Usage of the Fuzzy Laplace Transform Method for Solving

One-Dimensional Fuzzy Integral Equations

ABDULRAHMAN A. SHARIF1, MAHA M. HAMOOD2, AMOL D. KHANDAGALE3

1Department of Mathematics, Hodeidah University, Al-Hudaydah, YEMEN.

2Department of Mathematics, Taiz University, Taiz, YEMEN.

3Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, INDIA.

Abstract: - In this paper, we propose the solution of fuzzy Volterra and Fredholm integral equations with convolu-

tion type kernel using fuzzy Laplace transform method (FLTM) under Hukuhara differentiability. It is shown that

FLTM is a simple and reliable approach for solving such equations analytically. Finally, the method is illustrated

with few examples to show the ability of the proposed method.

Key-Words: - Fuzzy integral equation, Fuzzy Laplace transform method, Fuzzy convolution.

Received: July 21, 2021. Revised: February 22, 2022. Accepted: March 15, 2022. Published: April 26, 2022.

1 Introduction

Modeling many problems of science, engineering,

physics, and other disciplines-leads to linear and non-

linear Fredholm and Volterra integral equations of the

second kind. These are usually difficult to solve an-

alytically and in many cases the solution must be ap-

proximated. Therefore, in recent years several numer-

ical approaches have been proposed [5, 6, 7, 20].

The basic idea and arithmetics of fuzzy sets were

first introduced by Zadeh in [25]. The concept of

fuzzy derivatives and fuzzy integration were studied

in [10, 12] and then some generalization have been

investigated in [9, 10]. The topic of fuzzy integral

equations has been rapidly grown recent years.

Abbasbandy et. al [1] proposed a numerical algo-

rithm for solving linear Fredholm fuzzy integral equa-

tions of the second kind by using parametric form of

fuzzy number and converting a linear fuzzy Fredholm

integral equation to two linear systems of integral

equation of the second kind in crisp case. Babolian

et. al [3] proposed another numerical procedure for

solving fuzzy linear Fredholm integral of the second

kind using Adomian method. Moreover, Friedman et.

al [11] proposed an embedding method to solve fuzzy

Volterra and Fredholm integral equations. However,

there are several research papers about obtaining the

numerical integration of fuzzy-valued functions and

solving fuzzy Volterra and Fredholm integral equa-

tions [2, 4, 8, 13, 14, 15, 16, 17, 18, 22, 24].

The fuzzy Laplace transform method solves FDEs

and corresponding fuzzy initial and boundary value

problems [2]. In this way fuzzy Laplace transforms

reduce the problem of solving a FDE to an algebraic

problem [19]. This switching from operations of cal-

culus to algebraic operations on transforms is called

operational calculus, a very important area of applied

mathematics, and for engineers, the fuzzy Laplace

transform method is practically the most important

operational method.

Recently, Allahviranloo and Barkhordari in [2] pro-

posed fuzzy Laplace transforms for solving first or-

der fuzzy differential equations under generalized H-

differentiability. By such benefits, we develop fuzzy

Laplace transform method to solve fuzzy convolu-

tion Volterra integral equation (FCVIE) of the second

kind. So, the original FCVIE is converted to two crisp

convolution integral equations in order to determine

the lower and upper function of solution, using fuzzy

convolution operator.

The paper is organized as follows. In section 2,

some basic definitions which will be used later in the

paper are provided. In section 3, the fuzzy Laplace

transform is studied. In section 4, the fuzzy convolu-

tion Volterra and Fredholm integral equations of the

second kind with fuzzy convolution kernel is stud-

ied. Then, the fuzzy Laplace transforms are applied to

solve such special fuzzy integral equation in section

5. Illustrative examples are also considered to show

the ability of the proposed method in section 6, and

the conclusion is drawn in section 7.

2 Preliminaries

In this section, we will recall some basics def-

initions and theorems needed throughout the paper

such as fuzzy number, fuzzy-valued function and the

derivative of the fuzzy-valued functions [9, 10, 12, 23,

25].

We denote the set of all real numbers by R.

A fuzzy number is a mapping u:R−→ [0,1]

with the following properties:

• (a) uis upper semi-continuous,

• (b) uis fuzzy convex, i.e., u(λx + (1 −λ)y)≥

min{u(x), u(y)}for all x, y ∈R, λ ∈[0,1],

• (c) uis normal, i.e., ∃x0∈Rfor which u(x0) =

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• (d) suppu ={x∈R|u(x)>0}is the support

of the u, and its closure cl(suppu)is compact.

Let Ebe the set of all fuzzy numbers on R. The

α−levelset of a fuzzy number u∈E, 0≤α≤

1,denoted by [u]α, is defined as

[u]α={x∈R|u(x)≥α}if 0< α ≤1

cl(suppu)if α = 0.

It is clear that the α−level set of a fuzzy num-

ber is a closed and bounded interval [u(α), u(α)],

where u(α)denotes the left-hand endpoint of [u]α

and u(α)denotes the right-hand endpoint of [u]α.

Since each y∈Rcan be regarded as a fuzzy number

˜ydefined by

˜y(t) = 1if t =y

0if t =y.

An equivalent parametric definition is also given in

[12] as:

Definition 1. A fuzzy number u in parametric form is

a pair (u, u)of functions u(α),u(α),0≤α≤1,

which satisfy the following requirements:

•1.u(α)is a bounded non-decreasing left con-

tinuous function in (0,1],and right continuous at

•2.u(α), is a bounded non-increasing left contin-

uous function in (0,1],and right continuous at

•3.u(α)≤u(α),0≤α≤1.

A crisp number αis simply represented by u(α) =

u(α) = α, 0≤α≤1. We recall that for a < b <

cwhich a, b, c ∈R,the triangular fuzzy number

u= (a, b, c)determined by a, b, c is given such that

represented by u(α) = a+ (b−a)αand u(α) =

c−(c−b)αare the endpoints of the alpha-level sets,

for all α∈[0,1].The Hausdorff distance between

fuzzy numbers given by d:E×E−→ R+∪ {0}.

d(u, v) = sup

α∈[0,1]

max{|u(α)−v(α)|,|u(α)−v(α)|}

where u= (u(α), u(α)), v = (v(α), v(α)) ⊂Ris

utilized in [10]. Then, it is easy to see that dis a metric

in Eand has the following properties

•(i)d(u+w, v +w) = d(u, v),∀u, v, ω ∈E,

•(ii)d(ku, kv) = |k|d(u, v),∀k∈R, u, v ∈E,

•(iii)d(u+v, w +e)≤d(u, w) +

d(v, e),∀u, v, w, e ∈E,

•(iv) (d, E),is a complete metric space

Definition 2. [8], Let f:R−→ Ebe a fuzzy

valued function. If for arbitrary fixed t0∈Rand

ϵ > 0, a δ > 0such that

|t−t0|< δ =⇒d(f(t), f(t0)) < ϵ.

fis said to be continuous

Theorem 1. [12] Let f(x)be a fuzzy-valued function

on [a, ∞)and it is represented by (f(x, α), f(x, α)).

For any fixed r∈[0,1],assume f(x, α)and

overlinef (x, α)are Riemann integrable on [a, b]for

every b≥a, and assume there are two positive M(α)

and M(α)) such that Rb

a|f(x, α)|dx ≤M(α)and

a|f(x, α)|dx ≤M(α)for every b≥a. Then f(x)

is improper fuzzy Riemann integrable on [a, ∞)and

the improper fuzzy Riemann integral is a fuzzy num-

ber. Further more, we have:

Z∞

f(x)dx =Z∞

f(x, α)dx, Z∞

f(x, α)|dx.

Proposition 1. [12] If each of f(x)and g(x)is

fuzzy-valued function and fuzzy Riemman integrable

on I= [a, ∞)then f(x) + g(x)is fuzzy Riemman

integrable on I. Moreover, we have

(f(x) + g(x))dx =Z1

f(x)dx +Z1

g(x)dx.

3 Fuzzy Laplace transforms

Suppose that fis a fuzzy-valued function and s is a

real parameter. We define the fuzzy Laplace trans-

form of fas following:

Definition 3. The fuzzy Laplace transform of fuzzy-

valued function f(t)is defined as following

F(s) = L(f(t)) = Z∞

e−stf(t)dt (1)

=lim

τ−→∞ Zτ

e−stf(t)dt,

whenever the limits exist.

The L(f)be also used to denote the fuzzy Laplace

transform of fuzzy-valued function f(t), and the in-

tegral is the fuzzy Riemann improper integral. The

symbol Lis the fuzzy Laplace transformation, which

acts on fuzzy-valued function f=f(t)and gener-

ates a new fuzzy-valued function, F(s) = L(f(t)).

Consider fuzzy-valued function f, then the lower and

upper fuzzy Laplace transform of this function are de-

noted, based on the lower and upper of fuzzy-valued

function fand 0≤α≤1as following:

F(s;α) = L(f(t;α)) = [L(f(t;α)),L(f(t;α)),

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Abdulrahman A. Sharif, Maha M. Hamood, Amol D. Khandagale

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where

L(f(t;α)) = Z∞

e−stf(t;α))dt

=lim

τ−→∞ Zτ

e−stf(t;α))dt,

L(f(t;α)) = Z∞

e−stf(t;α))dt

=lim

τ−→∞ Zτ

e−stf(t;α))dt.

4 Fuzzy Volterra and Fredholm

integral equations

In this section, we investigate the solution of fuzzy

convolution Fredholm and Volterra equations of the

second kind [21].

4.1 Fuzzy Fredholm integral equations

The fuzzy convolution Fredholm integral equation of

the second kind is defined as

˜x(t) = ˜

f(t) + ZT

k(s−t)˜x(s)ds, t ∈[0, T ],(2)

where ˜x(t) = ˜x(t;α)=[x(t;α), x(t;α)],˜

f(t) =

f(t;α) = [f(t;α), f(t;α)],and ˜

k(s−t)is an arbi-

trary given fuzzy-valued convolution kernel function

and ˜

fis a continuous fuzzy-valued function. The so-

lution of Eq. (2) can be obtained by solving the fol-

lowing system integral equations:

x(t;α) = f(t;α) + ZT

k(s−t;α)x(s;α)ds, (3)

x(t;α) = f(t;α) + ZT

k(s−t;α)x(s;α)ds. (4)

We adopt fuzzy Laplace transform to solve the given

problem such that by taking fuzzy Laplace transform

on both sides of Eqs. (3)-(4) and using fuzzy convo-

lution, we get solution of Eq. (2) directly.

4.2 Fuzzy Volterra integral equations

The fuzzy convolution Volterra integral equation of

the second kind is defined a

˜x(t) = ˜

f(t) + Zt

k(s−t)˜x(s)ds, t ∈[0, T ],(5)

where ˜

x(t) = ˜

x(t;α)=[x(t;α),x(t;α)],˜

f(t) =

f(t;α) = [f(t;α), f(t;α)],and ˜

k(s−t)is an arbi-

trary given fuzzy-valued convolution kernel function

and ˜

fis a continuous fuzzy-valued function. The so-

lution of Eq. (5) can be obtained by solving the fol-

lowing system integral equations:

x(t;α) = f(t;α) + Zt

k(s−t;α)x(s;α)ds, (6)

x(t;α) = f(t;α) + Zt

k(s−t;α)x(s;α)ds. (7)

We adopt fuzzy Laplace transform to solve the given

problem such that by taking fuzzy Laplace transform

on both sides of Eqs. (6)-(7) and using fuzzy convo-

lution, we get solution of Eq. (5) directly. So, the

concept of fuzzy convolution must be introduced.

4.3 Fuzzy convolution

The convolution of two fuzzy-valued functions f and

g defined for t > 0by

(f∗g)(t) = Zt

f(τ).g(t−τ)dτ, (8)

which of course exists if fand gare, say, piece-wise

continuous. Substituting u=t−τgive

(f∗g)(t) = Zt

f(τ).g(t−u)du = (g∗f)(t)(9)

that is, the fuzzy convolution is commutative. Other

basic properties of the fuzzy convolution are as fol-

lows:

(i)c(f∗g) = cf ∗g=f∗cg, c is constant

(ii)f∗(g∗h) = (f∗g)∗h(associative property)

Property (i) is routine to verify. As for (ii)

[f∗(g∗h)](t)

=Zt

f(τ)(g∗h)(t−τ)dτ

=Zt

f(τ)Zt−τ

g(x)h(t−τ−x)dxdτ

=Zt

0Zt−τ

f(τ)g(u−τ)h(t−τ)dxdτ

= [(f∗g)∗h](t),

while having reverse the order of integration. One of

the very significant properties possessed by the fuzzy

convolution in connection with the fuzzy Laplace

transform is that the fuzzy Laplace transform of the

convolution of two fuzzy-valued functions is the

product of their fuzzy Laplace transform.

Theorem 2. (Convolution Theorem), If fand g

are piecewise continuous fuzzy-valued functions on

[0,∞]and of exponential order p, then

L{(f∗g)(t)}=L{(f(t)}.L{(g(t)}=F(s).G(s).(10)

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Proof. Let us start with the produce

L{(f(t)}.L{(g(t)}

=Z∞

e−sτ f(τ)dτ.Z∞

e−sug(u)du

=Z∞

0Z∞

e−s(τ+u)f(τ)g(u)dudτ,

substituting t=τ+u, and noting that τis fixed in

the interior integrals, so that du =dt, we have

L{(f(t)}.L{(g(t)}(11)

=Z∞

0Z∞

e−stf(τ)g(t−τ)dtdτ.

If we define g(t) = e

0for t < 0, then g(t−τ) =

0for t < τ and we can write (11) as

L{(f(t)}.L{(g(t)}

=Z∞

0Z∞

e−stf(τ)g(t−τ)dtdτ

Due to the hypotheses on f, g, the fuzzy Laplace in-

tegrals of f, g converge absolutely and hen

Z∞

0Z∞

|e−st)f(τ)g(t−τ)|dτ,

converges. This fact allows us to reverse the order of

integration, so that

L{(f(t)}.L{(g(t)}

=Z∞

0Z∞

e−stf(τ)g(t−τ)dτdt

=Z∞

0Zt

e−stf(τ)g(t−τ)dτdt

=Z∞

e−stZt

f(τ)g(t−τ)dτdt

=L{(f∗g)(t)}.

Please notice that in the fuzzy case, we should investi-

gate more accurately than the deterministic case. So,

mentioned calculation is assumed valid under suitable

conditions.

5 Fuzzy Laplace transform method

for FIE

Here, we shall obtain the solution of fuzzy convolu-

tion Fredholm integral equation using fuzzy Laplace

transform. Indeed, our method is constructed on

applying fuzzy convolution. Consider the original

Eq.(2), then by taking fuzzy Laplace transform on

both sides of it we get the following:

L{(˜x(t)}=L{(˜

f(t)}+LnZT

k(s−t)˜x(s)dso,

then, we get by using fuzzy convolution and definition

of fuzzy Laplace transform:

L{x(t;α)}=L{f(t;α)}+L{k(t;α)}L{x(t;α)},

and

L{x(t;α)}=L{f(t;α)}+L{k(t;α)}L{x(t;α)}.

Now, we should discuss L{k(t;α)}L{x(t;α)}and

Lk(t;α)Lx(t;α).

To this end, we have the following cases:

Case 1- if k(t;α)and x(t;α)are positive, then we get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Case 2- if k(t;α)is positive and x(t;α)is negative,

then we get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Case 3- if k(t;α)is negative and x(t;α)is positive,

then we get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Case 4- if k(t;α)and x(t;α)are negative, then we

get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Notice that, it is assumed that zero does not exist in

support. We obtain explicit formula for Case 1 and

the others are similar. Indeed, we can write case 1 in

a compact form

L{x(t;α)}=L{f(t;α)}

1−L{k(t;α)}

and

L{x(t;α)}=L{f(t;α)}

1−L{k(t;α)}.

Finally, using the inverse of fuzzy Laplace transform

we get the solution:

x(t;α) = L−1L{f(t;α)}

1−L{k(t;α)}

and

x(t;α) = L−1L{f(t;α)}

1−L{k(t;α)}

for all 0≤α≤1,provided that a fuzzy valued

function is define.

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5.1 Fuzzy Laplace transform method for

VIE

Here, we shall obtain the solution of fuzzy convo-

lution Fredholm and Volterra integral equations us-

ing fuzzy Laplace transform. Indeed, our method is

constructed on applying fuzzy convolution. Consider

the original Eq.(2) and Eq.(5), then by taking fuzzy

Laplace transform on both sides of it we get the fol-

lowing:

L{(˜x(t)}=L{(˜

f(t)}+LnZT

k(s−t)˜x(s)dso,

and

L{(˜x(t)}=L{(˜

f(t)}+LnZt

k(s−t)˜x(s)dso,

then, we get by using fuzzy convolution and definition

of fuzzy Laplace transform:

L{x(t;α)}=L{f(t;α)}+L{k(t;α)}L{x(t;α)},

and

L{x(t;α)}=L{f(t;α)}+L{k(t;α)}L{x(t;α)}.

Now, we should discuss L{k(t;α)}L{x(t;α)}and

Lk(t;α)Lx(t;α).

To this end, we have the following cases:

Case 1- if k(t;α)and x(t;α)are positive, then we get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Case 2- if k(t;α)is positive and x(t;α)is negative,

then we get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Case 3- if k(t;α)is negative and x(t;α)is positive,

then we get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Case 4- if k(t;α)and x(t;α)are negative, then we

get

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)},

L{k(t;α)}L{x(t;α)}=L{k(t;α)}L{x(t;α)}.

Notice that, it is assumed that zero does not exist in

support. We obtain explicit formula for Case 1 and

the others are similar. Indeed, we can write case 1 in

a compact form

L{x(t;α)}=L{f(t;α)}

1−L{k(t;α)}

and

L{x(t;α)}=L{f(t;α)}

1−L{k(t;α)}.

Finally, using the inverse of fuzzy Laplace transform

we get the solution:

x(t;α) = L−1L{f(t;α)}

1−L{k(t;α)}

and

x(t;α) = L−1L{f(t;α)}

1−L{k(t;α)}

for all 0≤α≤1,provided that a fuzzy valued

function is define.

6 Examples

In this section, we give some examples to obtain the

solution of fuzzy convolution Volterra and Fredholm

integral equations of the second kind.

Example 1. Consider the following fuzzy Volterra

integral equation

˜x(t) = (α, 2−α).et+Zt

sin(t−τ).˜x(τ)dτ.

We apply the fuzzy Laplace transform to both sides of

the equation, so tha

L{˜x(t)}=L{(α, 2−α).et}+L{sin(t)}.L{˜x(t)}

i.e

L{x(t;α)}=L{(α).et}+L{sin(t)}.L{x(t;α)},

L{x(t;α)}=L{(2 −α).et}+L{sin(t)}.L{x(t;α)},

Hence . we get

L{x(t;α)}= (α)2

s−1−1

s2−1

s,0≤α≤1

L{x(t;r)}= (2 −α)2

s−1−1

s2−1

s,0≤α≤1

By taking the inverse of fuzzy Laplace transform on

both sides of above relations, we have the following

x(t;α)=(α)(2et−t−1),

x(t;α) = (2 −α)(2et−t−1).

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Example 2. Consider the following fuzzy Fredholm

integral equation

˜x(t) = 1

2(α+ 1),1

2(3 −α)).t +Z2

4(t−τ).˜x(τ)dτ.

We apply the fuzzy Laplace transform to both sides of

the equation, so that

L{˜x(t)}=1

2L{((α+ 1),(3 −α)).t}+1

4L{t}.L{˜x(t)}

i.e

L{x(t;α)}=1

2L{(α+ 1).t}+1

4L{t}.L{x(t;α)},

L{x(t;α)}=1

2L{(3 −α).t}+1

4L{t}.L{x(t;α)}.

Hence we get

L{x(t;α)}=2(α+ 1)

4s2−1,

L{x(t;r)}=2(3 −α)

4s2−1.

By taking the inverse of fuzzy Laplace transform on

both sides of above relations, we have the following

x(t;α) = (α+ 1).sinh(t

2),

x(t;α) = (3 −α).sinh(t

2).

Example 3. Consider the following fuzzy Volterra

integral equation

˜x(t) = (1 + α, 3−α).cosh t+Zt

et−τ.˜x(τ)dτ.

Similarly, by taking fuzzy Laplace transform on both

sides of equation, we get the

L{˜x(t)}=L{(1 + α, 3−α).cosh(t)}

+L{et}.L{˜x(t)}

i.e

L{x(t;α)}=L{(1 + α).cosh t}+L{et}.L{x(t;α)},

L{x(t;α)}=L{(3 −α).cosh t}+L{et}.L{x(t;α)},

i.e

L{x(t;α)}=(1 + α)s

(s+ 1)(s−2),

L{x(t;α)}=(3 −α)s

(s+ 1)(s−2),0≤α≤1.

Hence, we get

L{x(t;r)}= (1 + α)2

3(s−2) +1

3(s+ 1),

L{x(t;α)}= (3 −α)2

3(s−2) +1

3(s+ 1).

Finally, by applying the inverse of fuzzy Laplace

transform, we get the following:

x(t;α) = (1 + α)( 2

3e2t+1

3e−t),0≤α≤1

x(t;α) = (3 −α)(2

3e2t+1

3e−t),0≤α≤1.

7 Conclusion

In the present paper, the fuzzy Laplace transform

method was applied to approximate the solution of

fuzzy Volterra and Fredholm integral equations. We

transformed our problem to a system of algebraic

equations so that by solving this system we obtained

the solution of this kind of equations by considering

the type of differentiability. Finally, the solution ob-

tained using the suggested method shows that this ap-

proach can solve the problem effectively.

An interesting extension of our study would be to

discuss method with neural networks and finite-time

stability for the Volterra and Fredholm integral equa-

tions. This topic will be the subject of a forthcoming

paper.

References:

[1] Abbasbandy, S., Babolian, E. and Alavi, M. Nu-

merical method for solving linear Fredholm fuzzy

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EQUATIONS

DOI: 10.37394/232021.2022.2.6

Abdulrahman A. Sharif, Maha M. Hamood, Amol D. Khandagale

E-ISSN: 2732-9976

Volume 2, 2022