike variational approach (FALVA) is extended and some applications to Physics discussed.Abstract and Applied Analysis 03/2012; 2012, Special Issue. · 1.27 Impact Factor
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arXiv:1110.4141v1 [math.OC] 18 Oct 2011
Submitted 26Sept2011; accepted 18Oct2011;
to Advances in Harmonic Analysis and Operator Theory,
The Stefan Samko Anniversary Volume
(Eds: A. Almeida, L. Castro, F.O. Speck),
Operator Theory: Advances and Applications, Birkh¨ auser Verlag.
http://www.springer.com/series/4850
Fractional variational calculus of variable order
T. Odzijewicz, A. B. Malinowska and D. F. M. Torres
To Professor Stefan Samko on the occasion of his 70th birthday
Abstract. We study the fundamental problem of the calculus of variations
with variable order fractional operators. Fractional integrals are considered in
the sense of Riemann–Liouville while derivatives are of Caputo type.
Mathematics Subject Classification (2010). 26A33; 34A08; 49K05.
Keywords. Fractional operators; fractional integration and differentiation of
variable order; fractional variational analysis; Euler–Lagrange equations.
1. Introduction
Fractional calculus is a discipline that studies integrals and derivatives of non
integer (real or complex) order [13,16,23,30]. The field was born in 1695 when
Leibniz asked in his letter to L’Hˆ opital what should be the meaning of a derivative
of noninteger order. After the letters between Leibniz and L’Hˆ opital, fractional
calculus became an ongoing topic with many wellknown mathematicians con
tributing to its theory [32]. The subject is nowadays very active due to its many
applications in mechanics, chemistry, biology, economics, and control theory [33].
In 1993, Samko and Ross proposed an interesting generalization to fractional
calculus [31] (see also Samko’s paper [29] of 1995). They introduced the study
of fractional integration and differentiation when the order is not a constant but
a function. More precisely, they considered an extension of Riemann–Liouville
and Fourier definitions [28,29, 31]. Afterwards, several works were dedicated to
variable order fractional operators, their applications and interpretations (see, e.g.,
[1,7,17]). In particular, Samko’s variable order fractional calculus turns out to be
very useful in mechanics and in the theory of viscous flows [7,9,17,22,24,25].
In this note we develop the fractional calculus of variations via Samko’s vari
able order approach. The fractional variational calculus was born in 19961997
Part of the first author’s Ph.D., which is carried out at the University of Aveiro under the
Doctoral Programme in Mathematics and Applications of Universities of Aveiro and Minho.
Page 2
2T. Odzijewicz, A. B. Malinowska and D. F. M. Torres
with the works of Riewe in mechanics [26,27], and is now under strong current re
search (see [2,3,5,6,8,10,11,14,15,18–21] and references therein). However, to the
best of the authors knowledge, results for the variable order case are a rarity and
reduce to those in [4]. Motivated by the advancements of [3,21], and in contrast
with [4], we consider here fractional problems of the calculus of variations where
the Lagrangian depends on classical integer order derivatives and both on variable
order fractional derivatives and integrals.
The paper is organized as follows. In §2 a brief review to the variable order
fractional calculus is given. Our results are then formulated and proved in §3: we
show the boundedness of the variable order Riemann–Liouville fractional integral
in the space L1[a,b] (Theorem 3.1); integration by parts formulas for variable order
fractional operators (Theorems 3.2 and 3.3); and a necessary optimality condition
for our general fundamental problem of the variable order fractional variational
calculus (Theorem 3.5). An illustrative example is discussed in §4.
2. Preliminaries
The reference book for fractional analysis and its applications is [30]. Here we recall
the necessary definitions for the variable order fractional calculus. Throughout the
text α denotes a given absolutely continuous function.
Definition 2.1 (Left and right Riemann–Liouville integrals of variable order). Let
0 < α(t) < 1 and f ∈ L1[a,b]. Then,
aIα(t)
t
f(t) =
1
Γ(α(t))
t ?
a
(t − τ)α(t)−1f(τ)dτ
(t > a)
is called the left Riemann–Liouville integral of variable fractional order α(t), while
tIα(t)
b
f(t) =
1
Γ(α(t))
b
?
t
(τ − t)α(t)−1f(τ)dτ
(t < b)
denotes the right Riemann–Liouville integral of variable fractional order α(t).
Definition 2.2 (Left and right Riemann–Liouville derivatives of variable order). Let
0 < α(t) < 1. IfaI1−α(t)
t
f ∈ AC[a,b], then the left Riemann–Liouville derivative
of variable fractional order α(t) is defined as
aDα(t)
t
f(t) =
1
Γ(1 − α(t))
d
dt
t ?
a
(t − τ)−α(t)f(τ)dτ
(t > a)
Page 3
Fractional variational calculus of variable order3
while the right Riemann–Liouville derivative of variable fractional order α(t) is
defined for functions f such thattI1−α(t)
b
f ∈ AC[a,b] by
tDα(t)
b
f(t) =
−1
Γ(1 − α(t))
d
dt
b
?
t
(τ − t)−α(t)f(τ)dτ
(t < b).
Definition 2.3 (Left and right Caputo derivatives of variable fractional order). Let
0 < α(t) < 1. If f ∈ AC[a,b], then the left Caputo derivative of variable fractional
order α(t) is defined as
C
aDα(t)
t
f(t) =
1
Γ(1 − α(t))
t ?
a
(t − τ)−α(t)d
dτf(τ)dτ
(t > a)
while the right Caputo derivative of variable fractional order α(t) is given by
C
tDα(t)
b
f(t) =
−1
Γ(1 − α(t))
b
?
t
(τ − t)−α(t)d
dτf(τ)dτ
(t < b).
The following result will be useful in the proof of Theorems 3.1 and 3.2.
Theorem 2.4 (cf. [12]). If x ∈ [0,1], the Gamma function satisfies the inequalities
x2+ 1
x + 1
≤ Γ(x + 1) ≤x2+ 2
x + 2.
(2.1)
3. Main results
In §3.1 we prove that the variable fractional order Riemann–Liouville integral
aIα(t)
t
is bounded on the space L1[a,b] (Theorem 3.1). In §3.2 we obtain a formula
of integration by parts for Riemann–Liouville integrals of variable order (Theo
rem 3.2) and a formula of integration by parts for derivatives of variable fractional
order (Theorem 3.3). Finally, in §3.3 we use the obtained formulas of integration
by parts to derive a necessary optimality condition for a general problem of the
calculus of variations involving variable order fractional operators (Theorem 3.5).
3.1. Boundedness
The following result allow us to consider problems of the calculus of variations with
a Lagrangian depending on left Riemann–Liouville integrals of variable order.
Theorem 3.1. Let
and for all t ∈ [a,b]. The Riemann–Liouville integralaIα(t)
variable fractional order α(t) is a linear and bounded operator.
1
n< α(t) < 1 for a certain n ∈ N greater or equal than two
t
: L1[a,b] → L1[a,b] of
Page 4
4T. Odzijewicz, A. B. Malinowska and D. F. M. Torres
Proof. The operator is obviously linear. Let1
n< α(t) < 1, f ∈ L1[a,b], and define
F(τ,t) :=
????
1
Γ(α(t))(t − τ)α(t)−1??? · f(τ)
1
n< α(t) < 1, then
if τ ≤ t;
0 if τ > t
for all (τ,t) ∈ ∆ = [a,b] × [a,b]. Since
1. for τ + 1 ≤ t we have ln(t − τ) ≥ 0 and (t − τ)α(t)−1< 1;
2. for τ ≤ t < τ + 1 we have ln(t − τ) < 0 and (t − τ)α(t)−1< (t − τ)
1
n−1.
Therefore,
?b
a
??b
?b
?b
a
F(τ,t)dt
?
dτ =
?b
1
a
f(τ)
??b
τ
????
1
Γ(α(t))(t − τ)α(t)−1
????dt
?
dτ
=
a
f(τ)
??τ+1
??τ+1
τ
????
????
Γ(α(t))(t − τ)α(t)−1
????dt +
????dt +
?b
τ+1
????
1
Γ(α(t))(t − τ)α(t)−1
????dt
?
dτ
<
a
f(τ)
τ
1
Γ(α(t))(t − τ)
1
n−1
?b
τ+1
????
1
Γ(α(t))
????dt
?
dτ.
Moreover, by inequality (2.1), one has
?b
a
f(τ)
??τ+1
??τ+1
??τ+1
τ
????
1
Γ(α(t))(t − τ)
1
n−1
????dt +
?b
τ+1
????
1
Γ(α(t))
????dt
(α(t) − 1)2+ 1dt
?
dτ
<
?b
?b
a
f(τ)
τ
α(t)
(α(t) − 1)2+ 1(t − τ)
1
n−1dt +
?b
?b
τ+1
α(t)
?
dτ
<
a
f(τ)
τ
(t − τ)
1
n−1dt + b − τ − 1
?
dτ =
a
f(τ)(n + b − τ − 1)dτ
< (n + b − a)?f? < ∞.
It follows from Fubini’s theorem that F is integrable in the square ∆ and
???aIα(t)
t
f
??? =
?b
?b
?b
a
????
1
Γ(α(t))
??t
??b
?t
1
a
(t − τ)α(t)−1f(τ)dτ
????dt
≤
a
a
????
F(τ,t)dτ
Γ(α(t))(t − τ)α(t)−1f(τ)
?
< (n + b − a)?f?.
????dτ
?
dt
=
aa
dt
Therefore,aIα(t)
t
: L1[a,b] → L1[a,b] and
???aIα(t)
t
??? < n + b − a.
?
Page 5
Fractional variational calculus of variable order5
3.2. Integration by parts formulas
The integration by parts formulas we now obtain have an important role in the
proof of the generalized Euler–Lagrange equation (3.5). We note that in Theo
rem 3.2 the lefthand side of (3.1) involves a left integral of variable order while
on the righthand side it appears a right integral.
Theorem 3.2. Let 0 < α(t) < 1 and f,g ∈ C[a,b]. Then,
?b
a
g(t)aIα(t)
t
f(t)dt =
?b
a
f(t)tIα(t)
b
g(t)dt.
(3.1)
Proof. Let 0 < α(t) < 1, f,g ∈ C[a,b], and define
F(τ,t) :=
? ???
1
Γ(α(t))(t − τ)α(t)−1g(t)f(τ)
0
???
if τ ≤ t;
if τ > t
for all (τ,t) ∈ ∆ = [a,b] × [a,b]. Since f and g are continuous functions on [a,b],
they are bounded on [a,b], i.e., there exist C1,C2> 0 such that g(t) ≤ C1and
f(t) ≤ C2, t ∈ [a,b]. Therefore,
?b
a
??b
a
F(τ,t)dτ
?
dt =
?b
a
??t
?b
?b
a
????
1
Γ(α(t))(t − τ)α(t)−1g(t)f(τ)
????dτ
?
dt
≤ C1C2
a
??t
??t
a
????
Γ(α(t))(t − τ)α(t)−1dτ
1
Γ(α(t))(t − τ)α(t)−1
????dτ
?
dt
= C1C2
a
a
1
?
dt.
Moreover,
?t
a
(t − τ)α(t)−1dτ = (t − a)α(t) 1
α(t).
Then, by (2.1), one has
C1C2
?b
a
1
Γ(α(t) + 1)(t − a)α(t)dt ≤ C1C2
?b
?b
a
α(t) + 1
α(t)2+ 1(t − a)α(t)dt
< C1C2
a
(t − a)α(t)dt.
Since 0 < α(t) < 1, then
1. for 1 ≤ t − a we have ln(t − a) ≥ 0 and (t − a)α(t)< (t − a);
2. for 1 > t − a we have ln(t − a) < 0 and (t − a)α(t)< 1.
Page 6
6T. Odzijewicz, A. B. Malinowska and D. F. M. Torres
Therefore,
C1C2
?b
a
(t − a)α(t)dt = C1C2
??a+1
a
(t − a)α(t)dt +
?b
= C1C21 + (b − a)2
a+1
(t − a)α(t)dt
?
< C1C2
??a+1
a
dt +
?b
a+1
(t − a)dt
?
2
< ∞.
Hence, one can use the Fubini theorem to change the order of integration:
?b
a
g(t)aIα(t)
t
f(t)dt =
?b
a
??t
a
g(t)f(τ)
1
Γ(α(t))(t − τ)α(t)−1dτ
?
?
dt
=
?b
a
??b
τ
g(t)f(τ)
1
Γ(α(t))(t − τ)α(t)−1dtdτ =
?b
a
f(τ)τIα(t)
b
g(τ)dτ.
?
In our second formula (3.2) of fractional integration by parts, the lefthand
side contains a left Caputo derivative of variable fractional order α(t), while on
the righthand side it appears right Riemann–Liouville integrals of variable order
1 − α(t) and a right Riemann–Liouville derivative of variable order α(t).
Theorem 3.3. Let 0 < α(t) < 1. If α, f andtI1−α(t)
b
g ∈ AC[a,b], then
?b
a
g(t)C
aDα(t)
t
f(t)dt = f(t)tI1−α(t)
b
g(t)
???
b
a
−
?b
a
f(t)
?
α′(t)ψ(1 − α(t))tI1−α(t)
b
g(t) −tDα(t)
b
g(t)
?
dt,
(3.2)
where ψ is the digamma function, i.e., ψ(x) =
d
dxlnΓ(x) =Γ′(x)
Γ(x).
Proof. By Definition 2.3, it follows thatC
Theorem 3.2 and integration by parts for classical (integer) derivatives, we obtain
aDα(t)
t
f(t) = aI1−α(t)
t
d
dtf(t). Applying
?b
a
g(t)C
aDα(t)
t
f(t)dt =
?b
b
a−
a
g(t)aI1−α(t)
t
d
dtf(t)dt =
?b
a
d
dtf(t)tI1−α(t)
b
g(t)dt
= f(t)tI1−α(t)
b
g(t)
???
???
?b
?b
a
f(t)d
dttI1−α(t)
b
g(t)dt
= f(t)tI1−α(t)
b
g(t)
b
a−
a
f(t)
?
α′(t)ψ(1 − α(t))tI1−α(t)
b
g(t) −tDα(t)
b
g(t)
?
dt.
?
Page 7
Fractional variational calculus of variable order7
3.3. A fundamental variational problem of variable fractional order
We consider the problem of extremizing (minimizing or maximizing) the functional
J[y] =
b
?
a
F
?
t,y(t),y′(t),C
aDα(t)
t
y(t),aIβ(t)
t
y(t)
?
dt
(3.3)
subject to the boundary conditions
y(a) = ya,y(b) = yb,
(3.4)
where α and β are absolutely continuous functions of t ∈ [a,b] taking values in
[0,1]. For simplicity of notation, we introduce the operator {·,·,·} defined by
{y,α,β}(t) =
?
t,y(t),y′(t),C
aDα(t)
t
y(t),aIβ(t)
t
y(t)
?
.
We assume that F ∈ C1?[a,b] × R4;R?, t ?→ ∂4F {y,α,β}(t) has absolutely con
b
continuous variable order fractional integraltIβ(t)
continuous usual derivative
dt.
tinuous integraltI1−α(t)
and continuous derivativetDα(t)
b
, t ?→ ∂5F {y,α,β}(t) has
, and t ?→ ∂3F {y,α,β}(t) has
b
d
Definition 3.4. A Lipshitz function y ∈ Lip([a,b];R) is said to be admissible for
the variational problem (3.3)–(3.4), ifC
on the interval [a,b], and y satisfies the given boundary conditions (3.4).
aDα(t)
t
y andaIβ(t)
t
y exist and are continuous
Theorem 3.5. Let y be a solution to problem (3.3)–(3.4). Then, y satisfies the
generalized Euler–Lagrange equation
∂2F {y,α,β}(t) −d
dt∂3F {y,α,β}(t) +tIβ(t)
∂4F {y,α,β}(t) +tDα(t)
b
∂5F {y,α,β}(t)
− α′(t)ψ(1 − α(t))tI1−α(t)
bb
∂4F {y,α,β}(t) = 0 (3.5)
for all t ∈ [a,b].
Proof. Suppose that y is an extremizer of J. Consider the value of J at a nearby
function ˆ y(t) = y(t)+εη(t), where ε ∈ R is a small parameter and η ∈ Lip([a,b];R)
is an arbitrary function satisfying η(a) = η(b) = 0 and such thatC
aIβ(t)
t
ˆ y(t) are continuous. Let
?b
A necessary condition for y to be an extremizer is given by
aDα(t)
t
ˆ y(t) and
J(ε) = J[ˆ y] =
a
F {ˆ y,α,β}(t)dt.
dJ
dε
????
ε=0
= 0 ⇔
b
?
a
?
∂2F {y,α,β}(t) · η(t) + ∂3F {y,α,β}(t)d
dtη(t)
+ ∂4F {y,α,β}(t)C
aDα(t)
t
η(t) + ∂5F {y,α,β}(t) ·aIβ(t)
t
η(t)
?
dt = 0.
(3.6)
Page 8
8T. Odzijewicz, A. B. Malinowska and D. F. M. Torres
Using the classical and the generalized fractional integration by parts formulas of
Theorems 3.2 and 3.3, we obtain
?b
a
∂3Fdη
dtdt = ∂3Fηb
a−
?b
a
?
ηd
dt∂3F
?
dt,
b
?
a
∂4FC
aDα(t)
t
ηdt = ηtI1−α(t)
b
∂4F
???
b
a
−
b
?
a
η
?
α′(t)ψ(1 − α(t))tI1−α(t)
b
∂4F −tDα(t)
b
∂4F
?
dt,
and
b
?
a
∂5FaIβ(t)
t
ηdt =
b
?
a
ηtIβ(t)
b
∂5Fdt.
Because η(a) = η(b) = 0, (3.6) simplifies to
?b
a
η(t)
?
∂2F {y,α,β}(t) −d
dt∂3F {y,α,β}(t)
− α′(t)ψ(1 − α(t))tI1−α(t)
b
∂4F {y,α,β}(t) +tDα(t)
b
∂4F {y,α,β}(t)
+tIβ(t)
b
∂5F {y,α,β}(t)
?
dt = 0.
One obtains (3.5) by the fundamental lemma of the calculus of variations (see,
e.g., [34]).
?
4. An illustrative example
Let 0 < α(t) < 1 and γ > −1. In Example 1 we make use of the identity
aIα(t)
t
[(t − a)γ] =Γ(γ + 1)(t − a)γ+α(t)
Γ(γ + α(t) + 1)
(4.1)
that one can find in the Samko and Ross paper [31].
Example 1. Let J be the functional defined by
J[y] =
b
?
a
?
1 +
Γ(α(t) + 3)
2Γ(3)(t − a)2+α(t)
?
aIα(t)
t
y(t))
?2
−aIα(t)
t
y(t)dt
Page 9
Fractional variational calculus of variable order9
with endpoint conditions y(a) = 0 and y(b) = (b − a)2. If y is an extremal for J,
then the following necessary optimality condition is satisfied:
2 1 +
2Γ(3)(t−a)2+α(t)
tIα(t)
b
Γ(α(t)+3)
Γ(3)(t−a)2+α(t) aIα(t)
t
y(t) − 1
?
Γ(α(t)+3)
?
aIα(t)
t
y(t)
?2
−aIα(t)
t
y(t)
= 0.
(4.2)
By identity (4.1), function
y(t) = (t − a)2
(4.3)
is a solution to the variable order fractional differential equation
y(t) =Γ(3)(t − a)2+α(t)
aIα(t)
t
Γ(α(t) + 3)
.
Therefore, it is a solution to the Euler–Lagrange equation (4.2). Note that (4.3) is
a locally Lipschitz function. Indeed,
??(t1− a)2− (t2− a)2??= (t1− a) + (t2− a) · (t1− a) − (t2− a)
Acknowledgments
Work supported by FEDER funds through COMPETE — Operational Programme
Factors of Competitiveness (“Programa Operacional Factores de Competitivi
dade”) and by Portuguese funds through the Center for Research and Develop
ment in Mathematics and Applications (University of Aveiro) and the Portuguese
Foundation for Science and Technology (“FCT — Funda¸ c˜ ao para a Ciˆ encia e a
Tecnologia”), within project PEstC/MAT/UI4106/2011with COMPETE number
FCOMP010124FEDER022690.Odzijewicz was also supported by FCT through
the Ph.D. fellowship SFRH/BD/33865/2009; Malinowska by Bia? lystok Univer
sity of Technology grant S/WI/02/2011; and Torres by FCT through the project
PTDC/MAT/113470/2009.
= t1+ t2− 2a · t1− t2 ≤ (t1 + t2 + 2a)t1− t2 ≤ 2(a + b)t1− t2.
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T. Odzijewicz
Center for Research and Development in Mathematics and Applications
Department of Mathematics, University of Aveiro, 3810193 Aveiro, Portugal
email: tatianao@ua.pt
A. B. Malinowska
Faculty of Computer Science, Bia? lystok University of Technology
15351 Bia? lystok, Poland
email: a.malinowska@pb.edu.pl
D. F. M. Torres
Center for Research and Development in Mathematics and Applications
Department of Mathematics, University of Aveiro, 3810193 Aveiro, Portugal
email: delfim@ua.pt
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