We introduce some generalized quadrature rules to approximate two-dimensional, Henstock integral of fuzzy-number-valued functions. We also give error bounds for mappings of bounded variation in terms of uniform modulus of continuity. Moreover, we propose an iterative procedure based on quadrature formula to solve two-dimensional linear fuzzy Fredholm integral equations of the second kind (
The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh and others. The topic of fuzzy integrations is discussed in [
Since many real-valued problems in engineering and mechanics can be brought in the form of two-dimensional fuzzy integral equations, it is important that we develop quadrature rules and numerical methods for such integral equations. In this paper, we introduce two-dimensional fuzzy integrals and propose some generalized quadrature rules and their dependent theorems for mappings of bounded variation. Also, we present the conditions for existence of unique solution for
In this section, we review some necessary basic definitions on fuzzy numbers, fuzzy-number-valued functions, and fuzzy integrals.
A fuzzy number is a function the support
The set of all fuzzy numbers is denoted by
For any
Moreover, the addition and scalar multiplication of fuzzy numbers in
Also, according to [ with respect to
For arbitrary fuzzy numbers
Throughout this paper, we denote that
In [
Suppose that
Also, if
The following properties hold:
Let
The function
If the above
In [
If
In [
If
Regarding [
A function
A function
(i) It is easy to see that
(ii) Let
A function
A function
We see that if
Indeed, we have
In this section, we present some quadrature rules for 2D Henstock integral. The following theorem gives a unified approach to quadrature rules in 2D Henstock integrals.
Let
It is known that the Henstock integrals are additive related to interval. This leads us to
From part (i) of Theorem
From the above inequality, we infer some generalization of well-known trapezoidal-type, midpoint-type, and three-point-type inequalities with error estimations.
Assume that
for any
for any
for any
(i) Taking in the previous theorem that
(ii) Taking that
(iii) Considering
Let
(i) If we take
(ii) Taking
(iii) It is easy to see that the inequality follows from the corresponding assertion (iii) of the previous corollary by taking
The next corollaries present simpler error estimation for the inequality stated in Theorem
Let
Considering Theorem
Let
Since
If
Let
If we define
If
Analogous to the proof of Theorem
Here, we consider the two-dimensional fuzzy Fredholm integral equations as follows:
Now, we will prove the existence and uniqueness of the solution of (
Sufficient conditions for the existence of a unique solution of (
Let
Moreover, the following error bound holds:
To prove this theorem, we investigate the conditions of the Banach fixed point principle. We first show that
This shows that
Now, we prove that the operator
Therefore, we obtained
Now, we introduce a numerical method to solve (
Here, we obtain an error estimate between the exact solution and the approximate solution for the given fuzzy Fredholm integral equation (
Consider the
Considering iterative procedure (
Using part (ii) of Corollary
By part (ii) of Theorem
Now, since
Since
The proposed iterative method of successive approximations was tested on three numerical examples to provide the accuracy and the convergence of the method and illustrate the correctness of the theoretical results. In these examples, we assumed that
Assume that
To obtain numerical solution, we apply the proposed method. To compare numerical and exact solutions, see Table
Numerical results on the level sets for Example
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0.00 | 0.000000 | 0.000657 | 0.000000 | 0.000661 | 0.000000 | 0.000008 | 0.000000 | 0.000003 |
0.25 | 0.000067 | 0.000427 | 0.000051 | 0.000617 | 0.000011 | 0.000053 | 0.000001 | 0.000005 |
0.50 | 0.000086 | 0.000586 | 0.000024 | 0.000258 | 0.000066 | 0.000815 | 0.000043 | 0.000001 |
0.75 | 0.000150 | 0.000423 | 0.000022 | 0.000367 | 0.000069 | 0.000499 | 0.000008 | 0.000006 |
1.00 | 0.000229 | 0.000329 | 0.000131 | 0.000221 | 0.000154 | 0.000154 | 0.000005 | 0.000005 |
Consider (
Numerical results on the level sets for Example
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0.0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
0.2 | 0.000003 | 0.000008 | 0.000000 | 0.000002 |
0.4 | 0.000007 | 0.000011 | 0.000000 | 0.000008 |
0.6 | 0.000004 | 0.000006 | 0.000001 | 0.000000 |
0.8 | 0.000000 | 0.000005 | 0.000000 | 0.000000 |
1.0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 |
The integral equation (
Numerical results on the level sets for Example
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0.0 | 0.000047 | 0.000028 | 0.000008 | 0.000003 |
0.2 | 0.000039 | 0.000086 | 0.000006 | 0.000009 |
0.4 | 0.000009 | 0.000008 | 0.000003 | 0.000000 |
0.6 | 0.000004 | 0.000005 | 0.000001 | 0.000000 |
0.8 | 0.000012 | 0.000033 | 0.000007 | 0.000002 |
1.0 | 0.000006 | 0.000004 | 0.000000 | 0.000000 |
In this paper, we introduced 2D fuzzy mappings and defined 2D fuzzy integrals. Quadrature rules to approximate the solution of 2D fuzzy integrals are given. We established the theorem of existence of unique solution of
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the editor and anonymous referees for various suggestions which have led to an improvement in both the quality and clarity of the paper.