#### On the fundamental solutions of a discontinuous fractional boundary value problem

Yakar and Akdogan Advances in Difference Equations
On the fundamental solutions of a discontinuous fractional boundary value problem
Ali Yakar
Zulfigar Akdogan
The main purpose of this study is to investigate a fractional discontinuous Sturm-Liouville problem with transmission conditions. We shall consider a fractional boundary value problem involving an operator with two parts. It is shown that the eigenvalues and corresponding eigenfunctions of the main problem coincide with the eigenvalues and corresponding eigenfunctions of the constructed operator in Hilbert spaces.
fractional calculus; fractional Sturm-Liouville problem; fractional boundary conditions; fractional transmission condition
1 Introduction
Fractional calculus has been a well-known topic since it was initiated in the seventeenth
century and studied by many great mathematicians of the time. The history of fractional
calculus can be found in [, ]. It has been shown that in many applications, fractional
derivatives based models provide more accurate solutions for real processes of
anomalous systems than the integer order derivatives based models do [–]. After realizing the
success of fractional differential equations in modeling real world problems, much work
has been done in this branch of mathematics in recent decades. For some recent
contributions about fractional differential equations and fractional dynamic systems, see [–]
and the references therein.
The studies of Sturm-Liouville (S-L) problems have attracted the attention of many
mathematicians and physicists since they have many useful applications in branches of
science, theoretical and applied mathematics. In particular, [–] contains many
references to problems in physics and mechanics.
On the other hand, the general theory and methods of boundary value problems with
continuous coefficients are highly developed, very little is known about a general
character of similar problems with discontinuities. In view of demands of modern
technology, engineering and physics, Sturm-Liouville type problems with transmission
conditions have become a very important place of research in recent years. Discontinuous
boundary value problems with transmission conditions and their applications to the
corresponding initial-boundary value problems for parabolic equations have been
investigated by Mukthrov et al. in [–]. Also, some problems with transmission conditions
which arise in mechanics (thermal conduction problem for a thin laminated plate) were
studied in the article []. Akdogan et al. and Demirci et al. [–] studied a
SturmLiouville problem with discontinuities in the case when an eigenparameter appears not
only in the differential equation but also in both the boundary and transmission
conditions.
As can be seen in the references cited above, there have been a lot of research papers
on Sturm-Liouville problems but the authors generally take into account the case where
classical integer order derivatives exist. In recent years, some previous works involving
fractional operators for S-L problems have been published. Riverouse et al. [] use some
fractional composition operators to propose a fractional approach to the ordinary
SturmLiouville problem and investigate the eigenvalues and eigenfunctions associated to these
operators. In [], the authors consider a new conformable fractional derivative and
apply a functional compression-expansion fixed point theorem to prove the existence of a
positive solution for fractional boundary value problem with S-L boundary conditions.
In [], utilizing the Legendre integral transform, the authors demonstrate some
applications of their results by solving both fractional ordinary and partial differential equations.
Zayernouri and Karniadakis [] investigated two classes of fractional Sturm-Liouville
eigenvalue problems on a compact interval [a, b] in more detail. They both obtained some
explicit forms for the eigensolutions of these problems and derived some useful
spectral properties of the obtained eigensolutions. In [], Klimek et al. apply the methods
of fractional variational analysis and prove the existence of a countable set of orthogonal
solutions and corresponding eigenvalues for a regular fractional Sturm-Liouville
problem.
The main aim of this paper is to extend some results of fractional Sturm-Liouville
problems to the case of discontinuous fractional Sturm-Liouville problems. Namely, we deal
with a discontinuous fractional Sturm-Liouville problem with transmission conditions. To
do so, we define an operator A in the Hilbert space L[–, ], the eigenvalues and
corresponding eigenfunctions of which coincide with the eigenvalues and corresponding
eigenfunctions of the boundary value problem respectively. Then, we establish the
characteristic function and prove that the eigenvalues of the considered problem coincide with the
roots of this characteristic function.
The paper is organized as follows: In Section , some of the basic properties of the
Riemann-Liouville and Caputo fractional derivatives are given, and we also prove a useful
lemma. Section presents the construction of a discontinuous fractional S-L problem.
We introduce the operator theoretical form in the Hilbert space L[–, ] and discuss the
characteristic function in the subsequent sections.
2 Some auxiliary definitions and results
In this section, we shall recall some basic definitions and facts which are necessary for the
development of the paper (see also [, ]).
Definition (Left and right Riemann-Liouville fractional integrals) Let [a, b] ⊂ ,
Re(α) > and f ∈ L[a, b]. Then the left and right Riemann-Liouville fractional integrals
Iaα+ and Ibα– of order α are given by
Iaα+f (x) =
Ibα– f (x) =
x f (t) dt
(α) a (x – t)–α , x ∈ (a, b],
b f (t) dt
(α) x (t – x)–α , x ∈ [a, b).
Definition (Left and right Riemann-Liouville (R-L) fractional derivatives) Let [a, b] ⊂
, Re(α) ∈ (, ) and f ∈ L[a, b]. The left R-L fractional derivative of order α of function
f , denoted by Daα+f , is defined as
∀x ∈ (a, b],
Daα+f (x) := DIa–+αf (x).
Similarly, the right Riemann-Liouville fractional derivative of order α of function f ,
denoted by Dbα– f , is
∀x ∈ [a, b),
Dbα– f (x) := –DIb––αf (x),
where D = ddx is the usual differential operator.
Definition (Left and right Caputo fractional derivatives) Let [a, b] ⊂ , Re(α) ∈ (, )
and f ∈ L[a, b]. The left and right Caputo fractional derivatives of order α are
∀x ∈ (a, b], c α
Da+f (x) := Ia–+αDf (x)
∀x ∈ [a, b), c α
Db– f (x) := –Ib––αDf (x),
respectively.
and
and
Daα+Iaα+f (x) = f (x),
Dbα– Ibα– f (x) = f (x)
Iaα+Daα+f (x) = f (x) –
Ibα– Dbα– f (x) = f (x) –
(x – a)α–
(α)
(b – x)α–
(α)
Ia–+αf (a),
Ia–+αf (b),
where α ∈ (, ).
Property The following property shows that the Riemann-Liouville derivative is the left
inverse of the Riemann-Liouville integral, but we cannot claim that it is the right inverse.
Property For certain classes of functions, the Caputo fractional derivatives are the
inverse operators of the Riemann-Liouville fractional integrals.
and
cDaα+Iaα+f (x) = f (x),
cDbα– Ibα– f (x) = f (x)
Iaα+cDaα+f (x) = f (x) – f (a),
Ibα– cDbα– f (x) = f (x) – f (b).
In classical calculus, the integration by parts formula relates the integral of a product
of functions to the integral of their derivative and antiderivative. As we can see below,
this formula also works for fractional derivatives; however, it changes the type of
differentiation: left Riemann-Liouville fractional derivatives are transformed to right Caputo
fractional derivatives. For more detail see, for example, [].
Property Assume that < α < , f ∈ AC[a, b] and g ∈ Lp(a, b) ( ≤ p ≤ ∞). Then the
following integration by parts formula holds:
a
b
a
b
f (x)Daα+g(x) dx =
g(x)cDbα– f (x) dx + f (x)Ia–+αg(x) xx==ba.
Now, we state and prove the following lemma which is going to be used in the next
section.
Lemma Let f ∈ L(a, b) and α ∈ (, ), then
. Iaα+ cDbα– f (x) = Mg (x) + (–)α(f (x) – f (b)),
. Iaα+ cDbα– f (x) = (–)α–Iaα+ Nf (x) + (–)α(f (x) – f (a)),
where Mg (x) = (α) ab(x – t)α–g(t) dt, Nf (x) = (–α) ab(x – t)–αf (t) dt and g(x) = cDbα– f (x).
Proof In view of Definition , we have
Mg (x) =
(α) a
x
(x – t)α–g(t) dt +
(x – t)α–g(t) dt
(α) x
b
= Iaα+ g(x) + (–)α–Ibα– g(x).
Then it leads to
Iaα+ g(x) = Mg (x) + (–)αIbα– g(x).
To prove (), by Definition , we obtain
Nf (x) =
( – α) a
x
c α
= Da+ f (x) +
(x – t)–αf (t) dt +
( – α) x
b
= Da+ f (x) + (–)–αcDbα– f (x),
c α
( – α) x
b
(x – t)–αf (t) dt
(t – x)–α(–)–α+ –f (t) dt
which gives
c α c α
Db– f (x) = Nf (x) – Da+ f (x) (–)α–.
α
By applying the fractional operator Ia+ to both sides, we get
Ia+ Db– f (x) = Ia+ Nf (x) – Da+ f (x) (–)α–
α c α α c α
= (–)α–Iaα+ Nf (x) + (–) Ia+ Da+ f (x)
α α c α
= (–)α–Iaα+ Nf (x) + (–) f (x) – f (a) ,
α
which completes the proof.
3 Fractional Sturm-Liouville problem with transmission conditions
Let the operator Lα,x be defined as
Lα,x :=
Lα,xu + λu =
⎨⎧ cDα– p(x)D–+ + q(x), x ∈ [–, );
α
⎩ cDα– p(x)D+ + q(x),
α
x ∈ (, ].
on x ∈ [–, ) ∪ (, ] with fractional boundary conditions
–α α
L(u) := cI–+ u(–) + cD–+ u(–) = ,
L(u) := dI–+αu() + dD+ u() =
α
and fractional transmission conditions at the inner point x =
L(u) := hI–+ u(–) + I–+αu(+) = ,
–α
α α
L(u) := D–+ u(–) + hD+ u(+) = ,
where < α ≤ , λ is complex eigenparameter;
p(x) =
⎧
⎨ p, x ∈ [–, );
⎩ p, x ∈ (, ].
q(x) is real-valued and continuous in both [–, ) and (, ], it also has finite limits q(±) :=
limx→± q(x), c + c = , d + d = , and p, p, h, h are positive real numbers.
4 The operator formulation of the problem
Let us consider the inner-product in the Hilbert space L(–, ) as follows:
h
p –
h
p
f , g =
f (x)g(x) dx +
f (x)g(x) dx,
where F := f (x), G := g(x) ∈ L(–, ). In this Hilbert space we define the operator A with
domain
()
⎧⎪ f : f (x) and Dαf (x), cDαf (x) are absolutely continuous ⎪⎫
D(A) := ⎨ on [–, ) ∪ (, ], and f (±), Dαf (±), I–αf (±) have ⎬
⎪⎩ finite limits, Lif = , i = , , , . ⎪⎭
and action law
Af := Lα,xf .
Au = λu.
Thus, problem ()-() can be written in the operator form as
Note that by eigenvalues and eigenfunctions of problem ()-() we mean eigenvalues and
eigenelements of the operator A, respectively.
Theorem The linear operator A is symmetric.
Proof For each f , g ∈ Dom(A), using () we write
Af , g = h
p –
= h
p –
Af (x)g(x) dx + h
p
Af (x)g(x) dx
cDα– pDα–+ f (x) g(x) dx + h
p –
+ h
p
cDα– pDα+ f (x) g(x) dx + h
p
q(x)f (x)g(x) dx
q(x)f (x)g(x) dx.
By applying Property , we get
Af , g = h
f (x)cDα– Dα–+ g(x) dx + Dα–+ g(x)I––+αf (x) – – Dα–+ f (x)I––+αg(x) –
+ h
f (x)cDα– Dα+ g(x) dx + Dα+ g(x)I–+αf (x) – Dα+ f (x)I–+αg(x)
–
+ h
p –
q(x)f (x)g(x) dx + h
p
q(x)f (x)g(x) dx
= f , Ag + h Dα–+ g(x)I––+αf (x) – – Dα–+ f (x)I––+αg(x) –
+ h Dα+ g(x)I–+αf (x) – Dα+ f (x)I–+αg(x) .
By considering the fractional transmission conditions ()-(), we have
Af , g = f , Ag
that the operator A is symmetric.
Corollary All eigenvalues of problem ()-() are real.
Corollary Let λ and λ be two different eigenvalues of problem ()-(). Then the
corresponding eigenfunctions f and g of this problem satisfy the following equality:
h
p –
Af (x)g(x) dx + h
p
Af (x)g(x) dx = .
As a consequence, the eigenfunctions of problem ()-() corresponding to the different
eigenvalues are orthogonal to the inner product () in the Hilbert space L(–, ).
Naturally, we can now assume that all eigenfunctions of problem ()-() are
realvalued.
Lemma The equivalent integral form of equation () with fractional conditions ()-()
is given as
u(x) = u(x) +
where u(x) = xα(–α) (–hI––+αu(–)) + Iα+ (– h Dα–+ u(–)).
Proof Let us consider equation ()
cDα– pDα+ u(x) + λ + q(x) u(x) = , x ∈ (, ],
integral operators Iα+ acting on this equation and by Lemma , we obtain
Iα+ cDα– pDα+ u(x) + Iα+ λ + q(x) u(x) =
and
pDα+ u(x) = Iα+ Nu(x) + pDα+ u(+) + (–)–αIα+ λ + q(x) u(x).
Applying Iα+ on both sides of () and using conditions ()-(), we find
u(x) =
–hI––+αu(–) + Iα+ –
h
Dα–+ u(–) + p I+αNu(x)
I+α λ + q(y) u(x).
xα–
(α)
+
(–)–α
p
Then we reach
u(x) = u(x) + p I+α Nu(x) + (–)–α λ + q(x) u(x) ,
()
()
()
We next define um(x, λ) to construct the successive approximations
x
(x – y)α– Num– (y) + (–)–α λ + q(y) um–(y) dy.
Remark The corresponding classical Sturm-Liouville problem with integer orders is
covered by the case α = .
Lemma Let Q := maxx∈(,] |q(x)|, PR := max|λ|≤R P(λ) and P(λ) := maxx∈(,] |u(x, λ)|,
kα := /[( – α) ( – α)]. Then the following estimate
um(x, λ) – um–(x, λ) ≤ PR
|λ| + kα + Q
p (α + )
m
()
holds for all m.
Proof Let us apply the mathematical induction for m. In what follows, for convenience we
shall use the notation K = / (α + ).
For m = , we have
u(x, λ) – u(x, λ) =
p I+α Nu (x) + (–)–α λ + q(x) u(x, λ) .
By using Lemma . in [], we have
By using Corollary . in [], we have
u(x, λ) – u(x, λ) ≤ p K Nu (x) + (–)–α λ + q(x) u(x, λ)
≤ p K
Nu (x) +
λ + q(x) u(x, λ) .
u(x, λ) – u(x, λ) ≤ pK kα u(x, λ) + |λ| + Q
u(x, λ)
≤ KpPR kα + |λ| + Q .
Suppose that () holds for m – , i.e.,
um–(x, λ) – um–(x, λ) ≤ PR pK |λ| + kα + Q
m–
.
Then we have
um(x, λ) – um–(x, λ)
=
K
≤ p
p I+α Num––um– (x) + (–)α λ + q(x) um–(x, λ) – um–(x, λ)
Num––um– (x) +
λ + q(x) um–(x, λ) – um–(x, λ)
um–(x, λ) – um–(x, λ)
The proof is completed.
Lemma The following IVP
cDα– pD–+ u(x) + q(x) + λ u(x) = , x ∈ [–, ],
α
has a unique solution on [–, ] provided that
K |λ| + kα + Q < .
p
+ |λ| + Q
um–(x, λ) – um–(x, λ)
Proof If we use a similar way in Lemma , we get a corresponding integral equation of the
problem as follows:
u(x) = u(x) + p I–α+ Nu(x) + (–)–α λ + q(x) u(x) ,
where u(x) = (x(++)αα) (–c) + (x+)α–
(α) c.
Let us construct the integral equation by
φ = T φ,
then we have
where the mapping T is defined as
Tf = u + p I–α+ Nf + (–)–α(λ + q)f ,
Tf – Tg =
p I–α+ (Nf – Ng ) + (–)–α(λ + q)(f – g) .
By applying Lemma . in [], we get
Tf – Tg ≤ pK (Nf – Ng ) + (–)–α(λ + q)(f – g)
≤ pK (Nf – Ng ) + (λ + q)(f – g) .
By relation () we have then
Nf – Ng = D–+ (f – g) + (–)–αcDα– (f – g),
c α
Nf – Ng ≤ kα f – g + kα f – g
= kα f – g ,
Tf – Tg ≤ pK |λ| + kα + Q f – g .
where we have used Corollary . in []. If we substitute the last inequality into (), we
find
By condition (), the mapping T is a contraction on the space C[–, ], · .
Consequently, there exists a unique solution of equation (). The proof is complete.
Theorem For any λ ∈ C satisfying Kpi–(|λ| + kα + Q) < (i = , ), the differential
equation () has a unique solution which satisfies fractional boundary condition () and
fractional transmission conditions ()-().
Proof Consider the following problem for each λ ∈ C:
Lα,xu(x) + λu(x) = , x ∈ [–, ),
cDα– pD–+ u(x) + q(x) + λ u(x) = , x ∈ [–, ),
α
I––+αu(–) = c,
α
D–+ u(–) = –c.
By considering Lemma , the initial value problem has a unique solution φ(x, λ).
Next, take into account the differential equation
Lα,xu(x) + λu(x) = , x ∈ (, ],
cDα– pDα+ u(x) + q(x) + λ u(x) = , x ∈ (, ],
I–+αu(+) = –hI––+αφ(–),
Dα+ u(+) = –
Dα–+ φ(–).
h
un(x, λ) = u(x, λ)
+
We establish the sequence {un(x, λ)} for x ∈ (, ] and n = , , . . . such that
(x – y)α– Nun– (y) + (–)–α λ + q(y) un–(y, λ) dy,
()
()
()
()
()
()
()
()
()
According to estimate () in Lemma , for < x ≤ , the absolute value of its terms is
less than the corresponding terms of the convergent numeric series
Hence, series () converges uniformly. Obviously, each term (uj(x, λ) – uj–(x, λ)) of series
() is continuous on x ∈ (, ]. Therefore, the sum of series () is continuous on x ∈ (, ]
and
φ(x, λ) = nl→im∞ un(x, λ) = u(x, λ) + u∗(x, λ)
is continuous on x ∈ (, ].
The uniform convergency of the sequence un(x, λ) allows us to take n → ∞ in the
relation (). This gives equations () showing that φ(x, λ), the limit function of the process
defined by () and (), is the solution of (). Furthermore, it is trivial that φ(x, λ)
satisfies the initial conditions ()-(). Finally, the function φ(x, λ) given by
where
u(x, λ) = I–α+ kDα–+ φ(–, λ) , x ∈ (, ].
Obviously, each of the functions un(x, λ) is an entire function of λ for each x ∈ (, ].
Now let us consider the series
u∗(x, λ) = lim un(x, λ) – u(x, λ) =
n→∞
uj(x, λ) – uj–(x, λ) .
∞
j=
φ(x, λ) = ⎨⎧ φ(x, λ), x ∈ [–, ),
⎩ φ(x, λ), x ∈ (, ]
satisfies the differential equation (), fractional boundary condition () and fractional
transmission conditions () and ().
In a similar manner, we can prove the following theorem.
Theorem For any λ ∈ C, the differential equation
Lα,xu(x) + λu(x) = , x ∈ [–, ) ∪ (, ]
has a unique solution
χ (x, λ) = ⎨⎧ χ(x, λ), x ∈ [–, ),
⎩ χ(x, λ), x ∈ (, ]
satisfying fractional boundary condition () and fractional transmission conditions () and
() for each x ∈ [–, ) ∪ (, ].
Remark If λ is not eigenvalue, then φ and χ are linearly independent solutions of
equation () in the interval [–, ). Similarly, φ and χ are linearly independent solutions
of equation () in the interval (, ]. Then it is obvious that the four functions u˜, u˜, u˜,
u˜ which are defined by
u˜ =
u˜ =
⎧⎨ φ(x, λ), x ∈ [–, ),
⎩ ,
x ∈ (, ],
⎨⎧ , x ∈ [–, ),
⎩ φ(x, λ), x ∈ (, ],
u˜ =
u˜ =
⎧⎨ χ(x, λ), x ∈ [–, ),
⎩ ,
x ∈ (, ],
⎨⎧ , x ∈ [–, ),
⎩ χ(x, λ), x ∈ (, ]
χ(x, λ) = αφ(x, λ).
χ (x, λ) = ⎨⎧ χ(x, λ), x ∈ [–, ),
⎩ χ(x, λ), x ∈ (, ]
are linearly independent solutions of equation () in whole [–, ) ∪ (, ].
To prove this fact, suppose if possible that λ = λ is not an eigenvalue but the
corresponding solutions φ(x, λ) and χ(x, λ) are linearly dependent. Then there is a constant
α = such that
From this equality obviously follows that the solution χ(x, λ) also satisfies the first
boundary condition. Consequently, the solution
satisfies also the first boundary condition. Therefore, χ (x, λ) satisfies all boundary and
transmission conditions, that is, χ (x, λ) is an eigenfunction for the considered problem
()-(), and consequently λ is an eigenvalue. Thus we have a contradiction which
completes the proof.
Let us consider the fractional Wronskians
ωi(λ) := WF φi(x, λ), χi(x, λ) , i = ,
:= I––+αφi(x, λ)Dα+ χi(x, λ) – I–+αχi(x, λ)Dα–+ φi(x, λ)
which are independent of x and are entire functions. The short calculation gives
ω(λ) = ω(λ).
ω(λ) := ω(λ) = ω(λ).
WF (λ) = –hω(λ),
Now we may introduce to the consideration the characteristic function
Lemma The fractional Wronskian WF satisfies the following relation:
()
where
L(φ)
WF (λ) = L(φ)
L(φ)
L(φ)
L(χ)
L(χ)
L(χ)
L(χ)
L(φ)
L(φ)
L(φ)
L(φ)
L(χ)
L(χ) .
L(χ)
L(χ)
Proof Employing the definitions of the functions φi(x, λ) and χi(x, λ), i = , , we obtain
WF (λ) =
ω(λ)
hI––+αφ(–, λ) hI––+αχ(–, λ)
Dα–+ φ(–, λ) Dα–+ χ(–, λ)
ω(λ)
I–+αφ(+, λ) I–+αχ(+, λ)
hDα+ φ(+, λ) hDα+ χ(+, λ)
= ω(λ)ω(λ) hI––+αφ(–, λ)
α
D–+ φ(–, λ)
I–+αχ(+, λ)
hDα+ χ(+, λ)
= –hω(λ)ω(λ)
= –hω(λ).
φ(x, λ) = cχ(x, λ); x ∈ (, ]
φ(x, λ) = ⎨⎧ φ(x, λ), x ∈ [–, ),
⎩ φ(x, λ), x ∈ (, ]
Corollary The zeros of the function WF (λ) consist of the zeros of the characteristic
function ω(λ).
Theorem The eigenvalues of fractional boundary value problem ()-() are the same as
the roots of the characteristic function ω(λ).
Proof Let λ = λ be a root of the characteristic function ω(λ), hence ω(λ) = . It follows
that φ and χ are linearly dependent, that is,
for some c = . As a result, the function φ(x, λ) satisfies fractional boundary condition
(). So, φ(x, λ) which is given by
satisfies the main problem ()-(). So, the function φ(x, λ) is an eigenfunction of problem
()-() corresponding to the eigenvalue λ.
Let λ = λ be an eigenvalue and u(x, λ) be any corresponding eigenfunction. It must be
proved that ω(λ) = . Let us suppose that ω(λ) = . Then, since ω(λ) = and ω(λ) =
, there exist constants ci, i = , , , , at least one of which is not zero, such that
⎧
u(x, λ) = ⎨ cφ(x, λ) + cχ(x, λ), x ∈ [–, ),
⎩ cφ(x, λ) + cχ(x, λ), x ∈ (, ]
since ω(λ) = and ω(λ) = .
Since the eigenfunction u(x, λ) satisfies both fractional boundary and fractional
transmission conditions ()-(), we have
Liu(·, λ) = ,
for i = , , , .
Also since at least one of the constants ci, i = , , , , is not zero,
det Liu(·, λ) = ,
that is, WF (λ) = . But, by Lemma , WF (λ) = . This contradiction completes the proof.
Acknowledgements
We are immensely grateful to Prof. Oktay Muhtarog˘ lu for his expert advice and for sharing his pearls of wisdom with us
throughout preparing this research, and we also thank the anonymous reviewers for their valuable comments.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
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