1. Introduction
One of the key tools for many fields of applied mathematics is the integral equation. Many branches of science and engineering naturally contain integral equations [
1]. Functional equations, such as partial differential equations, integral and integro-differential equations, stochastic equations, and others are frequently produced when real-world issues are mathematically modeled. Integro-differential equations are the common component of mathematical descriptions of physical events; they may be found in fluid dynamics, biological models, and chemical kinetics. Integro-differential equations arise in many numerous physical processes, including the formation of glass-forming process [
2], nano-hydrodynamics [
3], drop wise condensation [
4], wind ripple in the desert [
5] and biological model [
6].
Many researchers are now focusing their research on the investigation of fuzzy integral equations and fuzzy differential equations. Zadeh [
7,
8] was the first to present the idea of fuzzy sets. By Kaleva and Seikkala [
9,
10], fuzzy integral equations were first developed. Many scholars have recently concentrated their attention on this area and written numerous studies that are available in the literature [
11]. Fuzzy integral equations have been solved using a variety of analytical techniques, including the Adomian decomposition approach [
12,
13], homotopy analysis method [
14], homotopy perturbation method [
15], Laplace transform method [
16] and Sumudu decomposition method [
17]. Numerous numerical methods are available to solve fuzzy integral problems (see [
18,
19,
20]).
In this work, we consider the following system of integro-differential equations as [
6]
with initial conditions
where
are given functions,
are unknown functions and
.
Here, the numbers of two distinct species at time z are and , where the first species grows and the second shrinks. In the event that they exist together, assuming that the second species would consume the first, there will be a rise in the second species’ rate, or , which depends on both the first species’ historical values and its current populations, or . The coefficients of increase and decrease for the first and second species, respectively, are and . The values of the parameters and the kernels , are dependent on the species.
In our work, we have considered this model in fuzzy sense, i.e., if and are fuzzy functions, these functions can be expressed in parametric form as , , and , respectively.
The classical higher order Taylor series approach, which needs symbolic computation of the required derivatives of the data function and is computationally costly for higher order, is different from the differential transform method. The approximate solution is assessed using the finite Taylor series via the differential transformation technique. However, the differential transform technique does not compute the derivative directly; rather, the relative derivatives are generated through an iterative process. Allahviranloo et al. [
21] have suggested fuzzy differential transform method (FDTM) in order to solve first order fuzzy differential equations under generalized differentiability. The fuzzy integro-differential equations, higher order fuzzy differential equations, fuzzy boundary value problems, etc. may all be simply added to the scope of this technique. In this article, the above said biological model has been solved by fuzzy differential transform method. The FDTM has been applied by many authors to solve integral equations and integro-differential equations [
22,
23,
24].
This paper has been organized as follows: in
Section 2, some fundamental terminologies and outcomes that will be utilized later are brought. For the purpose of solving a fuzzy system of integral equations, a fuzzy differential transform method is presented in
Section 3. In
Section 4, we study the main result. The proposed approach is used to resolve three illustrative cases in
Section 5.
Section 6 draws conclusions.
2. Preliminaries
The most fundamental fuzzy calculus notations are introduced in this section. To begin, a fuzzy number is defined.
Definition 1 ([19]). An ordered pair of functions ; that match the following conditions can be used to describe a fuzzy number v.
- 1.
is a left continuous monotonic bounded increasing function.
- 2.
is a is a left continuous monotonic bounded decreasing function.
- 3.
.
For arbitrary , and , we define addition and scalar multiplication by k as
- a.
- b.
- c.
,
Since each
can be regarded as a fuzzy number
defined by
Let
be the set of all upper semicontinuous normal convex fuzzy numbers with bounded
-level intervals, the Hausdorff distance between fuzzy numbers given by
such that
It is easy to see that
D is a metric in
and has the following properties [
11].
- a.
;
- b.
- c.
- d.
is a complete metric space.
Definition 2. Let be a fuzzy valued function. If for arbitrary fixed and , there exists a such that is said to be continuous. It is well-known that the H-derivative (differentiability in the sense of Hukuhara) for fuzzy mappings was initially introduced by Puri and Ralescu [25] which is based on the H-difference of sets, as follows: Definition 3. Let . If there exists such that , then z is called the H-difference of x and y, and it is denoted by .
In this paper we consider the following definition which was introduced by Chalco–Cano and Román-Flores [
26].
Definition 4. Let and . We say that f is differential at , If there exists an element , such that for all ; and the limits (in the metric D) .
In this paper, we consider the nonlinear system of fuzzy integro-differential equation given in Equation (
1), where
are fuzzy valued functions, and the signs of
and
do not change in
. Let
Equation (
1) can be transformed into two systems as
3. Fuzzy Differential Transform Method
Let us consider
is differentiable of order
k over time
z, then
and
are called lower and upper spectrum of
at
, respectively. So if
be differentiable, then
can be represented as
The mentioned equations are known as the inverse transformation of
and
, respectively. The more conceptual definitions and theorems related to fuzzy differential transform are available in [
11].
Theorem 1 ([11]). Let’s assume that and are fuzzy-valued functions and and , respectively, represent their fuzzy differential transformations. Then
If , then ,
If , then .
If , then .
Provided the Hukuhara difference exists.
Theorem 2 ([11]). Consider the fuzzy-valued functions and , then , where and are the differential transformations of h and l, respectively.
Theorem 3 ([11]). Let us consider , then where and are the fuzzy differential transformations of fuzzy-valued functions h and l, respectively.
Theorem 4 ([11]). Assume that and are fuzzy differential transformations of and respectively. Under (i)—differentiability of f, if then
.
And under (ii)—differentiability of f we have
.
Theorem 5 ([11]). Suppose that and are the fuzzy differential transformations of the functions and (is a positive real valued function), respectively. If , then
4. Main Results
We demonstrate a few theorems in this section that allow the FDTM to be extended to the systems (1) and (
2).
Theorem 6. If is the fuzzy differential transform of at , then the fuzzy differential transform of at is defined as and .
Proof. From the above, we get
and
From the above, we get . □
Theorem 7. If is the fuzzy differential transform at , then the fuzzy differential transform of at is and .
Theorem 8. If , then for the fuzzy differential transform of in , we have and and if , then for the fuzzy differential transform of in , we have and .
Proof. For , we have and
Again, we have and .
Theorem 9. If , then the fuzzy differential transform of in is of the formand If , then the fuzzy differential transform of in is of the formandwhere and are the fuzzy differential transforms of functions and in , respectively. Proof. By putting the value of
at
in
By putting the value of
at
in
Now we consider the following system of fuzzy integro-differential equations as
and
with initial conditions
where
are given functions and
are unknown functions.
Therefore, the systems (
4) and (
5) can be written as
Using Theorems 6–9, the fuzzy differential transform of the systems (
6) and (
7) can be reduced as
and
where,
and
denote the fuzzy differential transforms of the functions
and
, respectively. Similarly,
and
denote the fuzzy differential transforms of the functions
and
, respectively.
By substituting
and
using Theorem 8 and
and
using Theorem 9, we obtain
for
with the initial conditions
. If we set
N instead of
∞, a nonlinear algebraic system of equations is obtained and by solving this system, the unknowns
and
are obtained. Finally, we get the approximate solution of (
4) and (
5) as