International Journal of Analysis and Applications

ISSN 2291-8639

Volume 13, Number 1 (2017), 82-92

http://www.etamaths.com

ON THE BANACH SPACE TECHNIQUES IN THE EXISTENCE AND

UNIQUENESS OF THE FUZZY FRACTIONAL KLEIN-GORDON EQUATION’S

SOLUTION

A. EBADIAN1, M. SHAMS YOUSEFI2, F. FARAHROOZ1,∗ AND M. NAJAND FOUMANI2

Abstract. In this paper, we study the existence and uniqueness of the solution of all fuzzy fractional

differential equations, which are equivalent to the fuzzy integral equation. We use the Banach space

techniques in this study. Also we will show that the fuzzy fractional Klein-Gordon equation (FFKGE)

is equivalent to a fuzzy integral equation. We use parametric form of FFKGE with respect to definition

and give new homotopy analysis method to obtain the approximate solution of this equation.

1. Introduction

In many cases of the modeling of real world phenomena, fuzzy initial value problems appear naturally,

because information about the behavior of a dynamical system is uncertain. In order to obtain a more

adequate model, we have to take into account these uncertainties. On the other hand, fractional calculus

found many applications in various fields of physical sciences such as viscoelasticity, diffusion, control,

relaxation, processes and so on [1].

The Klein-Gordon equation, which is denoted by KGE in this paper, is nowadays regarded as the

relativistic form of the schrödinger equation. It affords appropriate description for spin zero particles.

Since the solution of the KGE is often a complicated problem, use of pure mathematical methods is

required.

Ebaid, applied Exp-function method for solving KGE [2]. Raicher et.al used a novel solution to

the KGE in the presence of a strong rotating electric field [3]. But discussion on the fuzzy fractional

Klein-Gordon equation (FFKGE) has not been done.

We consider the FFKGE with boundary conditions as follows

∂2γŨ(x,t)

∂t2γ
=
∂2Ũ(x,t)

∂x2
+ Ũ(x,t) 0 < x < 1 , 0 < t < 1 , 0 < γ ≤ 1

where γ is a parameter describing the order of fractional time derivative and , Ũ(x, 0) = K̃(1 + sin x),K̃ =

(0.25β,−0.25β), 0 ≤ β ≤ 1 and Ũt(x, 0) = 0, 0 < x < 1,
where Ũ(x,t) : (0, 1) × [0, 1) −→ RF is fuzzy number-valued function and RF is the set of all fuzzy
numbers.

In this paper we give a new homotopy analysis transform method for solving FFKGE. We use the

parametric form of the above equation and find the approximate solution of this equation. The existence

and uniqueness of the solution and convergence of the proposed method are proved in details. For this

purpose we show that the FFKGE is equivalent to fuzzy integral equation. The concept of conformable

fuzzy fractional derivative will define in this paper. Also, we define the fuzzy Banach space. Since the

fixed point theorems in Banach spaces are powerful tools to prove existence and uniqueness of solution

for integral equations, so in this study, we use fixed point theorem and introduce a contraction operator

on a suitable Banach space.

The paper is organized as follows: in Sect. 2.2 we present some concepts and results about the fuzzy

number. We explain the fractional transform and it is applied for FFKGE, then the equivalency to the

Received 13th August, 2016; accepted 7th October, 2016; published 3rd January, 2017.

2010 Mathematics Subject Classification. 34A12; 26A33.
Key words and phrases. fractional calculus; convergence; existence; homotopy analysis method; uniqueness; Klein-

Gordon equation.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

82



ON THE BANACH SPACE TECHNIQUES 83

fuzzy integral equation is also proved in Sect. 3. The existence and uniqueness of the solution for fuzzy

integral equation is discussed in Sect. 4, where we define the CF ([0, 1]) and its properties and use the

functional analysis methods. In Sect. 5 the Homotopy analysis transform method is applied to solve this

equation.

2. Preliminaries

We now recall some definitions and symbols needed through the paper. We follow [4] in definitions

and notations.

Definition 2.1. A fuzzy number is a function u : R → [0, 1] satisfying the following properties:
a. u is upper semicontinuous on R,
b. u(x) = 0 outside of some interval [c,d],

c. there are the real numbers a and b with c ≤ a ≤ b ≤ d, such that u is increasing on [c,a],
decreasing on [b,d] and u(x) = 1 for each x ∈ [a,b],

d. u is fuzzy convex set (that is u(λx + (1 −λ)y) ≥ min{u(x),u(y)}, ∀x,y ∈ R,λ ∈ [0, 1])

The set of all fuzzy numbers is denoted by RF .
In the following we introduce a concept that will be very efficient and useful to use and identification of

fuzzy numbers.

Definition 2.2. For any u ∈ RF the α−cut set of u is denoted by [u]α and defined by [u]α = {x ∈
R | u(x) ≥ α}, where 0 ≤ α ≤ 1. The notation,

[u]α = [uα,uα]; α ∈ [0, 1]

refers to the lower and upper branches on u, in other words

uα = min{x | x ∈ uα}, uα = max{x | x ∈ uα}

An arbitrary fuzzy number u is represented, in parametric form, by an ordered pair of functions

u = (u,u), which define the end points of the α−cuts, satisfying the three conditions:
a. u is a bounded non-decreasing left continuous function on (0, 1], and right continuous at 0,

b. u is a bounded non-increasing left continuous function on (0, 1], and right continuous at 0,

c. u(r) ≤ u(r), 0 ≤ r ≤ 1.
For arbitrary u = (u,u), v = (v, v̄) and k ≥ 0, addition (u + v) and multiplication by k as (u + v)(r) =
u(r) + v(r), (u + v)(r) = u(r) + v̄(r), ku(r) = ku(r), ku(r) = ku(r),k ≥ 0, and ku(r) = kū(r),ku(r) =
ku(r), k < 0 are defined.

It is well-known that the addition and multiplication operations of real numbers can be extended to RF .
In other words, for any u,v ∈ RF and λ ∈ R, we define uniquely the sum u⊕v and the product λ�u by

[u⊕v]α = [u]α ⊕ [v]α, [λ�u]α = λ[u]α, ∀α ∈ [0, 1].

	 is the Hukuhara difference (H-difference), it means that w 	 v = u if and only if u ⊕ v = w for all
u,v,w ∈ RF .

Definition 2.3. For arbitrary fuzzy number u = (u(r),u(r)), v = (v(r),v(r)) the Hausdorff distance

between these fuzzy numbers given by D : RF ×RF → R+ ∪{0},

D(u,v) = sup
r∈[0,1]

max{|u(r) −v(r)|, |u(r) − v̄(r)|},

where D is a metric on RF and has the following properties (see [1]).

a. D(u⊕w,v ⊕w) = D(u,v),∀u,v,w ∈ RF ,
b. D(k �u,k �v) = |k|D(u,v),∀k ∈ R,u,v ∈ RF ,
c. D(u⊕v,w ⊕e) ≤ D(u,w) + D(v,e),∀u,v,w,e ∈ RF ,
d. (RF ,D) is a complete metric space.



84 A. EBADIAN, M. SHAMS YOUSEFI, F. FARAHROOZ AND M. NAJAND FOUMANI

Definition 2.4. The function f : T −→ RF is called a fuzzy function, and the α−cut set of f is
represented by

f(t; α) = [f(t; α),f(t; α)]; ∀α ∈ [0, 1],

where f(t; α) = f(t)
α
, f(t; α) = f(t)

α
.

A fuzzy function may have fuzzy domain and fuzzy range. So the function f : RF −→ RF is also a
fuzzy function.

Definition 2.5. Let f : R → RF be a fuzzy function. If for an arbitrary fixed number t0 ∈ R and ε > 0,
exists δ > 0 such that

|t− t0| < δ =⇒ D(f(t),f(t0)) < ε t ∈ RF ,
then f is said to be continuous at t0.

Definition 2.6. The fuzzy function f : R −→ RF is called to be fuzzy bounded if there exists M > 0 such
that ‖ f ‖F.u:=sup D(f(u), 0̂) ≤ M, (u ∈ R).

Proposition 2.1. Let f : [a,b] ⊆ R −→ RF be a fuzzy continuous function. Then it is fuzzy bounded.

In the following we consider the concept of integral of a fuzzy function. The integration on the α-cut

of fuzzy function is also defined.

Definition 2.7. let f : [a,b] −→ RF be a fuzzy function. For each partition p = {x1,x2, · · · ,xm} of [a,b]
and for arbitrary xi−1 ≤ ξi ≤ xi, 2 ≤ i ≤ m, let

Rp =

m∑
i=2

f(ξi)(xi −xi−1).

The define integral of f(x) over [a,b] is ,∫ b
a

f(x; y) = lim Rp, max |xi −xi−1| −→ 0

provided that this limit exists in metric D. If the function f is continuous in the metric D, its definite

integral exists [5].

Furthermore: ∫ b
a

f(x; α) =

∫ b
a

f(x; α),

and ∫ b
a

f(x; α) =

∫ b
a

f̄(x; α)

More details about the properties of the fuzzy integral are given in [5].

Now, we want to introduce a new definition of fuzzy fractional derivative as [8]:

Definition 2.8. Let f : [a,b] −→ RF . The fuzzy γ-fractional integral of fuzzy-valued function f is defined
as follows:

(Iγf)(x) =

∫ x
a

f(t)

t1−γ
dt, x > a , 0 < γ < 1.

let us consider the α−cut representation of fuzzy-valued function f is f(x; α) = [f(x; α), f̄(x; α)], for
0 ≤ α ≤ 1, then we indicate the fuzzy γ − fractional integral of fuzzy-valued function f based on its
lower and upper functions as follows:

Theorem 2.1. Let f : [a,b] −→ RF . The fuzzy γ−fractional integral of fuzzy-valued function f can be
expressed as follows:

(Iγf)(x; α) = [(Iγf)(x; α), (Iγf̄)(x; α)], 0 < α < 1

where

(Iγf)(x; α) =

∫ x
a

f(t; α)

t1−γ
dt, (Iγf̄)(x; α) =

∫ x
a

f̄(t; α)

t1−γ
dt.



ON THE BANACH SPACE TECHNIQUES 85

Theorem 2.2. For γ ∈ [0, 1] and f : [a,b] −→ RF

D
γ
t f(t) = lim

ε→0

f(t + εt1−γ) 	f(t)
ε

.

For t > 0, γ ∈ (0, 1); Dγt f(t) is called the conformable fuzzy fractional derivative of f of order γ [9, 10].

Using this kind of fractional derivative and some useful formulas can convert differential equations into

integer-order differential equations.

Some properties for the suggested conformable fuzzy fractional derivative given in [8] are as follows:

D
γ
t (t

η) = ηtη−γ,η ∈ R, (2.1)

D
γ
t (f(t)g(t)) = g(t)D

γ
t f(t) ⊕f(t)D

γ
t g(t). (2.2)

D
γ
t f[g(t)] = f

′
g[g(t)]D

γ
t g(t) = D

γ
gf[g(t)](g

′(t))γ. (2.3)

3. The fractional transform

In this section, we reduce FFKGE to an ordinary differential equation, then we show that this equation

is equivalent to fuzzy integral equation. We consider the FFKGE with boundary conditions as follows

∂2γŨ(x,t)

∂t2γ
=
∂2Ũ(x,t)

∂x2
+ Ũ(x,t) 0 < x < 1 , 0 < t < 1 , 0 < γ ≤ 1. (3.1)

Now, we introduce the following transformations :

Ū(x,t) = Ū(ξ), ξ = ax +
btγ

γ
a,b > 0, a +

b

γ
< 1

So, we can say that 0< ξ <1 by using (1) and (3) and by substituting into equation (3.1) it is derived

that

b2U
′′
−a2U

′′
−U = 0. (3.2)

Now, we show that this fuzzy equation is equivalent to fuzzy integral equation as form:

U
′′
(ξ) = f̄(ξ) =⇒ U

′
(ξ) = U

′
(0) +

∫ ξ
0

f̄(z)dz

=⇒ U(ξ) = U(0) + U
′
(0)ξ +

∫ ξ
0

∫ ξ
0

f̄(z)dzdξ,

on the other hand, ∫ ξ
0

· · ·
∫ ξ

0

f(ξ)(dξ)n =
1

(n− 1)!

∫ ξ
0

(ξ −z)n−1f(z)dz,

therefore,

U(ξ) = U(0) + U
′
(0)ξ +

∫ ξ
0

(ξ −z)f̄(z)dz.

By substituting into equation (8) we have

(b2 −a2)f̄(ξ) −U(0) −U
′
(0)ξ −

∫ ξ
0

(ξ −z)f̄(z)dz = 0,

f̄(ξ) =

[
−(U(0) + U

′
(0)ξ)

b2 −a2

]
︸ ︷︷ ︸

ḡ(ξ)

+

∫ ξ
0

(ξ −z)
b2 −a2︸ ︷︷ ︸
K(ξ,z)

f̄(z)dz = 0,

f̄(ξ) = ḡ(ξ) +

∫ ξ
0

K(ξ,z)f̄(z)dz.

similarly,

f(ξ) = g(ξ) +

∫ ξ
0

K(ξ,z)f(z)dz.



86 A. EBADIAN, M. SHAMS YOUSEFI, F. FARAHROOZ AND M. NAJAND FOUMANI

4. Existence And convergence Analysis

In this section, we prove the existence and uniqueness of the solution and convergence of the method

by using the following assumptions. We consider fuzzy integral equation as follow:

f(ξ) = g(ξ) +

∫ ξ
0

K(ξ,z)f(z)dz,

where k is an arbitrary positive kernel on [0, 1] × [0,ξ] and functions f,g : [0, 1] −→ RF are continuous
fuzzy number-valued functions. We assume that K is continuous and therefore it is uniformly bounded

so there exists M1 > 0 such that

|K(ξ,z)| ≤ M1 0 ≤ ξ ≤ 1, 0 ≤ z ≤ ξ.

Now consider the set,

CF ([0, 1]) =

{
f : [0, 1] −→ RF ; f is continuous

}
,

which is the space of fuzzy continuous function because for f,g ∈ CF ([0, 1]) and α ∈ R,αf + g is
continuous. Regarding to Def. 2.6 we define the fuzzy uniform norm as form

‖f‖F.u := sup
ξ∈[0,1]

D(f(ξ), 0̂).

In the next theorem we show that CF ([0, 1]) is a Banach space.

Theorem 4.1. (CF ([0, 1]),‖.‖F.u) is a Banach space.

Proof. Let {fn}∞n=1 be a cauchy sequence in CF ([0, 1]). Then for each ε > 0 there exists M ∈ N such
that D(fn(ξ),fm(ξ)) < ε for all n,m ≥ M, and for all ξ ∈ [0, 1].
That is,

sup D(fn(ξ),fm(ξ)) =‖ fn −fm ‖F.u< ε for all n,m ≥ M.

This implies that for each ξ ∈ [0, 1], {fn(ξ)} is a cauchy sequence in the complete metric space RF . So
there exists a function f such that fn(ξ) −→ f(ξ) for all ξ ∈ [0, 1]. It means that the pointwise limit
function f(ξ) = limn→∞fn(ξ) exists. At first, we want to prove that {fn} also converges uniformly to f,
that is ‖fn −f‖F.u −→ 0 (n →∞).
In other words for each ε > 0 we need to find M such that ‖fn −f‖F.u ≤ ε for n > M.
For this, let ε > 0 and then fix M such that ‖fn −fm‖F.u < ε2 for all n,m ≥ M. We can do this since
{fn} is a cauchy sequence. Using the triangle inequality, we have

‖fn −f‖F.u ≤‖fn −fM‖F.u + ‖f −fM‖F.u.

As we know that for n ≥ M, we have ‖fn −fM‖F.u < ε2 . Therefore

‖f −fM‖ = lim
n→∞

‖fn −fM‖F.u <
ε

2
.

So ‖fn −f‖F.u ≤ ε2 +
ε
2

= ε for n > M which means ‖fn −f‖F.u −→ 0 (n →∞).
Now, we have to show that f is continuous.

So let ε > 0 and ξ1 ∈ [0, 1], we want to find δ > 0 such that for an arbitrary fixed number ξ2;
D(f(ξ1),f(ξ2)) < ε when |ξ1 − ξ2|, with using the triangle inequality

D(f(ξ1),f(ξ2)) ≤ D(f(ξ1),fn(ξ1)) + D(fn(ξ1),fn(ξ2)) + D(fn(ξ2),f(ξ2))

for some n.

Now, we pick n such that D(f(ξ1),fn(ξ1)) <
ε
3

and D(fn(ξ2),f(ξ2)) <
ε
3
, on the other hand fn is

continuous. So, D(fn(ξ1),fn(ξ2)) <
ε
3

and we have D(f(ξ1),f(ξ2)) ≤ ε3 +
ε
3

+ ε
3

= ε.

Hence f ∈ CF ([0, 1]), so CF ([0, 1]) is a Banach space. �

Now we define the operator T as

T(f)(ξ) = g(ξ) +

∫ ξ
0

K(ξ,z)f(z) dz, ∀ξ ∈ [0, 1], ∀f ∈ CF ([0, 1]), g : [0, 1] → RF .



ON THE BANACH SPACE TECHNIQUES 87

T(f)(ξ) can be represented as form T(f)(ξ) = (T(f)(ξ),T(f)(ξ)) where,

T(f)(ξ) = g(ξ) +

∫ ξ
0

K(ξ,z)f(z) dz, T(f)(ξ) = g(ξ) +

∫ ξ
0

K(ξ,z)f(z) dz.

Sufficient conditions for the existence of a unique solution for the above integral equation will be given

in the following.

Theorem 4.2. let K = K(ξ,z) be continuous and positive for 0 ≤ ξ ≤ 1 , 0 ≤ z ≤ ξ and f,g : [0, 1] −→
RF be fuzzy continuous functions on [0, 1]. If M1 ξ < 1, than the Homotopy analysis method

f0(ξ) = g(ξ),

fm(ξ) = g(ξ) +

∫ ξ
0

K(ξ,z)fm−1(z)dz m ≥ 1.

convergence to the unique solution f.

Proof. First we show that T(CF ([0, 1])) ⊆ CF ([0, 1])). Since g is continuous on the compact set [0, 1], so
it is uniformly continuous. Therefore

∀ε1 > 0 ∃ρ1 > 0 s.t. |ξ1 − ξ2| < ρ1 =⇒ D(g(ξ1),g(ξ2)) < ε1.

It means; sup max{|g(ξ1) −g(ξ2)|, |g(ξ1) −g(ξ2)|} < ε1 0 ≤ ξ1,ξ2 ≤ 1,
consequently,

|g(ξ1) −g(ξ2)| < ε1 , |g(ξ1) −g(ξ2)| < ε1.
As mentioned above f is continuous thus f is bounded. It means

∃ M2 > 0 s.t. |f| ≤ M2,

we must show that,

∀ε > 0 ∃ ρ > 0 s.t. |ξ1 − ξ2| < ρ =⇒‖T(f)(ξ1) −T(f)(ξ2)‖F.u < ε.

Since

‖T(f)(ξ1) −T(f)(ξ2)‖F.u = sup
ξ1,ξ2∈[0,1]

max {|T(f)(ξ1) −T(f)(ξ2)|, |T(f)(ξ1) −T(f)(ξ2)|}

It is enough to show that,

|T(f)(ξ1) −T(f)(ξ2)| < ε , |T(f)(ξ1) −T(f)(ξ2)| < ε

|T(f)(ξ1) −T(f)(ξ2)| ≤ |g(ξ1) −g(ξ2)| + |
∫ ξ1

0

K(ξ1,z)f(z)dz −
∫ ξ2

0

K(ξ2,z)f(z)dz|

≤ ε1 + |
∫ ξ1

0

K(ξ1,z)f(z)dz +

∫ 0
ξ2

K(ξ2,z)f(z)dz|

≤ ε1 +
∫ ξ1

0

|K(ξ1,z)||f(z)|dz +
∫ 0
ξ2

|K(ξ2,z)||f(z)|dz

≤ ε1 +
∫ ξ1

0

M1|f(z)|dz +
∫ 0
ξ2

M1|f(z)|dz

= ε1 +

∫ ξ1
ξ2

M1|f(z)|dz

≤ ε1 + M1.M2
∫ ξ1
ξ2

dz

= ε1 + M1M2(ξ1 − ξ2).

Choosing ε1 =
ε

2
, ξ1 − ξ2 =

ξ

2M1M2
, we have,

|T(f)(ξ1) −T(f)(ξ2)| < ε.

Similarly;

|T(f)(ξ1) −T(f)(ξ2)| < ε.



88 A. EBADIAN, M. SHAMS YOUSEFI, F. FARAHROOZ AND M. NAJAND FOUMANI

So T(CF ([0, 1])) ⊆ CF ([0, 1]).
Now, we show that the operator T is a contraction. So for f,h ∈ CF ([0, 1]) and ξ ∈ [0, 1]

D(T(f)(ξ),T(h)(ξ)) ≤ D(g(ξ),g(ξ)) + D(
∫ ξ

0

K(ξ,z)f(z)dz,

∫ ξ
0

K(ξ,z)h(z)dz)

=

∫ ξ
0

|K(ξ,z)|D(f(z),h(z))dz

≤ M1
∫ ξ

0

D(f(z),h(z))dz

= M1ξD(f,h),

therefore,

D(T(f)(ξ),T(h)(ξ)) ≤ M1ξ D(f,h).

After taking supremum we have

‖T(f)(ξ) −T(h)(ξ)‖F.u ≤ M1ξ ‖f −h‖F.u,

since M1ξ < 1 the operator T is a contraction on Banach space (CF ([0, 1]),‖ . ‖F.u) consequently, the
Banach’s fixed point theorem implies that this integral equation has a unique solution f in CF ([0, 1]). �

The existence and uniqueness of solution for integral equation was proved. We conclude that the

FFKGE also has a unique solution.

Corollary 4.1. The FFKGE has a unique solution.

Proof. Combine the Sect. 3 (equivalency the FFKGE and fuzzy integral equation) and Theorem. 4.2. �

5. The homotopy analysis transform method

The application of Homotopy analysis method in linear and nonlinear problems has been devoted by

scientists and engineers. The fundamental work was done by liao and He [6]. He’s technique in particular,

eliminated some of the traditional limitations of methods and was successfully applied to solve many

problems in various, fields including fluid mechanics, heat transfer and so on [7]. This method has a

significant advantage in that it provides an analytical approximate solution to a wide range of nonlinear

problems in applied science. To illustrate the basic idea of this method to solve the FFKGE. We consider

the parametric from of this equation as follows :

∂2γU(x,t)

∂t2γ
=
∂2U(x,t)

∂x2
+ U(x,t), (5.1)

∂2γU(x,t)

∂t2γ
=
∂2U(x,t)

∂x2
+ U(x,t).

Equation (5.1) can be written as

∂U(x,t)

∂t
=
∂1−2γ∂2U(x,t)

∂t1−2γ∂x2
+
∂1−2γ

∂t1−2γ
U(x,t).

Now the methodology consists of applying the laplace transform first on both sides of above equation,

we get

L[U(x,t)] −
U(x, 0)

s
−
Ut(x, 0)

s2
=

1

s2γ
L

[
∂1−2γ

∂t1−2γ
∂2U(x,t)

∂x2

]
+

1

s2γ
L

[
∂1−2γ

∂t1−2γ
U(x,t)

]
. (5.2)

Equation (5.2) can be written as a nonlinear operator form as follow:

N[U(x,t)] = 0,

where N is nonlinear operator, U(x,t) is unknown function and x,t are independent variables, U(x, 0) is

auxiliary parameter.



ON THE BANACH SPACE TECHNIQUES 89

Using q ∈ [0, 1] as an embedding parameter, then we have

N[φ̄(x,t; q)] = L[φ̄(x,t; q)] −
U(x, 0)

s
−
Ut(x, 0)

s2
=

1

s2γ
L

[
∂1−2γ

∂t1−2γ
∂2φ̄(x,t; q)

∂x2

]
+

1

s2γ
L

[
∂1−2γ

∂t1−2γ
φ̄(x,t; q)

]
,

where φ̄(x,t; q) is the real function of x,t and q. By means of generalizing the traditional homotopy

methods construct the zero-order deformation equation.

(1 −q)L[φ̄(x,t; q) −U0(x,t)] = qhN[φ(x,t; q)], (5.3)

where h is a nonzero auxiliary parameter. When q = 0 and q = 1, it holds

φ̄(x,t; 0) = U0(x,t), φ̄(x,t; 1) = U(x,t).

Expanding φ̄i(x,t; q) in Taylor’s series with respect to q we have

φ̄(x,t; q) = U0(x,t) + Σ
∞
m=1Um(x,t)q

m,

where

Um(x,t) =
1

m!

∂mφ̄(x,t; q)

∂qm

∣∣∣∣
q=0

.

Differentiating deformation (5.2), m times with respect to q, dividing by m! and setting q = 1, we have

the mth-order deformation

L[Um(x,t) −XmUm−1(x,t)] = hRm(Um−1(x,t)),

where

Rm(Um−1(x,t)) =
1

(m− 1)!
∂m−1N[φ̄(x,t; q)]

∂qm−1

∣∣∣∣
q=0

,

and

Xm =

{1 m>1,
0 m61.

Therefore for this equation,

Rm(Um−1(x,t)) = L[Um−1(x,t)] − (1 −Xm)
K(β)(1 + sin x)

s
=

1

s2γ
L[
∂1−2γ

∂t1−2γ
∂2Um−1(x,t)

∂x2
]

+
1

s2γ
L[
∂1−2γ

∂t1−2γ
Um−1(x,t)],

U0(x,t; β) = K(β)(1 + sin x),

m = 1 =⇒ R1(U0(x,t; β) =
−K(β)
s2γ+1

,

if h = −1 then U1(x,t; β) =
K(β)t2γ

Γ(2γ + 1)
.

By using this method,

U2(x,t; β) =
K(β)t4γ

Γ(4γ + 1)
,

U3(x,t; β) =
K(β)t6γ

Γ(6γ + 1)
,

U4(x,t; β) =
K(β)t8γ

Γ(8γ + 1)
,

U5(x,t; β) =
K(β)t10γ

Γ(10γ + 1)
.

Proceeding in this manner, the rest of the components Ūn(x,t; β) for n ≥ 5 can be completely obtained.
We get the approximated solution of fuzzy fractional differential equation as follow

U(x,t; β) = [(1 + sin x) +
t2γ

Γ(2γ + 1)
+

t4γ

Γ(4γ + 1)
+

t6γ

Γ(6γ + 1)
+

t8γ

Γ(8γ + 1)
+

t10γ

Γ(10γ + 1)
]K(β),



90 A. EBADIAN, M. SHAMS YOUSEFI, F. FARAHROOZ AND M. NAJAND FOUMANI

U(x,t; β) = [(1 + sin x) +
t2γ

Γ(2γ + 1)
+

t4γ

Γ(4γ + 1)
+

t6γ

Γ(6γ + 1)
+

t8γ

Γ(8γ + 1)
+

t10γ

Γ(10γ + 1)
]K(β).

Let K(β) = 0.25β and K(β) = −0.25β for 0 ≤ β ≤ 1
so the exact solution is given by

U(x,t; β) = (1 + sin x)K(β) + Σ∞r=1
t2rγ

Γ(2rγ + 1)
K(β),

U(x,t; β) = (1 + sin x)K(β) + Σ∞r=1
t2rγ

Γ(2rγ + 1)
K(β),

and

U(x,t; β) = (U(x,t; β),U(x,t; β)).

In the following tables and figures comparison between the exact solution and the different terms of

approximation solution for U,U is given by the homotopy analysis transform method at γ = 1
8
.

If we increase the computational process, the approximation solution will be closer to the exact solution.

Table. 1.

(x,t,β) Uapprox[5] Uapprox[15] Uapprox[25] Uexact
(0.2,0.2,0.1) 0.07711367095 0.08075077885 0.08075381528 0.08075381592

(0.2,0.4,0.1) 0.1004436229 0.1133482422 0.1134055766 0.1134055435

(0.2,0.6,0.1) 0.1498635015 0.1495343794 0.1498624662 0.1498635015

(0.2,0.8,0.1) 0.1409872942 0.1914962138 0.1926482315 0.1926555602

(0.4,0.2,0.1) 0.08188239622 0.08551950412 0.08552254055 0.08552254120

(0.6,0.2,0.1) 0.08626299950 0.08990010740 0.08990314382 0.08990314448

(0.8,0.2,0.1) 0.09008083995 0.09371794785 0.09372098428 0.09372098492

(0.2,0.2,0.2) 0.1542273419 0.1615015577 0.1615076306 0.1615076318

(0.2,0.4,0.2) 0.2008872458 0.2266964845 0.2268109531 0.2268110870

(0.2,0.6,0.2) 0.2425774142 0.2990687587 0.2997249324 0.2997270030

(0.2,0.8,0.2) 0.2819745885 0.3829924276 0.3852964630 0.3853111204

(0.4,0.2,0.2) 0.1637647924 0.1710390082 0.1710450811 0.1710450824

(0.6,0.2,0.2) 0.1725259990 0.1798002148 0.1798062876 0.1798062890

(0.8,0.2,0.2) 0.1801616799 0.1874358957 0.1874419686 0.1874419698

Table. 2.

(x,t,β) Uapprox[5] Uapprox[15] Uapprox[25] Uexact
(0.2,0.2,-0.1) -0.07711367095 -0.08075077885 -0.08075381528 -0.08075381592

(0.2,0.4,-0.1) -0.1004436229 -0.1133482422 -0.1134055766 -0.1134055435

(0.2,0.6,-0.1) -0.1498635015 -0.1495343794 -0.1498624662 -0.1498635015

(0.2,0.8,-0.1) -0.1409872942 -0.1914962138 -0.1926482315 -0.1926555602

(0.4,0.2,-0.1) -0.08188239622 -0.08551950412 -0.08552254055 -0.08552254120

(0.6,0.2,-0.1) -0.08626299950 -0.08990010740 -0.08990314382 -0.08990314448

(0.8,0.2,-0.1) -0.09008083995 -0.09371794785 -0.09372098428 -0.09372098492

(0.2,0.2,-0.2) -0.1542273419 -0.1615015577 -0.1615076306 -0.1615076318

(0.2,0.4,-0.2) -0.2008872458 -0.2266964845 -0.2268109531 -0.2268110870

(0.2,0.6,-0.2) -0.2425774142 -0.2990687587 -0.2997249324 -0.2997270030

(0.2,0.8,-0.2) -0.2819745885 -0.3829924276 -0.3852964630 -0.3853111204

(0.4,0.2,-0.2) -0.1637647924 -0.1710390082 -0.1710450811 -0.1710450824

(0.6,0.2,-0.2) -0.1725259990 -0.1798002148 -0.1798062876 -0.1798062890

(0.8,0.2,-0.2) -0.1801616799 -0.1874358957 -0.1874419686 -0.1874419698



ON THE BANACH SPACE TECHNIQUES 91

Figure 1. Comparison between the exact solution and the 5th-order of approximation

solution given by the homotopy analysis transform method(Uapprox[5](x,t,β))

Figure 2. Comparison between the exact solution and the 5th-order of approximation

solution given by the homotopy analysis transform method(Uapprox[5](x,t,β))

Figure 3. Comparison between the exact solution and the 10th-order of approximation

solution given by the homotopy analysis transform method(Uapprox[10](x,t,β))

Figure 4. Comparison between the exact solution and the 10th-order of approximation

solution given by the homotopy analysis transform method(Uapprox[10](x,t,β))

6. conclusion

The homotopy analysis method is applied for solving the FFKGE. This equation is equivalent to fuzzy

integral equation. The existence, uniqueness of the solution and convergence of this method are proved.



92 A. EBADIAN, M. SHAMS YOUSEFI, F. FARAHROOZ AND M. NAJAND FOUMANI

References

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and comparison with Adomian’s method. journal of computational an applied mathematics. 223 (2009), 278-290.

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electric field. Physics Letters B. 75 (2015), 76-81.

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university of California, Berkeley, California, 1974.

[5] R. Goetschel, W. Voxman, Elementary calculus. Fuzzy sets And system. 18(1986), 31-43.

[6] SJ. Liao , Introduction to the homotopy analysis method. Boca Raton: chapman and Hall / cRc, 2003.

[7] JY Park , Yc Kwan , JV Jeong , Existence of solutions of fuzzy integral equations in Banach spaces. Fuzzy sets And

systems. 72 (3) (1995), 373-378.

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math. 264 (2014), 65-70.

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1Department of Mathematics, Payame Noor University, PO. Box 19395-3697 Tehran, Iran

2Department of Mathematis, Faculty of science University of Guilan, Po Box 1914 Rasht, Iran

∗Corresponding author: f.farahrooz@yahoo.com


	1. Introduction
	2. Preliminaries
	3. The fractional transform
	4. Existence And convergence Analysis
	5. The homotopy analysis transform method
	6. conclusion
	References