Central European Journal of Physics

A fractional approach to the Sturm-Liouville problem--Manuscript Draft--

Manuscript Number: CEJP-D-13-00066R1

Full Title: A fractional approach to the Sturm-Liouville problem

Article Type: Topical Issue: Research Article

Section/Category: Theoretical and Mathematical Physics

Keywords: fractional operators; fractional spatial derivatives; Sturm-Liouville theory

Corresponding Author: M. Pilar VelascoCentro Universitario de la DefensaSPAIN

Corresponding Author SecondaryInformation:

Corresponding Author's Institution: Centro Universitario de la Defensa

Corresponding Author's SecondaryInstitution:

First Author: Margarita Rivero

First Author Secondary Information:

Order of Authors: Margarita Rivero

Juan Trujillo

M. Pilar Velasco

Order of Authors Secondary Information:

Abstract: The objective of this paper is to show an approach to the fractional version of theSturm-Liouville problem, by using different fractional operators that return to theordinary operator for integer order. For each fractional operator we study some of thebasic properties of the Sturm-Liouville theory. We analyze a particular example thatevidences the applicability of the fractional Sturm-Liouville theory.

Response to Reviewers: To Referee 2: We have introduced all the minor corrections proposed by the referee 1,in connection with the references.

To Referee 3: Also we have considered the corrections and suggestion proposed bythe referee 3, in particular we have extended the introduction and the presentation ofthe fractional calculus. Also we have included the references suggested by the refereeenlarging the list of references.

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Cent. Eur. J. Phys. • 1-15Author version

Central European Journal of Physics

A fractional approach to the Sturm-Liouville problem

Research Article

Margarita Rivero1∗, Juan J. Trujillo1†, M. Pilar Velasco2‡

1 Universidad de La LagunaFacultad de Matematicas38271 La Laguna, Tenerife, Spain

2 Centro Universitario de la DefensaArea de Matematicas, Estadıstica e Investigacion Operativa50090 Zaragoza, Spain

Abstract: The objective of this paper is to show an approach to the fractional version of the Sturm-Liouville problem,

by using different fractional operators that return to the ordinary operator for integer order. For eachfractional operator we study some of the basic properties of the Sturm-Liouville theory. We analyze a

particular example that evidences the applicability of the fractional Sturm-Liouville theory.

PACS (2008): 26A33, 35R11, 34B24

Keywords: fractional operators, fractional spatial derivatives, Sturm-Liouville theory

© Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1. Introduction

The Sturm-Liouville problem was firstly studied over 170 years ago and it has many applications in different areas

of science, for example, engineering and mathematics ([7], [8], [9]). The classical Sturm-Liouville problem for a

linear differential equation of second order is a boundary-value problem as the following one:

− d

dx

[u(x)

dy

dx

]+ v(x)y = λr(x)y, x ∈ [a, b] (1)

a1y(a) + a2y′(a) = 0 (2)

b1y(b) + b2y′(b) = 0 (3)

This differential equation used to be written as

L(y) = λr(x)y, (4)

∗ E-mail: mrivero@ull.es† E-mail: jtrujill@ullmat.es‡ E-mail: velascom@unizar.es

1

ManuscriptClick here to download Manuscript: Rivero,Trujillo,Velasco CEJP revision1.tex

A fractional approach to the Sturm-Liouville problem

considering that L[y] = −[u(x)y′]′ + v(x)y is a linear homogeneous differential operator with some important

properties associated to the Sturm-Liouville theory, where u, u′, v, r are continuous functions on the interval

[a, b], such as u, r are positive in that interval. These conditions are satisfied in many significant problems

in mathematical physics, for example, the equation y′′ + λy = 0 that is obtained by applying the method of

separating variables to the equation that studies the problem of heat conduction in a bar. Also, other differential

equations can be transformed into Sturm-Liouville equations, for example Bessel, Hermite, Jacobi and Legendre

equations.

Fractional Calculus is the emerging mathematical field devoted to study convolution-type pseudo-differential

operators, specifically integrals and derivatives of any arbitrary real or complex order, that generalize the ordinary

integrals and derivatives. In the last three decades, the interest in these operators and their applications has

became of great importance in many fields of science and engineering, for example mechanics, electricity, chemistry,

biology, economics, control theory and signal and image processing, due to the Fractional Calculus constitutes

a meeting place of multiple disciplines: stochastic processes, probability, integro-differential equations, integral

transforms, special functions, numerical analysis... These fractional operators are non-local and they have certain

capacity of memory associated their convolution kernel, so Fractional Calculus became a powerful framework to

model many real processes of anomalous systems ([5], [21], [22], [27], [29]) by using fractional ordinary or partial

differential equations and systems of these fractional equations ([3], [4], [6], [10]). In this sense, the introduction of

fractional operators for building a fractional Sturm-Liouville theory can be interesting to generalize the classical

theory and to give theoretical support to the numerical results obtained by several authors recently (for example,

[1], [2], [12], [13], [14], [15], [16], [20], [24]).

Some previous works have been published about the fractional Sturm-Liouville problem with a particular fractional

operator [18], [19], but the problem of this operator is that it does not include the classical operator for integer

order. In this paper, we propose other fractional operators to construct a fractional Sturm-Liouville theory that

generalizes the ordinary Sturm-Liouville faithfully and that returns to the classical theory for integer derivative

orders in the fractional operator. We investigate the eigenvalues and eigenfunctions associated to these operators

and also theirs properties, with the objective of applying this generalized Sturm-Liouville theory to fractional

partial differential equations. So the ordinary Sturm-Liouville problem is intimately related to the search of

solutions for partial differential equations, through the method of separation of variables. In this sense, the study

of a fractional Sturm-Liouville theory would allow to apply its results to more complex problems, for example

by giving support to solve fractional partial differential equations with methods of separation of variables where

eigenvalues and eigenfunctions of fractional Sturm-Liouville problems are introduced.

This paper is organized as follows. In Section 2, preliminaries definitions and properties about Fractional Calculus

are shown. In Section 3, we establish new fractional generalizations of the classic Sturm-Liouville problem and

we study the properties of the eigenfunctions and eigenvalues associated to the different fractional differential

operators of the new fractional Sturm-Liouville theory. An illustrative example of a Sturm-Liouville fractional

2

Margarita Rivero, Juan J. Trujillo, M. Pilar Velasco

problem is solved in Section 4. Finally, in Section 5 we introduce the conclusions of this theory.

2. Fractional preliminaries

There exist several definitions of fractional operators. In this section, we introduce the fractional derivatives and

integrals used in this work and some their properties (see also [11], [17], [23], [25], [26], [28], [30]).

Definition 1.Let α > 0, with n− 1 < α < n and n ∈ N, [a, b] ⊂ R and let f be a suitable real function (for example, it sufficesif f ∈ L1(a, b)). The Riemann-Liouville fractional operators are:

(Iαa+f)(x) =1

Γ(α)

∫ x

a

(x− t)α−1f(t) dt (x > a) (5)

(Dαa+f)(x) = Dn(In−αa+ f)(x) (x > a) (6)

(Iαb−f)(x) =1

Γ(α)

∫ b

x

(t− x)α−1f(t) dt (x < b) (7)

(Dαb−f)(x) = Dn(In−αb− f)(x), (x < b) (8)

where D is the usual differential operator.

Definition 2.Let α > 0, with n− 1 < α < n and n ∈ N, [a, b] ⊂ R and let f be a suitable real function (for example, it sufficesif f ∈ L1(a, b)). The Caputo fractional derivative is:

(CDαa+f)(x) = (In−αa+ Dnf)(x) (x > a) (9)

(CDαb−f)(x) = (In−αb− Dnf)(x) (x < b) (10)

The following identity is well-known for a suitable function f (for example, f n-times derivable):

(Dαa+f)(x) = (CDα

a+f)(x) +

n−1∑j=0

f (j)(a)

Γ(1 + j − α)(x− a)j−α. (11)

Thus, we have:

(CDαa+1) = 0 ; (Dα

a+1) =(x− a)−α

Γ(1− α)(12)

Definition 3.Let α > 0, with n − 1 < α < n and n ∈ N, and let f be a suitable real function (for example, it suffices iff ∈ L1(R)). The Liouville fractional operators have the following forms for x ∈ R:

(Iα+f)(x) =1

Γ(α)

∫ x

−∞(x− t)α−1f(t) dt (13)

(Dα+f)(x) = Dn(In−α+ f)(x) (14)

(Iα−f)(x) =1

Γ(α)

∫ ∞x

(t− x)α−1f(t) dt (15)

(Dα−f)(x) = (−D)n(In−α− f)(x), (16)

where D is the usual differential operator.

3

A fractional approach to the Sturm-Liouville problem

Next, some properties of these fractional operators are shown:

Property 1.Let f ∈ Lp(a, b), 1 ≤ p ≤ ∞, then for x in [a, b]:

(Dαa+I

αa+f)(x) = f(x) (17)

(Dαb−I

αb−f)(x) = f(x) (18)

Property 2.Let f ∈ L1(R), then for x in R:

(Dα+I

α+f)(x) = f(x) (19)

(Dα−I

α−f)(x) = f(x) (20)

Property 3.Let n− 1 < α < n, n ∈ N. Then

Iαa+(x− a)β−1 =Γ(β)

Γ(β + α)(x− a)β+α−1 (α > 0 and β > 0) (21)

Dαa+(x− a)β−1 =

Γ(β)

Γ(β − α)(x− a)β−α−1 (α ≤ 0 and β > 0) (22)

Iαb−(b− x)β−1 =Γ(β)

Γ(β + α)(b− x)β+α−1 (α > 0 and β > 0) (23)

Dαb−(b− x)β−1 =

Γ(β)

Γ(β − α)(b− x)β−α−1 (α ≤ 0 and β > 0) (24)

In particular, it is verified:

Dαa+(x− a)α−j = 0, j = 1, 2, . . . , n (25)

Dαb−(b− x)α−j = 0, j = 1, 2, . . . , n (26)

Property 4.Let λ > 0 and <(α) ≥ 0, then:

Dα−e−λx = λαe−λx (27)

Dα+e

λx = λαeλx (28)

The following rules for fractional integration by parts for the Riemann-Liouville, Caputo and Liouville fractional

derivatives hold:

Lemma 1.Let α > 0, p ≥ 1, q ≥ 1, and (1/p) + (1/q) ≤ 1 + α (p 6= 1 and q 6= 1 in the case (1/p) + (1/q) = 1 + α). Iff(x) ∈ Iαb−(Lp) and g(x) ∈ Iαa+(Lp) then

∫ b

a

f(x)(Dαa+g)(x)dx =

∫ b

a

g(x)(Dαb−f)(x)dx (29)

4

Margarita Rivero, Juan J. Trujillo, M. Pilar Velasco

Lemma 2.Let α > 0, m − 1 < α < m , p ≥ 1, q ≥ 1, and (1/p) + (1/q) ≤ 1 + α (p 6= 1 and q 6= 1 in the case(1/p) + (1/q) = 1 + α). If f(x) ∈ Iαb−(Lp) and g(x) ∈ Iαa+(Lp), then

∫ b

a

f(x)(CDαa+g)(x)dx =

∫ b

a

g(x)(Dαb−f)(x)dx−

m−1∑k=0

(−1)m−kg(k)(x)Dm−k−1Im−αb− f(x)|ba (30)

∫ b

a

f(x)(CDαb−g)(x)dx =

∫ b

a

g(x)(Dαa+f)(x)dx−

m−1∑k=0

(−1)kg(x)Dm−k−1Im−αa+ f(x)|ba (31)

In particular for 0 < α < 1:

∫ b

a

f(x)(CDαa+g)(x)dx =

∫ b

a

g(x)(Dαb−f)(x)dx+ g(x)I1−αb− f(x)|ba (32)∫ b

a

f(x)(CDαb−g)(x)dx =

∫ b

a

g(x)(Dαa+f)(x)dx− g(x)I1−αa+ f(x)|ba (33)

Lemma 3.Let α > 0, p > 1, q > 1, and (1/p) + (1/q) = 1 + α. If f(x) ∈ Iα−(Lp(R+)) and g(x) ∈ Iα+(Lp(R+)) then

∫ ∞−∞

f(x)(Dα+g)(x)dx =

∫ ∞−∞

g(x)(Dα−f)(x)dx (34)

3. Fractional Sturm-Liouville problem

We will consider a fractional lineal transform of 2α order, 12< α ≤ 1, as a differential operator in the form:

Lα : C2[a, b]→ C[a, b] ; Lα[y] = −Dα(u(x)Dαy) + v(x)y (35)

where Dα is a suitable fractional derivative and u, v are continuous functions in [a, b], a(x) 6= 0.

Definition 4.Let Lα : S ⊂ V → V , with S a subspace of the vector space V . Lα is symmetric respect to the internal productin V (and a self-adjoint operator) if:

< Lαf, g >=< f,Lαg > ; ∀f, g ∈ S (36)

In particular, for two real-valued functions f, g in the interval [a, b] the internal product is:

< f, g >=

∫ b

a

f(x)g(x)dx (37)

Definition 5.We say that a fractional problem with boundary conditions for differential equations of 2α order is constituted by:

• A fractional differential equation:

Lαy(x) = h(x) (38)

with Lα = −Dα(uDα) + v a fractional differential operator of 2α order and h ∈ C[a, b].

5

A fractional approach to the Sturm-Liouville problem

• Two boundary conditions:

α1y(a) + α2I1−α(uDαy)(x)|x=a = γ1 (39)

β1y(b) + β2I1−α(uDαy)(x)x=b = γ2 (40)

where αi, βi, γi, i = 1, 2, are constants.

First, we study the fractional Sturm-Liouville problem with Riemann-Liouville fractional operator.

Keeping in mind the rules of fractional integration by parts (34) for the Riemann-Liouville fractional derivative,

we can observe that the fractional Sturm-Liouville operator Lα = −Dαb−(uDα

a+) + v verifies the following identity

similar to the Lagrange identity in integral form for self-adjoint operators.

Lemma 4.Let α > 0, p ≥ 1, q ≥ 1, and (1/p) + (1/q) ≤ 1 + α (p 6= 1 and q 6= 1 in the case (1/p) + (1/q) = 1 + α). Iff(x) ∈ Iαb−(Lp) and p(x)Dα

a+g(x) ∈ Iαa+(Lp), then

∫ b

a

f(x)Lαg(x)dx =

∫ b

a

g(x)Lαf(x)dx (41)

Proof:

∫ b

a

f(x)Lαg(x)dx = −∫ b

a

fDαb−(uDα

a+g)(x)dx+

∫ b

a

v(x)f(x)g(x)dx =

= −∫ b

a

u(x)Dαa+g(x)Dα

a+f(x)dx+

∫ b

a

v(x)f(x)g(x)dx =

= −∫ b

a

gDαb−(uDα

a+f)(x)dx+

∫ b

a

v(x)f(x)g(x)dx =

=

∫ b

a

g(x)Lαf(x)dx (42)

�

Definition 6.We consider that a Riemann-Liouville fractional Sturm-Liouville problem is a fractional problem with boundaryconditions in the form:

−Dαb−(u(x)Dα

a+y)(x) + v(x)y(x) = λr(x)y(x), a < x < b,1

2< α ≤ 1 (43)

α1y(a) + α2I1−αb− (uDα

a+y)(x)|x=a = 0 (44)

β1y(b) + β2I1−αb− (uDα

a+y)(x)|x=b = 0 (45)

where Dαb− and Dα

a+ are the right-sided and left-sided Riemann-Liouville fractional derivatives respectively, Iαb−is the right-sided Riemann-Liouville fractional integral, Lα = −Dα

b−(uDαa+) + v is a self-adjoint operator, the

constants in the boundary conditions verify α21 + α2

2 6= 0, β21 + β2

2 6= 0 and u, v, r are continuous functions, suchthat u(x) > 0 and r(x) > 0 in x ∈ [a, b]. The function r is called the “weight” or “density” function and thevalues of λ for which there exist non-trivial solutions are called eigenvalues of the boundary value problem.

In these conditions, the mentioned problem verifies the following properties:

6

Margarita Rivero, Juan J. Trujillo, M. Pilar Velasco

Property 5.All of the eigenvalues of the fractional Sturm-Liouville problem are real.

Proof: Let us suppose that λ is a complex eigenvalue of the problem (43), (44), (45), with its corresponding

eigenfunction φ, possibly complex-valued. Then the eigenvalue and its eigenfunction verify

Lαφ(x) = λr(x)φ(x) (46)

and taking the complex conjugated it is verified

Lαφ(x) = λr(x)φ (47)

Now for the Lemma (4) we obtain

0 =

∫ b

a

(φ(x)Lαφ(x)− φ(x)Lαφ(x))dx = (λ− λ)

∫ b

a

r(x)φ(x)φ(x)dx = (λ− λ)

∫ b

a

r(x)|φ(x)|2dx (48)

Since this last integral is always positive, the conclusion is λ = λ, that is, the eigenvalue λ is real.

�

Property 6.If φ1 and φ2 are two eigenfunctions of the fractional Sturm-Liouville problem corresponding to eigenvalues λ1 andλ2, respectively, with λ1 6= λ2, then ∫ b

a

r(x)φ1(x)φ2(x)dx = 0, (49)

that is, the eigenfunctions corresponding to different eigenvalues have the property of orthogonality with respectto the weight function r.

Proof: Let λ1, λ2 be eigenvalues with the corresponding eigenvalues φ1 and φ2, and such that λ1 6= λ2. These

eigenvalues and eigenfunctions verify:

Lαφ1(x) = λ1r(x)φ1(x) (50)

Lαφ2(x) = λ2r(x)φ2(x) (51)

By multiplying the equations (50) and (51) with φ2 and φ1, subtracting these equations and integrating over the

interval [a, b], the following relation is obtained:

(λ1 − λ2)

∫ b

a

r(x)φ1(x)φ2(x)dx =

∫ b

a

(φ1(x)Lαφ2(x)− φ2(x)Lαφ1(x))dx (52)

and the right side of this relation values 0 by Lemma (4). Since λ1 6= λ2, the orthogonality for the eigenfunctions

φ1 and φ2 is verified.

7

A fractional approach to the Sturm-Liouville problem

�

Now we will prove under which conditions the eigenvalues of the fractional Sturm-Liouville problem (43)-(45) are

simple; that is, to each eigenvalue there corresponds only one linearly independent eigenfunction, apart from a

constant. For that we will use a similar technique to such given in [19].

Let λ an eigenvalues of the mentioned problem, and φ an eigenfunction for it. The equation (43) can write in the

following form

Dαb−

[u(x)Dα

a+

(−φ(x) + Iαa+

1

u(x)Iαb−Fλ(φ)

)]= 0, (53)

where Fλ(φ) = vφ−λrφ. Then, by considering the general solution of an equation of the typeDαb− [u(x)Dα

a+h(x)] =

0, the equation (53) can be written as

− φ(x) + Iαa+1

u(x)Iαb−Fλ(φ) = ξ1(x− a)α−1 + ξ2I

αa+

(b− x)α−1

Γ(α)u(x). (54)

Now applying the operator I1−αb− (uDαa+) to the above expression we get

I1−αb− (u(x)Dαa+φ(x)) + I1b−Fλ(φ) = ξ2. (55)

Therefore, taking into account such equation, the boundary conditions (44)-(45) can be wrtten in terms of the

constants ξ1, ξ2 as follow

α10 + α2

(−I1b−Fλ(φ)

∣∣x=a

+ ξ2)

= 0

β1

[ξ1(b− a)α−1 + ξ2

(Iαa+

(b− x)α−1

Γ(α)u(x)

)∣∣∣∣x=b

− Iαa+(

1u(x)

Iαb−Fλ(φ)

)]+ β2 = 0.

(56)

Moreover,

ξ1 = 1(b− a)α−1 I

αa+

(1

u(x)Iαb−Fλ(φ)

)∣∣∣∣x=b

−β2 + β1 I

αa+

(1

u(x)Iαb−Fλ(φ)

)∣∣∣∣x=b

β1(b− a)α−1

∫ baFλ(φ),

ξ2 =

∫ b

a

Fλ(φ),

(57)

and then

φ(x) = −Iαa+(

1

u(x)Iαb−Fλ(φ)

)+A(x)

∫ b

a

Fλ(φ) +B(x) Iαa+

(1

u(x)Iαb−Fλ(φ)

)∣∣∣∣x=b

(58)

where A(x) =

Iαa+( 1

u(x)Iαb−Fλ(φ)

)−β2 + β1 I

αa+

(1

u(x)Iαb−Fλ(φ)

)∣∣∣∣x=b

β1(b− a)α−1

B(x) =

(x− a)α−1

(b− a)α−1.

From the last expression and the Theorem 9 in [19], we obtain the following property.

8

Margarita Rivero, Juan J. Trujillo, M. Pilar Velasco

Property 7.Let 1/2 < α ≤ 1. Then, unique continuous eigenfunction φ for the problem (43)-(45) corresponding to eacheigenvalue exists, apart from a constant, under the following condition

||q + λr|| < mp

Mϕ +Bϕ(b) +A(b− a)mp, (59)

where ϕ(x) = Iαa+Iαb−1, and

A = ||A(x)||, B = ||B(x)||, mp = minx∈[a,b]

||p(x)||, Mϕ = ||ϕ(x)||. (60)

Remark 1.For this fractional operator, it is still an open problem to prove that the eigenvalues are separated and form aninfinite sequence, where the eigenvalues can be ordered according to increasing magnitude so that

λ1 < λ2 < · · · < λn < . . . , (61)

and λn →∞ as n→∞.

In these conditions, by keeping in mind the results of properties 5, 6 and 7, we could use the normalized eigen-

functions φ1, φ2, . . . , φn, . . . of the fractional Sturm-Liouville problem (43)-(45) to obtain a series expansion for a

function f :

f(x) =

∞∑n=1

cnφn(x) (62)

whose coefficients cm are given by

cm =

∫ b

a

r(x)f(x)φm(x)dx =< f, rφm > m = 1, 2, ... (63)

It is expected that under certain regularity conditions, for example f and f ′ be piecewise continuous on a ≤ x ≤ b,

the series expansion (62) will converge to f(x+)+f(x−)2

at each point in the open interval (a, b).

Remark 2.For α = 1, the fractional Sturm-Liouville problem (43)-(45) turns into the ordinary Sturm-Liouville problem, andthis fact could allow us to generalize the classical Fourier series expansion by sines and cosines, although for thisobjective it would be necessary to study the zeros of the eigenfunctions associated to the specific boundary valueproblem in each case.

Similar results can be given for the fractional Sturm-Liouville problem with Caputo and Riemann-Liouville frac-

tional derivatives.

By the rules for fractional integration by parts (30)-(33) for the Caputo fractional derivatives, the following

identity is hold for the operator Lα = −Dαb−(uCDα

a+) + v:

9

A fractional approach to the Sturm-Liouville problem

Lemma 5.Let α > 0, p ≥ 1, q ≥ 1, and (1/p) + (1/q) ≤ 1 + α (p 6= 1 and q 6= 1 in the case (1/p) + (1/q) = 1 + α). Iff(x) ∈ Iαb−(Lp) and u(x)Dα

a+g(x) ∈ Iαa+(Lp), then

∫ b

a

f(x)Lαg(x)dx =

∫ b

a

g(x)Lαf(x)dx+(g(x)I1−αb− f(x)− f(x)I1−αb− g(x)

)|ba (64)

Proof:

∫ b

a

f(x)Lαg(x)dx = −∫ b

a

f(x)Dαb−(uCDα

a+g)(x)dx+

∫ b

a

v(x)f(x)g(x)dx

= −∫ b

a

u(x)CDαa+g(x)CDα

a+f(x)dx− f(x)I1−αb− (uCDαa+g)|ba +

∫ b

a

v(x)f(x)g(x)dx

= −∫ b

a

gDαb−(uCDα

a+f)(x)dx+ g(x)I1−αb− (uCDαa+f)(x)|ba − f(x)I1−αb− (uCDα

a+g)(x)|ba +

∫ b

a

v(x)f(x)g(x)dx

=

∫ b

a

g(x)Lαf(x)dx+(g(x)I1−αb− (uCDα

a+f)(x)− f(x)I1−αb− (uCDαa+g)(x)

)|ba (65)

�

Definition 7.A Caputo fractional Sturm-Liouville problem is a fractional problem with boundary conditions in the form:

−Dαb−(uCDα

a+y)(x) + v(x)y(x) = λr(x)y(x), a < x < b,1

2< α ≤ 1 (66)

α1y(a) + α2I1−αb− (uCDα

a+y)(x)|x=a = 0 (67)

β1y(b) + β2I1−αb− (uCDα

a+y)(x)|x=b = 0 (68)

where Lα = −Dαb−(uCDα

a+) + v is a self-adjoint operator and the constants αi, βi and the functions u, v, r verifythe same conditions that in Definition 6.

And then, by using Lema 5 analogous properties to the properties 5, 6 and 7 of the problem with Riemann-Liouville

fractional operator can be easily proved.

Again, we can obtain similar results for the Liouville fractional Sturm-Liouville problem:

Let us define the operator Lα = −Dα−(uDα

+) + v. For this operator the following identity is verified by using the

rule for fractional integration by parts (34):

Lemma 6.Let α > 0, p > 1, q > 1, and (1/p) + (1/q) = 1 + α. If f(x) ∈ Iα−(Lp(R+)) and u(x)Dα

+g(x) ∈ Iα+(Lp(R+)), then

∫ ∞−∞

f(x)Lαg(x)dx =

∫ ∞−∞

g(x)Lαf(x)dx (69)

10

Margarita Rivero, Juan J. Trujillo, M. Pilar Velasco

Proof:

∫ ∞−∞

f(x)Lαg(x)dx = −∫ ∞−∞

fDα−(uDα

+g)(x)dx+

∫ ∞−∞

v(x)f(x)g(x) =

= −∫ ∞−∞

u(x)Dα+g(x)Dα

+f(x)dx+

∫ ∞−∞

v(x)f(x)g(x) =

= −∫ ∞−∞

gDα−(uDα

+f)(x)dx+

∫ ∞−∞

v(x)f(x)g(x) =

=

∫ ∞−∞

g(x)Lαf(x)dx (70)

�

Definition 8.We consider that a Liouville fractional Sturm-Liouville problem is a fractional problem with boundary conditionsin the form:

−Dα−(uDα

+y)(x) + v(x)y(x) = λr(x)y(x), a < x < b,1

2< α ≤ 1 (71)

α1y(a) + α2I1−α− (u(x)Dα

+y)(x)|x=a = 0 (72)

β1y(b) + β2I1−α− (u(x)Dα

+y)(x)|x=b = 0 (73)

where Lα = −Dα−(pDα

+) + q is a self-adjoint operator and the constants αi, βi and the functions u, v, r verify thesame conditions that in Definition 6.

So similar results to the properties 5, 6 and 7 of the Riemann-Liouville fractional Sturm-Liouville problem are

proved for the Liouville operator too, by considering that in this case the interval of integration is (−∞,∞).

4. Particular example

Let us consider Lα = D2α = Dα−D

α+ with 1

2< α ≤ 1, where Dα

− and Dα+ are Liouville fractional derivative, and

then let us find the solution of the problem:

D2αy(x) = λy(x), a < x < b (74)

with particular boundary conditions, for example

y(0) = 0 (75)

y(1) + I1−α− Dα+y(1) = 0 (76)

Remark 3.Note that for α = 1 we obtain the classical second order operator D2. However, for α = 1

2we have not the

operator D in general, except for continuous functions that is the case that we will consider.

If λ > 0 then the following properties are verified:

11

A fractional approach to the Sturm-Liouville problem

Property 8.

Dα+ cos(λx) = λα cos

(λx+

απ

2

)(77)

Dα+ sin(λx) = λα sin

(λx+

απ

2

)(78)

Property 9.

Dα−D

α+ cos(λx) = λ2α cos(λx) (79)

Dα−D

α+ sin(λx) = λ2α sin(λx) (80)

Proof:

Dα−D

α+ cos(λx) = λαDα

− cos(λx+

απ

2

)= λαDα

−

(cos(λx) cos

(απ2

)− sin(λx) sin

(απ2

))= λ2α

(cos(λx− απ

2

)cos(απ

2

)− sin

(λx− απ

2

)sin(απ

2

))= λ2α cos(λx)

The second equality is demonstrated of analogous form.

�

Then the equation (74) for λ > 0 has the general solution:

y = c1 cos(λ

12α x

)+ c2 sin

(λ

12α x

)(81)

Moreover, Lα is a self-adjoint operator, that is < Lαu, v >=< u,Lαv > for all u, v ∈ C(R) using Lemma 6. And

then if we consider the interval [a, b] the fractional Sturm-Liouville problem (74), with certain suitable boundary

conditions, verifies the properties 5, 6 and 7.

In particular, if we take the homogeneous boundary conditions

y(0) = 0 (82)

y(1) + I1−α− Dα+y(1) = 0 (83)

we have that c1 = 0 in (81) and the eigenvalues λn,α could be obtained by solving numerically the equation:

sinλ12α + λ1− 1

2α cosλ12α = 0 (84)

12

Margarita Rivero, Juan J. Trujillo, M. Pilar Velasco

It is easily checked that this equation gives us an infinite and enumerable set of separated eigenvalues, whose

associated eigenfunctions are:

φn,α(x) = sin(λ12αn,αx) (85)

With these eigenfunctions, the solution of the problem (74)-(76) has the form of series expansion such as

y(x) =

∞∑n=1

cn,αφn,α(x) (86)

where coefficients cm are given by

cm,α =

∫ ∞−∞

y(x)φm,α(x)dx =< y, rφm,α > m = 1, 2, ... (87)

5. Conclusions

In this paper we use different fractional composition operators involving Riemann-Liouville, Caputo or Liouville

fractional operators to propose a fractional approach to the ordinary Sturm-Liouville problem. The classical

properties of the Sturm-Liouville theory are analyzed for the case of the eigenvalues and eigenfunctions of our

fractional Sturm-liouville problems and we proved that in the fractional case the eigenfunctions are orthogonal and

the eigenvalues are real and simple. In these conditions, we conjecture that these eigenfunctions and eigenvalues

will allow to define a new kind of series expansion that depends on a fractional parameter. The applicability

of the results is evidenced with an example of a 2α-order fractional differential equation, 12< α ≤ 1, which is

related to other more complex problems, in particular to the ordinary heat equation and its resolution by means

of the method of separation of variables for the classical problem with α = 1. This leads us to think about the

utility of a fractional generalization of the Sturm-Liouville theory for giving support to the method of separation

of variables in the resolution of fractional partial differential equations, as we will study in future works.

Acknowledgments

The authors want to express their gratitude to FEDER and to Spain Government (Projects MTM2010-16499 and

AYA2009-14212-C05-05/ESP).

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