afsharinafar2012

Extended Midpoint method for solving fuzzy differential

equations

*Reza Afsharinafar, *Fudziah Ismail and **Ali Ahmadian Hosseini *Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM, Serdang, Selangor,

Malaysia. *Mathematics Department, Science Faculty, Universiti Putra Malaysia, 43400 UPM, Serdang,

Selangor, Malaysia ��

Abstract- we study fuzzy differential equations (FDEs) using the strongly generalized differentiability concept. Utilizing the characterization problem, we present approximate solutions of FDEs under Generalized differen-tiability by an equivalent system of ODEs. Then we extend midpoint approximation method and give its error, which guarantees pointwise convergence. An illustrative example is given.

Keywords: fuzzy differential equations; generalized differentiability; generalized characterization theorem; midpoint method

I. INTRODUCTION Fuzzy differential equations (FDEs) are

studied as a powerful tool for mathematical modeling of real world problems to make a suitable setting for modeling uncertainty or vagueness. There are several approaches to the study of fuzzy differential equations [7,18].

Hukuhara derivative, the most popular approach, of a set-valued mapping was introduced by Hukuhara in [8] and has been used in several approaches. To define fuzzy derivative Puri and Ralescu [11] generalized the H-derivative from set-valued mapping to the class of fuzzy mapping and it studied by several authors [15,9]. Under this setting, fuzzy initial value problem is studied and the existence and uniqueness theories for the Fuzzy Differential Equations (FDEs) is developed. However, It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes by [6]. Hence, in some cases, it suffers the disadvantage since the solution of FDE has increasing diameter of its support. So FDEs

was interpreted as family of differential inclusion to resolve this shortcoming in [8]. Although family of differential inclusion has been successfully adapted in several applications, it has a main disadvantage that the derivative of a fuzzy-number-valued function doesn’t exist. In this direction, the strongly generalized differentiability was introduced by Bede and Gal in [2] and studied in [3, 4, 16]. By this kind of derivative, a larger class of fuzzy-number-valued function has a derivative. Indeed, by using the strongly generalized differentiability the fuzzy initial value problem (FIVP) has solutions with decreasing length in their support. However, the unique-ness condition of solutions is lost.

The results of [3] inspired some authors that have been applied numerical methods for the solution of FIVP [12-14, 1, 7]. They replaced the fuzzy differential equation by its parametric form and then solved numerically the system consist of two classic ordinary differential equations with initial conditions. In this paper, under generalized differentiability, we generalize midpoint method to solve FDEs.

On the other hand, in [4] it was shown that FDEs could be translated to the system of ordinary differential equations (ODEs). They presented a characterization theorem, which is shown the equivalence ODE system of FDE. Indeed, it states that FDEs can be solved numerically by suitable numerical method.

So, after preliminary section, the characterization theorem is presented with the concept of the generalized differentiability.in section 4, we generalize midpoint method for solving FDEs and followed by a complete error analysis. Finally, a numerical example is presented.

2012 IEEE Symposium on Humanities, Science and Engineering Research

978-1-4673-1310-0/12/$31.00 ©2012 IEEE 1579

II. PRELIMINAREIS Let us denote �� the class of fuzzy subsets

of real axis (i.e. �� ) satisfying the following properties:

i. �� is normal, i.e. there exist !" � with �#!"$ % �, ii. �� is a convex fuzzy set (i.e. �#&! ' #� ( &$)$ * +,-.�#!$��#)$/ � �& �� !� ) �),

iii. �� is upper semicontinuous on �,

iv. 01.! �2�#!$ 3 �/, the closure of a subset is compact.

Then �� is called the space of fuzzy number. Obviously, � 4 �� and by conditions (i)-(iv) it follows that ��5 % .!6�2�#!$ *7/ is a non-empty bounded closed interval for all � 8 7 8 � while � belongs to ��.

Also, we denote explicitly the 7-level set of � by ��5� �5� in which, �5 and �5 refers the lower and upper branches of �, respectively.

For �� 96�� , and :6� , the sum � ; 9 and the product : < � are defined by �� ;9�5 % ��5 ' �9�5 , �: < ��5 % :��5 , �76��, where ��5 ' �9�5 means the usual addition of two intervals (subsets) of � and :��5 means the usual product between a scalar and a subset of �.

Defining the metric space =�� > �� ? @ .�/ by

=#�� 9$ % A�B5C�"�D�+E!.F�5 ( 95F� F�5 ( 95F/

#�� =$ is a complete metric space and the following properties are well known:

=#� ' G� 9 ' G$ % =#�� 9$� �� 9� G6�� =#H < �� H < 9$% 2H2=#�� 9$� �H6�� 96�� =#� ' 9�G ' I$ 8 =#�� G$' =#9� I$� �� 9� G� J6��

Definition 1: (see [10]) let !� )6�, K6� is called the H-difference of !� ) , if ! ' ) % K and it is denoted ! L ).

Definition 2: (see [11]) let M�N O �� be a fuzzy function. We say P is differentiable at &"6N if there exists an element PQ#&"$6�� such that the limits

1,+R�"SP#&" ' T$ L P#&"$T

and

1,+R�"SP#&"$ L P#&" ( T$T �

exist and are equal to PQ#&"$. Here the limit is taken in the metric space #�� =$� Note that the above definition of Hukuhara

derivative is restrictive; for instance in [3] the authors shown that if P#&$ % 0 < U#&$ where c is fuzzy number and U� #E� V$ � �? is a function with UQ#&$ W � , then P is not differentiable. To avoid that shortcoming, proposed a more general definition of a derivative for fuzzy-number-valued function by considering a lateral type of H-derivatives.

Definition 3: (see [3]) let P�N O �� and &"6N. We say that P is differentiable at &" if:

1. For all T 3 � sufficiently near to 0,there exist P#&" ' T$ L P#&"$�P#&"$ L P#&" ( T$ and the limits (in the metric D)

1,+R�"SP#&" ' T$ L P#&"$T %

1,+R�"SP#&"$ L P#&" ( T$T % PX#&"$

or 2. For all T 3 � sufficiently near to

0, there exist P#&" ' T$ L P#&"$�P#&"$ L P#&" ( T$ and the limits (in the metric D)

1,+R�"YP#&" ' T$ L P#&"$T %

1,+R�"YP#&"$ L P#&" ( T$T % PX#&"$

Remark 1: (see [3]) If F is differentiable in the senses (1) and (2) simultaneously, then for h > 0 sufficiently small, we have P#&" ' T$ % P#&"$ ' �D , P#&"$ %P#&" ( T$ ' �Z , P#&"$ % P#&" ' T$ ' 9D and P#&"$ % P#&" ' T$ ' 9Z , with �D� �Z� 9D� 9Z6�� . Thus P#&"$ % P#&"$ '#�Z ' 9D$, which implies two possibilities: �Z % 9D % [."/ if P#&"$ % [."/ ; or �Z % [.\/ % (9D, with E]�, if PX#&"$6�. Therefore, if there exists PX#&"$ in the first form (second form) with PX#&"$ ^ �, then PX#&"$ does not exist in the second form (first form, respectively).

The principal properties of defined derivatives are well known and can be found in


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[3, 16, 10]. In this paper, we make use of the following Theorem [16].

Theorem 1: Let P�N O �� and denote �P#&$�5 % �M5#&$� U5#&$� for each 76��. i. If P is differentiable in the first

form ((1)-differentiable) then M5 and U5 are differentiable.

ii. Functions and �PX#&$�5 %�MX5#&$� UX5#&$�. iii. If P is differentiable in the

second form ((2)-differentiable) then M5 and U5 are differentiable functions and

�PQ#&$�5 % _MQ5#&$� UQ5#&$`. Proof: see [16].

III. CHARACTERIZATION THEOREM FOR FDEs UNDER

GENERALIZED DIFFERENTIABILITY

Here we study the fuzzy initial value problem

!Q#&$ % M#&� !$� !#�$ % !" � ��

where M�N > �� O �� is a continuous fuzzy mapping and !" is a fuzzy number. The interval I may be [0,A] for some A > 0 or N % ��a$. Based on the Theorems [3] we can obtain some results on the existence of solutions of fuzzy differential equations.

Theorem 2: Let M�N > �� O �� be a continuous fuzzy function such that there exists H 3 � such that =#M#&� !$� M#&� K$$ 8 H=#!� K$ , �&6N , !� K6�� . Then problem (1) has two solutions (one (1)-differentiable and the other one (2)-differentiable) on I.

Proof: see [16].

Now by theorem (1) we translate the FIVP (1) into a system of ODEs. We have �!#&$�5 %�!5#&$� !5#&$� . If !#&$ is (1)-differentiable then by theorem (1), the FIVP (1) is equivalent to the following system of ODEs:

b!Q#&$ % M5c&� !5� !5d�!#�$ % !"�!Q#&$ % M5c&� !5� !5d�!#�$ % !"� (2)

Also, if !#&$ is (2)-differentiable then by theorem (1) the FIVP (1) is equivalent to the following system of ODEs:

b!Q#&$ % M5c&� !5� !5d�!#�$ % !"�!Q#&$ % M5c&� !5� !5d�!#�$ % !"� (3)

where

�M#&� !$�5 % �M5c&� !5� !5d� M5c&� !5� !5d�. As we can see in [16], first we ensure that �!5#&$� !5#&$� the solution of the ODEs

systems (2) and (3) are valid level sets and if �!X5#&$� !X5#&$� the 7-cut of derivative of !#&$ are valid level sets of a fuzzy valued function. Then, by stacking theorem (Representation Theorem) [9] we can construct a fuzzy solution e#f$ equivalent to (2) and (3) such that they are (1)-differentiable and (2)-differentiable, respectively.

By characterization theorems [4], which show that a fuzzy differential equation can be translated equivalently into a system of ODEs, we can use any numerical method for the system of ODES. Also, by next theorem the authors of [4] state that the FIVP (1) is equivalent to the system (2) or (3) in case of (1)-differentiability or (2)-differentiability, respectively.

Theorem 3: Let us consider the FIVP (1) where M�N > �� O �� is such that

i. �M#&� !$�5 % �M5c&� !5� !5d, M5c&� !5� !5dgii. M5 and M5 are equicontinuous;

iii. there exist h 3 � such that

iM5#&� !D� )D$ ( M5#&� !Z� )Z$i 8 h+E! .2!D ( !Z2� 2)D ( )Z2/� �76�� FM5#&� !D� )D$ ( M5#&� !Z� )Z$F 8 h+E! .2!D ( !Z2� 2)D ( )Z2/� �76��g then the FIVP (1) is equivalent to the system of ODEs (2) or (3) for (1)-differentiability and (2)-differentiability, respectively.

Proof: In [4] the equivalency for case of (1)-differentiability was proved. The result for (2)-differentiability is obtained by using theorem (1), too.

IV. MIDPOINT METHOD FOR SOLVING FDEs BY

CHARACTERIZATION THEOREM Some numerical methods for solving FDEs

under Hukuhara differentiability such as the Euler method and Taylor method, Adams-Bashford method, Adams Moulton method and Nyström method were presented in [12, 14, 1,


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7]. But, as we can see, in most cases we cannot solve the FDEs analytically. So a numerical approach is needed to solve this shortcoming.

In this section we generalize the midpoint method as a numerical method for solving FDE (1) by the Generalized Characterization theorem with its error analysis.

Let FIVP (1)

!Q#&$ % M#&� !$� !#�$ % !"

where N % �� j� and �M#&� !$�5 %_Pc&� !� !d� kc&� !� !d`. We split I into equally grid points � % &" W &D W l W &m % j . We denote exact solutions by �nD#&$�5 %onD#&� 7$� nD#&� 7$p and �nZ#&$�5 % �nZ#&� 7$ , nZ#&� 7$� which are approximated by some �)D#&$�5 % o)D#&� 7$� )D#&� 7$p and �)Z#&$�5 % o)Z#&� 7$� )Z#&� 7$p, respectively. And also, if we denote the exact and approximate solutions at &q % &" ' -T� T % rm � � 8 - 8 s by nDq#7$� nZq#7$� )Dq#7$ and )Zq#7$ , respectively.

Then based on the equations (2) and (3), to generalize midpoint method we proceed as follows:

• Translate fuzzy differential equation to its equivalent ordinary differential equations system based on (1)-differentiability or (2)-differenti-ability

• Solve numerically ODEs systems which consist of four classic ordinary differential equations with initial conditions

• Ensure that the solution and the derivative of the solution (!#&$) are valid level sets

• Based on the equations (2) and (3), for - % � , generalize Euler method and then for - 3 �, extend midpoint method for finding two fuzzy solutions of FDEs under generalized differentiability.

For - % �

tuvuw )Dq?D#7$ % )Dq#7$ ' TP�&q� )Dq#7$� )Dq#7$��)Dq?D#7$ % )Dq#7$ ' Tk o&q� )Dq#7$� )Dq#7$p �)D"#7$ % )"#7$�)D"#7$ % )"#7$�#x$

tuvuw )Zq?D#7$ % )Zq#7$ ' TP�&q� )Zq#7$� )Zq#7$��)Zq?D#7$ % )Zq#7$ ' Tk o&q� )Zq#7$� )Zq#7$p �)Z"#7$ % )"#7$�)Z"#7$ % )"#7$�#y$

for - 3 �

tuvuw )Dq?D#7$ % )DqzD#7$ ' TP�&q� )Dq#7$� )Dq#7$��)Dq?D#7$ % )DqzD#7$ ' Tk o&q� )Dq#7$� )Dq#7$p �)D"#7$ % )"#7$�)D"#7$ % )"#7$�#{$

tuvuw )Zq?D#7$ % )ZqzD#7$ ' TP�&q� )Zq#7$� )Zq#7$��)Zq?D#7$ % )ZqzD#7$ ' Tk o&q� )Zq#7$� )Zq#7$p �)Z"#7$ % )"#7$�)Z"#7$ % )"#7$�#|$

Remark: by theorem (1) we can see that the uniqueness of the solution of the fuzzy differential equations lost.

EXAMPLE: Consider the problem [4]

!Q#&$ % (} < !#&$�!#�$ % !"

where !" is a fuzzy number. Let } % �� N %�� and !" % �7 ( �� ( 7� . We can get the exact solution related to (1)-differentiability and (2)-differentiability solutions of the problem as follows, respectively:

!#&� 7$ % �#7 ( �$I~� #� ( 7$I~�� !#&� 7$ % �#7 ( �$Iz~� #� ( 7$Iz~�� To extend Euler and Midpoint methods to

generalized them we divide I into � % �� equally spaced subintervals and to get the solutions for case (1)-differentiability and case (2)-differentiability calculate for - % �

tuvuw)Dq?D

5 % )Dq5 ( T)Dq5�)Dq?D5 % )Dq5 ( T)Dq5�)D"5 % !"�)D"5 % !"�

and


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tuvuw)Zq?D

5 % )Zq5 ( T)Zq5�)Zq?D5 % )Zq5 ( T)Zq5�)Z"5 % !"�)Z"5 % !"�

and also for - 3 � compute

tuvuw)Dq?D

5 % )DqzD5 ( T)Dq5�)Dq?D5 % )DqzD5 ( T)Dq5�)D"5 % !"�)D"5 % !"�

and

tuvuw)Zq?D

5 % )ZqzD5 ( T)Zq5�)Zq?D5 % )ZqzD5 ( T)Zq5�)Z"5 % !"�)Z"5 % !"�

Following tables and Fig. 1 are shown a comparison between the lower and upper exact and approximate solutions at & % �� according to (2)-differentiability:

Table 1: Comparison between the lower exact solution and the lower approximate solution at & % ��

7 )D nD ID

0 -0.904843933909887 -0.904837418035960 0.651587392797026e-5

0.1 -0.814359540518899 -0.814353676232364 0.586428653515103e-5

0.2 -0.723875147127910 -0.723869934428768 0.521269914244282e-5

0.3 -0.633390753736921 -0.633386192625172 0.456111174951257e-5

0.4 -0.542906360345932 -0.542902450821576 0.390952435669334e-5

0.5 -0.452421966954944 -0.452418709017980 0.325793696398513e-5

0.6 -0.361937573563955 -0.361934967214384 0.260634957122141e-5

0.7 -0.271453180172966 -0.271451225410788 0.195476217834667e-5

0.8 -0.180968786781977 -0.180967483607192 0.130317478561071e-5

0.9 -0.090484393390989 -0.090483741803596 0.065158739280535e-5

1 0 0 0

Table 2: Comparison between the upper exact solution and the upper approximate solution at & % ��

� �Z �Z �Z

0 0.904843933909887 0.904843933909887 0.651587392797026e-5

0.1 0.814359540518899 0.814359540518899 0.586428653515103e-5

0.2 0.723875147127910 0.723875147127910 0.521269914244282e-5

0.3 0.633390753736921 0.633390753736921 0.456111174951257e-5

0.4 0.542906360345932 0.542906360345932 0.390952435669334e-5

0.5 0.452421966954944 0.452421966954944 0.325793696398513e-5

0.6 0.361937573563955 0.361937573563955 0.260634957122141e-5

0.7 0.271453180172966 0.271453180172966 0.195476217834667e-5

0.8 0.180968786781977 0.180968786781977 0.130317478561071e-5

0.9 0.090484393390989 0.090484393390989 0.065158739280535e-5

1 0 0 0


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Fig. 1. The comparison between the exact and approximate solutions

V. CONCLUSION A generalized stacking theorem has been

presented for the approximate solution of FDEs under generalized differentiability. As a matter of fact, by representation theorem a FDE can be transferred into two systems of ODEs, which can be solved by any suitable numerical method, and then by stacking theorem we can bunch the fuzzy solutions of FDE. As a future work, we will apply this method for fuzzy partial differential equations and also for second order fuzzy differential equations.

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