Numerical solution of fuzzy differential equations by two-
step modified Simpson rule
Ekhtiar Khodadadi
Department of Mathematics, Malekan Branch, Islamic Azad University, Malekan,
Contents:
Abstract
1. Introduction 2. Preliminaries
3. Two-step modified Simpson rule 4. Convergence and Stability
5. Numerical Results 6. Conclusion
2
Abstract
In this paper, a numerical explicit two-step modified Simpson rule for fuzzy first-order initial value problem is present, and their applicability is illustrated with an example.
Keywords: fuzzy differential equations; fuzzy Cauchy
problem; two-step methods; Midpoint rule; Trapezoidal rule;
1. Introduction
Fuzzy differential equations (FDEs) are applied in modeling problems in science and engineering.
Most of the science and engineering utilizations of FDEs require the solution of an FDE subject to some fuzzy initial conditions; therefore, a fuzzy initial value problem arises. The concept of fuzzy derivative was first introduced by Chang and Zadeh [1]. Later, Dubois and Prade [2] introduced the concept of fuzzy derivative based on the extension principle. Kandel and Byatt [3, 4]
presented the concept of fuzzy differential equation in 1987. The FDEs and the initial value problem were regularly tussled by Kaleva [5, 6].
There are several approaches for solving fuzzy differential equations are proposed in the literature.
S. Sindu Devi and K. Ganesan [7] proposed Simpson’s rule and Runge-Kutta method of order four for the numerical solution of fuzzy differential equations. Kanagarajan K. et al. [8] studied numerical solution of fuzzy differential equations by Modified two-step Simpson method and the dependency problem. M. Sh. Dahaghin et al. [9] analyzed the two-step method for numerical
solution of fuzzy ordinary differential equation.
42. Preliminaries
2.1. Notations and definitions
Definition 2.1. Let be a nonempty set. A fuzzy set in is characterized by its membership function , and is interpreted as the degree of membership of an element in fuzzy set for each .
Let us denote by the class of fuzzy subsets of the real axis, that is,
satisfying the following properties:
i. is normal, that is, there exists such that , ii. is a convex fuzzy set (i.e., ),
iii. is upper semi-continuous on ,
iv. is compact, where cl denotes the closure of a subset.
The space is called the space of fuzzy numbers. Obviously, . For , we denote
Then from (i)–(iv), it follows that the -level set is a nonempty compact interval for all . The notation
denotes explicitly the -level set of . The following remark shows when is a valid -level set.
6
Remark 2.2. The sufficient conditions for to define the parametric form of a fuzzy number are as follows:
i. is a bounded monotonic increasing (nondecreasing) left-continuous function on and right continuous for , ii. is a bounded monotonic decreasing (nonincreasing) left-continuous function on and right continuous for , iii. .
Let be a real interval. A mapping is called a fuzzy process and its -level set is denoted by
A triangular fuzzy number is a fuzzy set in that is characterized by an ordered triple with such that and . The -level set of a triangular fuzzy number is given by
for any .
Definition 2.3. Let . If there exists such that , then is called the H-difference of and , and it is denoted by .
In this paper the sign “” stands always for H-difference, and let us remark that . Usually we denote by , while stands for the H-difference.
8
Definition 2.4. Let be a fuzzy function. We say is Hukuhara differentiable at if:
i. There exists an element such that, for all sufficiently near , there are , and the limits (in -metric)
or
ii. There exists an element such that, for all sufficiently near , there are , and the limits (in -metric)
Here the limits are taken in the metric space .
Definition 2.5. Let . The fuzzy integral is defined by
provided the Lebesgue integrals on the right exist.
Remark 2.6. Let . If is Hukuhara differentiable and its Hukuhara derivative is integrable over , then
for all values of , , where .
10
2.2, Fuzzy Differential Equations
Consider the first-order fuzzy differential equation , where is a fuzzy function of , is a fuzzy function of crisp variable and fuzzy variable , and is Hukuhara fuzzy derivative of . If an initial value is given, a fuzzy Cauchy problem of first order will be obtained as follows:
Sufficient conditions for the existence of a unique solution to Eq. (3.1) are:
i. Continuity of ,
ii. Lipschitz condition .
By Theorem 5.2 in [8] we may replace Eq. (2.2) by equivalent system
which possesses a unique solution which is a fuzzy function, i.e. for each , the pair is a fuzzy number.
The parametric form of (2.3) is given by
for . In some cases the system given by (2.4) can be solved analytically [8]. In most cases analytical solutions may not be found, and a numerical approach must be considered. Some numerical methods such as the fuzzy Euler method, Nyström method, predictor corrector method, and Trapezoidal Rule presented in [6, 1, 5, 2, 3, 4]. In the following, we present a new method to numerical solution of FDE.
12
3. Two-step modified Simpson rule
In the interval we consider a set of discrete equally spaced grid points . The exact and approximate solutions at , are denoted by and , respectively.
The grid points at which the solution is calculated are
Let which is triangular fuzzy number. We have
where
for ,
we have
By fuzzy interpolation, Theorem 10 [1], we get
14
From (2.1) and (3.1) it follows that
where
According to (3.1), if (3.2), (3.3), (3.5) and (3.6) are situated in (3.8) and (3.3), (3.4), (3.6) and (3.7) ) are situated in (3.9), we obtain
16
and by calculating the integral of Lagrange Coefficients,
, , , , , we obtain
Then
Similarly, we obtain
Equations (3.10) and (3.11) are an implicit equation in term of . To avoid of solving such implicit equation we will substitute by in right hand side of (3.10) and (3.11) where .
18
Therefore,
But we have
where is in between and .
20
As the result of above we will have
where , and is in between and .
Thus, we have the following two-step explicit equation for calculation using and :
for .
22
4. Convergence and Stability
Suppose the exact solution is approximated by some . The exact and approximate solutions at , are denoted by and , respectively. Our next result determines the point wise convergence of the Modified Simpson approximates to the exact solution. The following lemma will be applied to show convergence of these approximates; that is,
Lemma 4.1. Let a sequence of numbers satisfy
for some given positive constant and . Then
Proof. See [6].
Lemma 4.2. Suppose that a sequence of non-negative numbers satisfy
for some given positive constants , and . Then, for , we have
Proof. It is obvious that . Therefore, we have:
If we set and , then
By using Lemma 4.1 with , the proof is completed.
24
Let and be the functions and of (2.3), where and are constants and . The domain where and are defined is therefore Now, we will present the convergence theorem.
Theorem 4.1. Let and belong to and suppose that the partial derivatives of and be bounded on . Then for arbitrary fixed the Modified Simpson approximations of Eq. (3.12) converge to the exact solutions , for .
Proof. It is sufficient to show
, . By Taylor’s theorem, we have
and
where , and is in between and . Consequently
26
� ( � � +1 ; � ) − � ( � � +1 ; � ) = � ( � � −1 ; � ) − � ( � � −1 ; � ) + h 3 � ( ( � � −1 , [ � ( � � −1 ; � ) , � ( � � −1 ; � ) ] ) − � ( � � −1 , [ � ( � � −1 ; � ) , � ( � � −1 ; � ) ] ) ) + 2h 3 ( � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) − � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) + 2h 3 ( � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) − � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) + h 3 ( � ( � � +1 , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) − � ( � � +1 , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) ) +h 3 � ( � ) ,
� ( � � +1 ; � ) − � ( � � +1 ; � ) = � ( � � −1 ; � ) − � ( � � −1 ; � ) + h 3 � ( ( � � −1 , [ � ( � � −1 ; � ) , � ( � � −1 ; � ) ] ) − � ( � � −1 , [ � ( � � −1 ; � ) , � ( � � −1 ; � ) ] ) ) + 2h 3 ( � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) − � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) + 2h 3 ( � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) − � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) + h 3 ( � ( � � +1 , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) − � ( � � +1 , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) , � � ( � ; � ) +h � ( � � , [ � ( � � ; � ) , � ( � � ; � ) ] ) ) ) +h 3 � ( � ) ,
28
where
Set , . Then
where and are upper bound for , respectively.
Set , then
30
If setting , we obtain,
where . By using lemma 4.2 we have:
where , , and , Because of and , for , we have
therefore, we have and consequently , in other words, and the proof is completed. Remark 3.1. Above theorem results that convergence order is .
32
5. Numerical Results
Example 5.1. Consider the initial value problem
, ,
the exact solution at for is given by .
A comparison between the approximate solutions by Modified Simpson rule, , Trapezoidal rule, , Midpoint rule, , and the exact solution, at with , is shown in Table 5.1 and Figure 5.1.
Table 5.1.
34
6. Conclusion
We have presented Modified Simpson rule for numerical solution of first-order fuzzy differential equations. To illustrate the efficiency of the new method, we have compared our method with the Midpoint rule and Trapezoidal rule in some examples. We have shown the global error in Modified Simpson rule is less than in Midpoint rule and more than in Trapezoidal rule.
36
