YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL VOLTERRA AND ABEL'S INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

This study outlines the local fractional integro–differential equations carried out by the local fractional calculus. The analytical solutions within local fractional Volterra and Abel’s integral equations via the Yang–Laplace transform are discussed. Some illustrative examples will be discussed. The obtained results show the simplicity and efficiency of the present technique with application to the problems for the local fractional integral equations. © 2018, Semnan University, Center of Excellence in Nonlinear Analysis and Applications. All rights reserved.

Full text

YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL
VOLTERRA AND ABEL’S INTEGRO-DIFFERENTIAL EQUATIONS
FUAT USTA, H¨
USEYIN BUDAK, AND MEHMET ZEKI SARIKAYA
Abstract. This study outlines the local fractional integro-differential equations carried out by the
local fractional calculus. The analytical solutions within local fractional Volterra and Abel’s integral
equations via the Yang-Laplace transform are discussed. Some illustrative examples will be discussed.
The obtained results show the simplicity and efficiency of the present technique with application to
the problems for the local fractional integral equations.
1. INTRODUCTION
Fractional derivatives and fractional calculus have a long history and there are a number of appli-
cations in applied mathematics and engineering. Finding the solution for the differential and integral
equations is one of the hot topics among the mathematicians and engineers. There are several an-
alytical and numerical techniques for solving them [9], such as the spectral Legendre-Gauss-Lobatto
collocation method [1], the shifted Jacobi-Gauss-Lobatto collocation method [2], the variation iteration
method [3], the heat-balance integral method [4], the Adomian decomposition method [7], the finite
element method [19], and the finite difference method [5].
Recently the theory of local fractional calculus as one of the practical tools to handle the fractal
and continuously non-differentiable functions, was successfully implemented in real world problems
which modelled by local fractional calculus. For detailed information about recent developments local
fractional equations, please refer to [6], [8], [10]-[18].
The main target of this paper is to take into account to application of Yang-Laplace transform to
Volterra and local fractional Abel’s integro-differential equations with local fractional derivative and
local fractional integral. The remainder of this work is organized as follows: in Section 2, we give a brief
description of the local fractional calculus, while, in Section 3, we show that for the local fractional
Volterra integral equations the Yang-Laplace transform method can be applied successfully. In Section
4, we show that how to applied the Yang-Laplace transform to local fractional Abel’s integral equations.
Some examples are given in Section 5, while some conclusions and further directions of research are
discussed in Section 6.
2. PRELIMINARIES
The objective of this section is to state the prerequisite definitions and also to summarized the
necessary equalities for local fractional calculus.
2.1. Local Fractional Calculus. Recall the set Rαof real line numbers and use the Gao-Yang-
Kang’s idea to describe the definition of the local fractional calculus, see [10, 11]. Recently, the theory
of Yang’s fractional sets [11] was introduced as follows. For 0 < α ≤1,we have the following α-type
sets:
ZαThe α-type set of integer is defined as the set {0α,±1α,±2α, ..., ±nα, ...}.
QαThe α-type set of the rational numbers is defined as the set {mα= (p/q)α:p, q ∈Z, q 6= 0}.
JαThe α-type set of the irrational numbers is defined as the set {mα6= (p/q)α:p, q ∈Z, q 6= 0}.
RαThe α-type set of the real line numbers is defined as the set Rα=Qα∪Jα.
Table 1. α−type sets
2000 Mathematics Subject Classification. 65R20, 45D05, 45E10, 26A33.
Key words and phrases. Local fractional calculus, Volterra and Abel’s integral equations, Yang-Laplace transform.
1

2 FUAT USTA, H¨
USEYIN BUDAK, AND MEHMET ZEKI SARIKAYA
If aα, bαand cαbelongs the set Rαof real line numbers, then
(i) aα+bαand aαbαbelongs the set Rα;
(ii) aα+bα=bα+aα= (a+b)α= (b+a)α;
(iii) aα+ (bα+cα)=(a+b)α+cα;
(iv) aαbα=bαaα= (ab)α= (ba)α;
(v) aα(bαcα) = (aαbα)cα;
(vi) aα(bα+cα) = aαbα+aαcα;
(vii) aα+ 0α= 0α+aα=aαand aα1α= 1αaα=aα.
The definition of the local fractional derivative and local fractional integral can be given as follows.
Definition 1. [11] A non-differentiable function f:R→Rα, x →f(x)is called to be local fractional
continuous at x0, if for any ε > 0,there exists δ > 0,such that
|f(x)−f(x0)|< εα
holds for |x−x0|< δ, where ε, δ ∈R. If f(x)is local continuous on the interval (a, b),we denote
f(x)∈Cα(a, b).
Definition 2. [11] The local fractional derivative of f(x)of order αat x=x0is defined by
f(α)(x0) = dαf(x)
dxαx=x0
= lim
x→x0
∆α(f(x)−f(x0))
(x−x0)α,
where ∆α(f(x)−f(x0)) e=Γ(α+ 1) (f(x)−f(x0)) .
If there exists f(k+1)α(x) =
k+1 times
z }| {
Dα
x...Dα
xf(x) for any x∈I⊆R, then we denoted f∈D(k+1)α(I),
where k= 0,1,2, ...
Definition 3. [11] Let f(x)∈Cα[a, b].Then the local fractional integral is defined by,
aIα
bf(x) = 1
Γ(α+ 1)
b
Z
a
f(t)(dt)α=1
Γ(α+ 1) lim
∆t→0
N−1
X
j=0
f(tj)(∆tj)α,
with ∆tj=tj+1 −tjand ∆t= max {∆t1,∆t2, ..., ∆tN−1},where [tj, tj+1], j = 0, ..., N −1and
a=t0< t1< ... < tN−1< tN=bis partition of interval [a, b].
Here, it follows that aIα
bf(x) = 0 if a=band aIα
bf(x) = −bIα
af(x) if a < b. If for any x∈[a, b],
there exists aIα
xf(x),then we denoted by f(x)∈Iα
x[a, b].
Lemma 1. [11](Local fractional integration is anti-differentiation) Suppose that f(x) = g(α)(x)∈
Cα[a, b],then we have
aIα
bf(x) = g(b)−g(a).
Lemma 2. (Local fractional integration by parts) Suppose that f(x), g(x)∈Dα[a, b]and f(α)(x),
g(α)(x)∈Cα[a, b],then we have
aIα
bf(x)g(α)(x) = f(x)g(x)|b
a−aIα
bf(α)(x)g(x).
Lemma 3. [11] We have the following properties of local fractional calculus
i) dαxkα
dxα=Γ(1 + kα)
Γ(1 + (k−1) α)x(k−1)α;
ii) 1
Γ(α+ 1)
b
R
a
xkα(dx)α=Γ(1 + kα)
Γ(1 + (k+ 1) α)b(k+1)α−a(k+1)α, k ∈R.
3. YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL
VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS
In local fractional Volterra integral equations, at least one of the limits of local fractional integration
must be a variable. There are two kinds of local fractional Volterra integral equations based on the

YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL VOLTERRA... 3
place of unknown function. For the first kind local fractional Volterra integral equations, the unknown
function u(x) appears only inside local fractional integral sign in the form:
m(x) = 1
Γ(1 + α)Zx
0
K(x, t)u(t)(dt)α,
such that m(x) : R→Rαand K(x, t) : R×R→Rα. On the other hand, local fractional Volterra
integral equations of the second kind, the unknown function u(x) appears both inside and outside the
local fractional integral sign. The second kind local fractional Volterra integral equation is represented
by the form:
u(x) = m(x) + λ
Γ(1 + α)Zx
0
K(x, t)u(t)(dt)α.
In addition to these the local fractional Volterra integro-differential equations of the second and first
kinds arise along the same line as local fractional Volterra integral equations with one or more of
local derivatives in addition to the local fractional integral operators. The local fractional Volterra
integro-differential equations of second and first kinds are respresented by respectively
u(nα)(x) = m(x) + λ
Γ(1 + α)Zx
0
K(x, t)u(t)(dt)α,
and
m(x) = 1
Γ(1 + α)Zx
0
K1(x, t)u(t)(dt)α+1
Γ(1 + α)Zx
0
K2(x, t)u(nα)(t)(dt)α.
The Yang-Laplace transform method is a spectacular technique that can be utilized for solving local
fractional Volterra integro-differential equations.
Theorem 1. (Yang-Laplace transform) Let f:R→Rα.The Yang-Laplace transform of f(x)is given
by
Lα{f(x)}=fα
s(s) = 1
Γ(1 + α)Z∞
0
Eα(−sαxα)f(x)(dx)α,0< α ≤1,
where Eα(xα) = P∞
k=0 xαk
Γ(1+kα)and xα, sα∈Rα.
The Yang-Laplace transform fα
s(s) may fail to exist. If f(x) has infinite discontinuities or if it grows
up swiftly, then fα
s(s) does not exist. Furthermore, a significant necessary condition for the existence
of the Yang-Laplace transform fα
s(s) is that fα
s(s) have to vanish as sapproaches at infinity.
Theorem 2. (Inverse Yang-Laplace transform) The inverse Yang-Laplace transform of fα
s(s)is given
by
L−1
α{fα
s(s)}=f(x) = 1
(2π)αZβ+i∞
β−i∞
Eα(sαxα)fα
s(s)(ds)α,0< α ≤1,
where Eα(xα) = P∞
k=0 xαk
Γ(1+kα),sα∈βα+iα∞αand Re(sα) = βα>0α.
Another significant theorem that will be used in solving local fractional Volterra integro-differential
equations is the convolution theorem.
Definition 4 (Convolution).The convolution of two functions is the function f∗gdefined by
(f∗g)(x) = 1
Γ(1 + α)Zx
0
f(t)g(x−t)(dt)α
As further results, the properties of the convolution of the non-differentiable functions for convenience
lead as
(1) (f∗g)(x) = (g∗f)(x)
(2) (f∗(g+h))(x) = ((f+g)∗h)(x)
In the convolution theorem for the Yang-Laplace transform, it was remarked that if the kernel
K(x, t) : R×R→Rαof the local fractional Volterra integral equation of second kind:
u(x) = m(x) + λ
Γ(1 + α)Zx
0
K(x, t)u(t)(dt)α,

4 FUAT USTA, H¨
USEYIN BUDAK, AND MEHMET ZEKI SARIKAYA
depends on the difference x−t, then it is called a difference kernel. Then the local fractional Volterra
integral equation of second kind owing difference kernel can be expressed as
u(x) = m(x) + λ
Γ(1 + α)Zx
0
K(x−t)u(t)(dt)α
3.1. Yang-Laplace transform method for local fractional Volterra Integral Equations of the
First Kind and Second Kind. Let consider two mappings f(x) and g(x) that possess the necessary
conditions of the existence of Yang-Laplace transform for each. Let the Yang-Laplace transforms for
the mappings f(x) and g(x) be given by Lα{f(x)}=fα
s(s) and Lα{g(x)}=gα
s(s). The Yang-Laplace
convolution product of these two mappings is described by
Lα{f(x)∗g(x)}=Lα1
Γ(1 + α)Zx
0
f(t)g(x−t)(dt)α=fα
s(s)gα
s(s)
Starting from this point of view, we will focus on special Volterra integral equations where the kernel
is a difference kernel. Note that we will perform the Yang-Laplace transform method and the inverse
Yang-Laplace transform method using the elementary transforms given in [11]. By taking Yang-Laplace
transform of both sides of local fractional Volterra integral equations we deduce that
(3.1) U(s) = M(s) + λK(s)U(s)
where U(s) = Lα{u(x)},M(s) = Lα{m(x)}and K(s) = Lα{K(x, t)}. Solving (3.1) for U(s) gives
(3.2) U(s) = M(s)
1−λK(s),1−λK(s)6= 0.
The solution u(x) is obtained by taking the inverse Yang-Laplace transform of both sides of (3.2) where
we find
(3.3) u(x) = L−1
αM(s)
1−λK(s).
Note that the right side of (3.3) can be solved by using the Yang-Laplace transform of ordinary
functions given in [11].
Remark 1. If the unknown function u(x)appears only under the integral sign of Volterra equation,
the local fractional integral equation is named a first kind local fractional Volterra integral equation.
Under this circumstances the Yang-Laplace transform technique solution of first kind local fractional
Volterra integral equation given as
u(x) = L−1
αM(s)
K(s).
where K(s)6= 0.
3.2. Yang-Laplace transform method for local fractional Volterra Integro-Differential Equa-
tions of the Second Kind. The local fractional Volterra integro-differential equation depend on the
difference kernel can be expressed as
(3.4) u(nα)(x) = m(x) + λ
Γ(1 + α)Zx
0
K(x−t)u(t)(dt)α.
In order to solve local fractional Volterra integro-differential equations by using the Yang-Laplace
transform method, it is key feature to use the Yang-Laplace transforms of the local fractional derivatives
of u(x). We can easily deduce that
Lα{u(nα)(x)}=snαLα{u(x)} − s(n−1)αu(0) −s(n−2)αu(α)(0) − · · · − u((k−1)α)(0).
The Yang-Laplace transform method can be performed likewise by following the same steps used before
in previous subsection. Namely we first perform the Yang-Laplace transform to both sides of (3.4),
use the proper Yang-Laplace transform for the local fractional derivative of u(x), and then solve for
U(s). After then we use the inverse Yang-Laplace transform of both sides of the resultant equation to
get the solution u(x) of the equation.

YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL VOLTERRA... 5
3.3. Yang-Laplace transform method for local fractional Volterra Integro-Differential Equa-
tions of the First Kind. The standard format of the local fractional Volterra integro-differential
equation of the first kind depend on difference kernel is given by
(3.5)
1
Γ(1 + α)Zx
0
K1(x−t)u(t)(dt)α+1
Γ(1 + α)Zx
0
K2(x−t)u(nα)(t)(dt)α=m(x), K2(x−t)6= 0,
where initial conditions are prescribed. Taking the Yang-Laplace transform of both sides of (3.5) gives
Lα{K1(x−t)∗u(t)}+Lα{K2(x−t)∗u(nα)(t)}=Lα{m(x)},
so that
K1(s)U(s) + K2(s)(snαLα{u(x)} − s(n−1)αu(0) −s(n−2)αu(α)(0) − · · · − u((k−1)α)(0)) = M(s)
where U(s) = Lα{u(x)},M(s) = Lα{m(x)},K1(s) = Lα{K(x, t)}and K2(s) = Lα{K(x, t)}. Using
the provided initial conditions and solving for U(s) we deduce that
(3.6) U(s) = M(s) + K2(s)(s(n−1)αu(0) −s(n−2)αu(α)(0) − · ·· − u((k−1)α)(0))
K1(s) + snαK2(s)
on the condition
K1(s) + snαK2(s)6= 0.
By taking the inverse Laplace transform of both sides of (3.6), the exact solution is easily obtained.
4. YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL ABEL’S
INTEGRAL EQUATIONS
Abel in 1823 examined the motion of a small particle that slides down along a smooth unknown
curve, in a vertical plane, under the affect of the gravity. The particle takes the time m(x) to move
from the highest point of vertical height xto the lowest point 0 on the curve. The Abel’s problem is
derived to find the equation of that curve.
4.1. Yang-Laplace transform method for local fractional Abel’s Integral Equations. Local
fractional version of Abel”s integral equations can be represented as
(4.1) m(x) = 1
Γ(1 + α)Zx
0
1
√xα−tαu(t)(dt)α
where m(x) is a prespecified data function, and u(x) is the solution that will be determined. It is to be
noted that Abel’s local fractional integral equation (4.1) is also named local fractional Volterra integral
equation of the first kind. Moreover the kernel K(x, t) in Abel’s local fractional integral equation (4.1)
is
K(x, t) = 1
√xα−tα
where
K(x, t)→ ∞ as t→x.
It is interesting to point out that although Abel’s local fractional integral equation is a local fractional
Volterra integral equation of the first kind, the solution technique of local fractional Volterra integral
equation of the first kind except Yang-Laplace transform method are not practicable here. For instance
the series solution cannot be performed in this case particularly if u(x) is not analytic. Furthermore,
converting Abel’s local fractional integral equation to a second kind local fractional Volterra equation
is not existing since we cannot use Leibnitz rule due to the singularity behaviour of the kernel in (4.1).
In order to solve local fractional Abel’s integral equations we need to take the Yang-Laplace trans-
form of both sides of (4.1) leads to
Lα{m(x)}=Lα{u(x)}Lα{x−α/2}
or equivalently
M(s) = U(s)Γ(1 −α
2)
sα
2

6 FUAT USTA, H¨
USEYIN BUDAK, AND MEHMET ZEKI SARIKAYA
that gives
(4.2) U(s) = M(s)sα
2
Γ(1 −α
2)
where Γ is the gamma function. The last equation (4.2) can be rewritten as
U(s) = sα
[Γ(1 −α
2)]2
Γ(1 −α
2)
sα
2M(s)
which can be rewritten by
(4.3) Lα{u(x)}=sα
[Γ(1 −α
2)]2Lα{v(x)}
where
v(x) = 1
Γ(1 + α)Zx
0
(xα−tα)−1/2m(t)(dt)α.
Using the fact that
Lα{v(α)(x)}=sαLα{v(x)} − y(0),
into (4.3) we deduce that
(4.4) Lα{u(x)}=1
[Γ(1 −α
2)]2Lα{vα(x)}.
Applying the inverse Yang-Laplace transform to both sides of (4.4) gives the formula
(4.5) u(x) = 1
[Γ(1 −α
2)]2
dα
dxαZx
0
1
√xα−tαm(t)(dt)α
that will be used for the identification of the solution u(x). Notice that the formula (4.5) will be
used for solving Abel’s local fractional integral equation, and it is not necessary to use Yang-Laplace
transform method for each problem. Abel’s problem given by (4.1) can be solved straight-forwardly
by using the formula (4.5) where the unknown function u(x) has been replaced by the given function
m(x).
4.2. Yang-Laplace transform method for the Generalized local fractional Abel’s Integral
Equations. The generalized local fractional Abel’s integral equations are the singular local fractional
integral equation given as
(4.6) m(x) = 1
Γ(1 + α)Zx
0
1
(xα−tα)θu(t)(dt)α,0< θ < 1,
where θare known constants such that 0 < θ < 1, m(x) is a predetermined data function, and u(x)
is the solution that will be determined. The Abel’s problem discussed previous subsection is a special
case of the generalized equation where θ=1
2.
To construct a formula that will be used for solving the generalized Abel’s local fractional integral
equation (4.6), we will perform the Yang-Laplace transform technique in the same way to the approach
followed before. By taking the Yang-Laplace transforms of both sides of (4.6) leads to
Lα{m(x)}=Lα{u(x)}Lα{x−αθ}
or equivalently
M(s) = U(s)Γ(1 −αθ)
s(1−θ)α
that gives
(4.7) U(s) = M(s)s(1−θ)α
Γ(1 −αθ)
where Γ is the gamma function. The last equation (4.7) can be rewritten as
U(s) = sα
Γ(1 −αθ)Γ(1 −α+αθ)
Γ(1 −α+αθ)
sαθ M(s)

YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL VOLTERRA... 7
which can be rewritten by
(4.8) Lα{u(x)}=sα
Γ(1 −αθ)Γ(1 −α+αθ)Lα{v(x)}
where
v(x) = 1
Γ(1 + α)Zx
0
1
(xα−tα)1−θm(t)(dt)α.
Using the fact that
Lα{v(α)(x)}=sαLα{v(x)} − y(0),
into (4.8) we deduce that
(4.9) Lα{u(x)}=1
Γ(1 −αθ)Γ(1 −α+αθ)Lα{vα(x)}.
Performing the inverse Yang-Laplace transform to both sides of (4.9) gives the formula
(4.10) u(x) = 1
Γ(1 −αθ)Γ(1 −α+αθ)
dα
dxαZx
0
1
(xα−tα)1−θm(t)(dt)α,0< θ < 1.
Notice that the exponent of the kernel of the generalized Abel’s local fractional integral equation is
−θ, but the exponent of the kernel of the formulae (4.10) is (θ−1). The unknown function in (4.6)
has been replaced by m(t) in (4.10).
4.3. Yang-Laplace transform method for the Main Generalized local fractional Abel’s
Integral Equations. It is helpful to present a further generalization to local fractional Abel’s integral
equation by considering a generalized singular kernel instead of K(x, t) = 1
√xα−tα. The generalized
version of kernel will be of the form
K(x, t) = 1
[n(x)−n(t)]θ,0< θ < 1,
where n(t) : R→Rα. The main generalized local fractional Abel’s integral equation is given by
(4.11) m(x) = 1
Γ(1 + α)Zx
0
1
[n(x)−n(t)]θu(t)(dt)α,0< θ < 1,
where n(t) : R→Rαis strictly monotonically increasing and differentiable function in some interval
0< t < b, and dα
dtαn(t) for every tin the interval. The solution u(x) of (4.11) is given by
u(x) = 1
Γ(1 + α)B(θ, 1−θ)
dα
dxαZx
0
1
[n(x)−n(t)](1−θ)m(t)dα
dtαn(t)(dt)α,0< θ < 1.
where
B(x, y) = 1
Γ(1 + α)Z1
0
t(x−1)α(1 −t)(y−1)α(dt)α.
To prove this formula, we consider the integral
Zx
0
m(y)dα
dtαn(y)
[n(x)−n(y)](1−θ)(dy)α
and substitute for m(x) from (4.11) to obtain
1
Γ(1 + α)Zx
0Zy
0
u(t)dα
dtαn(y)
[n(y)−n(t)]θ[n(x)−n(y)](1−θ)(dt)α(dy)α
where by changing the order of integration we find
1
Γ(1 + α)Zx
0
u(t)Zx
t
dα
dtαn(y)
[n(y)−n(t)]θ[n(x)−n(y)](1−θ)(dy)α(dt)α.
We can prove that
1
Γ(1 + α)Zx
t
dα
dtαn(y)
[n(y)−n(t)]θ[n(x)−n(y)](1−θ)(dy)α= Γ(1 + α)B(θ, 1−θ).

8 FUAT USTA, H¨
USEYIN BUDAK, AND MEHMET ZEKI SARIKAYA
This means that
(4.12) Zx
0
m(y)dα
dtαn(y)
[n(x)−n(y)](1−θ)(dy)α=B(θ, 1−θ)Zx
0
u(t)(dt)α.
Differentiating both sides of (4.12) gives
u(x) = 1
Γ(1 + α)B(θ, 1−θ)
dα
dxαZx
0
1
[n(x)−n(t)](1−θ)m(t)dα
dtαn(t)(dt)α,0< θ < 1.
5. EXAMPLES
The Yang-Laplace transform method for solving local fractional Volterra and Abel’s integral equa-
tions of the first and second kinds will be exemplified by running the following examples.
Example 1. Considering the following local fractional Volterra integral equation of second kind
u(x) = 1 + 1
Γ(1 + α)Zx
0
u(t)(dt)α.
The exact solution of Example 1 is the form
u(x) = Eα(xα).
Example 2. Considering the following local fractional Volterra integro-differential equation of second
kind
u(2α)(x) = −1α−xα
Γ(1 + α)+1
[Γ(1 + α)]2Zx
0
(xα−tα)u(t)(dt)α,
with the initial condition
u(0) = 1αand u(α)(0) = 1α.
The exact solution of Example 2 is the form
u(x) = sinα(xα) + cosα(xα).
Example 3. Considering the following local fractional Volterra integro-differential equation of first
kind
1
Γ(1 + α)Zx
0
cosα(xα−tα)u(t)(dt)α+1
Γ(1 + α)Zx
0
sinα(xα−tα)u(3α)(t)(dt)α= 1α+sinα(xα)+cosα(xα)
with the initial condition
u(0) = 1α, u(α)(0) = 1αand u(2α)(0) = −1α.
The exact solution of Example 3 is the form
u(x) = xα
Γ(1 + α)+ cosα(xα).
Example 4. Considering the following local fractional Abel’s integral equation
[Γ(1 −α
2)]2=Zx
0
1
√xα−tαu(t)(dt)α
The exact solution of Example 4 is the form
u(x) = 1
√xα.
6. Concluding Remarks
The main goal of the current study was to determine the solution of local fractional Volterra and
Abel’s integro-differential equations using the Yang-Laplace transform. The present study confirms
previous findings in case of α= 1 and contributes additional evidence. In order to validate our scheme
we presented some examples. It is recommended that further research be undertaken in the following
areas: system of local fractional integral equations and nonlinear local fractional integral equations.

YANG-LAPLACE TRANSFORM METHOD FOR LOCAL FRACTIONAL VOLTERRA... 9
References
[1] A. H. Bhrawy and D. Baleanu, A spectral Legendre-GaussLOBatto collocation method for a space-fractional ad-
vection diffusion equations with variable coefficients, Reports on Mathematical Physics, vol. 72, no. 2, pp. 219233,
2013.
[2] A. H. Bhrawy and M. A. Alghamdi, A shifted Jacobi-GaussLobatto collocation method for solving nonlinear frac-
tional Langevin equation involving two fractional orders in different intervals, Boundary Value Problems, vol. 2012,
article 62, 2012.
[3] J. Hristov, An exercise with the Hes variation iteration method to a fractional Bernoulli equation arising in a
transient conduction with a non-linear boundary heat flux, International Review of Chemical Engineering, vol. 4,
no. 5, pp. 489497, 2012.
[4] J. Hristov, Approximate solutions to fractional subdiffusion equations, European Physical Journal: Special Topics,
vol. 193, no. 1, pp. 229243, 2011.
[5] F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang, and Q. Liu, Numerical methods for solving the multi-term
time fractional wave-diffusion equation, Fractional Calculus and Applied Analysis, vol. 16, no. 1, pp. 925, 2013.
[6] Ma, X.J., H.M. Srivastava, D. Baleanu and X.J. Yang, A new Neumann series method for solving a family
of local fractional Fredholm and Volterra integral equations. Math. Problems Eng., 2013: 325121-325126. DOI:
10.1155/2013/325121
[7] S. Momani and Z. Odibat, Analytical solution of a timefractional Navier-Stokes equation by Adomian decomposition
method, Applied Mathematics and Computation, vol. 177, no. 2, pp. 488494, 2006.
[8] A. A. Neamah, Local fractional Variational iteration method for solving Volterra integro-differential equations
within local fractional operators, Journal of Mathematics and Statistics 10 (3): 401-407, 2014.
[9] A.M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press,
Beijing, and Springer-Verlag, Berlin, 2011.
[10] J. Yang, D. Baleanu and X. J. Yang, Analysis of fractal wave equations by local fractional Fourier series method,
Adv. Math. Phys. , 2013 (2013), Article ID 632309.
[11] X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, 2012.
[12] X. J. Yang, Local fractional integral equations and their applications, Advances in Computer Science and its Ap-
plications (ACSA) 1(4), 2012.
[13] X. J. Yang, Generalized local fractional Taylor’s formula with local fractional derivative, Journal of Expert Systems,
1(1) (2012) 26-30.
[14] X. J. Yang, Local fractional Fourier analysis, Advances in Mechanical Engineering and its Applications 1(1), 2012
12-16.
[15] X. J. Yang, Picards approximation method for solving a class of local fractional Volterra integral equations , Ad-
vances in Intelligent Transportation Systems, Vol. 1, No. 3, 2012
[16] X. J. Yang, Y. Zhang, A new successive approximation to nonhomogeneous local fractional Volterra equation,
Advances in Information Technology and Management,1(3) 2012 138-141
[17] X. J. Yang, Y. Zhang, A New Adomian Decomposition Procedure scheme for solving local fractional Volterra integral
equation, Advances in Information Technology and Management (AITM) Vol. 1, No. 4, 2012.
[18] J. Yang, D. Baleanu and X. J. Yang, A Local Fractional Variational Iteration Method for Laplace Equation within
Local Fractional Operators, Abstract and Applied Analysis, 2013 (2013), Article ID 202650
[19] Z. Zhao and C. Li, Fractional difference/finite element approximations for the time-space fractional telegraph
equation, Applied Mathematics and Computation, vol. 219, no. 6, pp. 29752988, 2012.
Department of Mathematics, Faculty of Science and Arts, D¨
uzce University, D¨
uzce-TURKEY
E-mail address:fuatusta@duzce.edu.tr
Department of Mathematics, Faculty of Science and Arts, D¨
uzce University, D¨
uzce-TURKEY
E-mail address:hsyn.budak@gmail.com
Department of Mathematics, Faculty of Science and Arts, D¨
uzce University, D¨
uzce-TURKEY
E-mail address:sarikayamz@gmail.com