Numerical Solution of Fuzzy Differential Equations by Two-Step Modified Simpson Rule

Numerical solution of fuzzy differential equations by two- step modified Simpson rule Ekhtiar Khodadadi Department of Mathematics, Malekan Branch, Islamic Azad University, Malekan, Iran. 1

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; Modified Simpson rule 3

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. 4

2. Preliminaries 2.1. Notations and definitions Definition 2.1.Let ? be a nonempty set. A fuzzy set ? in ? is characterized by its membership function ?:? 0,1 , 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, ?: 0,1 , satisfying the following properties: ? is normal, that is, there exists ?0 such that ? ?0 = 1, i. ii. ? is a convex fuzzy set (i.e., ? ??1+ 1 ? ?2 min ? ?1,? ?2 iii. ? is upper semi-continuous on , iv. cl x u x > 0 is compact, where cl denotes the closure of a subset. , ? 0,1 , ?1,?2 ), 5

The space ? is called the space of fuzzy numbers. Obviously, ?. For 0 < ? 1, we denote ??= ? ? ? ? , ?0= ? ? ? > 0 , Then from (i) (iv), it follows that the ?-level set ?? is a nonempty compact interval for all 0 ? 1. 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: ? ? is a bounded monotonic increasing (nondecreasing) left-continuous function on continuous for ? = 0, ii. ? ? is a bounded monotonic decreasing (nonincreasing) left-continuous function on continuous for ? = 0, iii. ? ? ? ? , 0 ? 1. 0,1 and right i. 0,1 and right 7

Let ? be a real interval. A mapping ?:? ? is called a fuzzy process and its ?-level set is denoted by ?= ? ?;? ,? ?;? ? ?,? ? ? , 0,1 . A triangular fuzzy number is a fuzzy set ? in ? that is characterized by an ordered triple ??,??,?? 3 with ?? ?? ?? such that ?0= ??,?? and ?1= ??. The ?-level set of a triangular fuzzy number ? is given by ??= ?? 1 ? ?? ??,??+ 1 ? ?? ?? , 2.1 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 ? ? ? + 1 ?. Usually we denote ? + 1 ? by ? ?, while ? ? stands for the H-difference. 8

Definition 2.4. Let ?:? ? be a fuzzy function. We say ? is Hukuhara differentiable at ?0 ? if: There exists an element ? ?0 ? such that, for all > 0 sufficiently near 0, there are ? ?0+ ? ?0, ? ?0 ? ?0 and the limits (in ? -metric) i. ? ?0+ ? ?0 ? ?0 ? ?0 = ? ?0, lim 0+ = lim 0+ or There exists an element ? ?0 ? such that, for all < 0 sufficiently near 0, there are ? ?0+ ? ?0, ? ?0 ? ?0 and the limits (in ? -metric) i. ? ?0+ ? ?0 ? ?0 ? ?0 = ? ?0. lim 0 = lim 0 Here the limits are taken in the metric space ?,? . 9

?? ? ?? is defined by Definition 2.5. Let ?,? ?. The fuzzy integral ? ? ? ? ? ? ? ?? = ? ?;? ??, ? ?;? ?? , ? ? ? provided the Lebesgue integrals on the right exist. Remark 2.6. Let ?,? ?. If ?:? ? is Hukuhara differentiable and its Hukuhara derivative ? is integrable over ?,? , then ? ? ? ??, ? ? = ? ?0 + ?0 for all values of ?0, ?, where ? ?0 ? ?. 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 ? ?0 = ?0 ? is given, a fuzzy Cauchy problem of first order will be obtained as follows: ? ? = ? ?,? ? , ? ?0 = ?0 ?, ?0 ? ?, 2.2) Sufficient conditions for the existence of a unique solution to Eq. (3.1) are: i. ii. Lipschitz condition ? ? ?,? ,? ?,? Continuity of ?, ?? ?,? , ? > 0. By Theorem 5.2 in [8] we may replace Eq. (2.2) by equivalent system ? ? = ? ?,? ? = ? ?,?,? , ? ?0 = ?0, 2.3) ? ? = ? ?,? ? = ? ?,?,? , ? ?0 = ?0. 11

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 2.3 ? ?;? = ? ?,? ?;? = ? ?,? ?;? ,? ?;? = min ? ?,? ? ? ?;? ,? ?;? , ? ?0;? = ?0? , 2.4) ? ?;? = ? ?,? ?;? = ? ?,? ?;? ,? ?;? = mat ? ?,? ? ? ?;? ,? ?;? , ? ?0;? = ?0? . for 0 ? 1. 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, Nystr m 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 ?0,? we consider a set of discrete equally spaced grid points ?0< ?1< ?2< < ??= ?. The exact ?= ? ??;? ,? ??;? and ?? ?= ??? ,??? , and approximate solutions at ??, 0 ? ?, are denoted by ? ?? respectively. The grid points at which the solution is calculated are =? ?0 , ??= ?0+ ? , ? = 0,1,2, ,?. ? Let ? ?? = ? ??;? ,? ??;? , 0 ? ? which ? ??,?? is triangular fuzzy number. We have ??+1 ? ?,? ? ??, ? ??+1 = ? ?? 1 + 3.1) ?? 1 where ? ?,? ? = ???,? ?;? ,???,? ?;? ,???,? ?;? . 13 for ?? 1 ? ?? ??+1,

we have ? ?? ? ??+1 ?? 1 ??+1 ?? 1? = 0, ?? 1 ?? ? ?? 1 ? ??+1 ?? ?? 1 ?? ??+1 ??? = 0, ? ?? 1 ? ?? ??+1 ?? 1 ??+1 ?? ??+1? = 0. By fuzzy interpolation, Theorem 10 [1], we get ???,? ?;? = ?? 1? ???? 1,? ?? 1;? + ??? ????,? ??;? + ??+1? ????+1,? ??+1;? , 3.5) ???,? ?;? = ?? 1? ???? 1,? ?? 1;? + ??? ????,? ??;? + ??+1? ????+1,? ??+1;? , 3.6) ???,? ?;? = ?? 1? ???? 1,? ?? 1;? + ??? ????,? ??;? + ??+1? ????+1,? ??+1;? , 3.7) 14

2.1 3.1 From (2.1) and (3.1) it follows that ?= ? ??+1;? ,? ??+1;? ? ??+1 , where ? ??+1;? ?? ??+1 ????,? ?;? + 1 ? ???,? ? ;? ?? + ????,? ?;? + 1 ? ???,? ? ;? ??, 3.8) = ? ?? 1;? + ?? 1 ?? ? ??+1;? ?? ??+1 ????,? ?;? + 1 ? ???,? ?;? ????,? ?;? + 1 ? ???,? ?;? = ? ?? 1;? + ?? + ??. 3.9) ?? 1 ?? 15

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, ?? ?? ?? ?? 1? ?? =5 ??? ?? =2 ??+1? ?? = 12, ?? 1 3, ?? 1 12, ?? 1 ??+1?? 1? ?? = ??+1??? ?? =2 ??+1??+1? ?? =5 12, ?? 3, ?? 12, ?? we obtain ? ??+1;? = ? ?? 1;? +5 +2 12? ??+1,? ??+1;? 12? ?? 1,? ?? 1;? 12? ?? 1,? ?? 1;? +5 12? ??+1,? ??+1;? 3? ??,? ??;? +2 3? ??,? ??;? = ? ?? 1;? + ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??+1;? . 3 17

Then ? ??+1;? 5 90?4?1,? ?1 = ? ?? 1;? + ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??+1;? 3 , ?1 ?1 ??+1. 3.10) Similarly, we obtain ? ??+1;? = ? ?? 1;? + 5 90? 4 ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??+1;? ?1,? ?1 3 , ?1 ?1 ??+1. 3.11) Equations (3.10) and (3.11) are an implicit equation in term of ? ??+1;? . To avoid of solving such implicit equation we will substitute ? ??+1;? by ? ??;? + ? ??,? ?? 2 ??,??+1. + 2 ? ?2,? ?2 in right hand side of (3.10) and (3.11) where ?2 18

Therefore, ? ??+1;? = ? ?? 1;? + 2 + 2? ?2,? ?2 ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??;? + ? ??,? ??;? 3 5 90?4?1,? ?1 , ?? 1 ?1 ??+1, ?? ?2 ??+1, ? ??+1;? = ? ?? 1;? + 2 + 2? ?2,? ?2 ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??;? + ? ??,? ??;? 3 5 90? 4 ?1,? ?1 , ?? 1 ?1 ??+1, ?? ?2 ??+1. 19

But we have + 2 + 2 ? ??+1,? ??;? + ? ??,? ??;? = ? ??+1,? ??;? + ? ??,? ??;? ? ?2,? ?2 ????+1,?3, 2? ?2,? ?2 2 + 2 where ?3 is in between ? ??;? + ? ??,? ??;? and ? ??;? + ? ??,? ??;? ? ?2,? ?2 . 2 20

As the result of above we will have ? ??+1;? = ? ?? 1;? + ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??;? + ? ??,? ??;? 3 + 3 ????+1,?3 5 90?4?1,? ?1 6? ?2,? ?2 , ? ??+1;? = ? ?? 1;? + ? ?? 1,? ?? 1;? + 2? ??,? ??;? + 2? ??,? ??;? + ? ??+1,? ??;? + ? ??,? ??;? 3 + 3 ????+1,?3 5 4 6? ?2,? ?2 90? ?1,? ?1 , where ?? 1 ?1 ??+1, ?? ?2 ??+1 and ?3 is in between ? ??;? + ? ??,? ??;? and ? ??;? + ? ??,? ??;? + 2 2 ? ?2,? ?2 . 21

Thus, we have the following two-step explicit equation for calculation ? ??+1;? using ? ??1;? and ? ??;? : for 0 ? ?. 22

4. Convergence and Stability Suppose the exact solution ? ?;? ,? ?;? is approximated by some ? ?;? ,? ?;? . The exact and approximate ?= ??? ,??? and ?? ?= ??? ,??? , respectively. Our next solutions at ??, 0 ? ?, are denoted by ?? result determines the point wise convergence of the Modified Simpsonapproximates to the exact solution. The following lemma will be applied to show convergence of these approximates; that is, lim ? ?, ;? = ? ?;? , lim ? ?, ;? = ? ?;? . 4.1) ? Lemma 4.1. Let a sequence of numbers ?? ?=0 satisfy ??+1 ? ??+ ?, 0 ? ? 1, 4.2) for some given positive constant ? and ?. Then ??+1 ?? ???+ ??? ? 1 , ? ? ? 1. ? 1 Proof. See [6]. 23

? Lemma 4.2. Suppose that a sequence of non-negative numbers ?? ?=0 satisfy ??+1 ? ??+ ? ?? 1+ ?, 0 ? ? 1, ?2+4?+? 2 for some given positive constants ? , ? and ?. Then, for ? = , we have ??+1+ ? ? ?? ???1+ ? ? ?0 + ??? 1 ? 1, ?2+4?+? 2 ?2+4? ? 2 Proof. It is obvious that ? = . Therefore, we have: ?2+ 4? ? 2 ?2+ 4? + ? 2 ?2+ 4? ? 2 ??+1+ ?? ??+ ?? 1 + ?, ?2+4? ? 2 ?2+4?+? 2 If we set ??+1= ??+1+ ?? and ? = , then ??+1 ???+ ?, 1 ? ? 1. Lemma 4.1 By using Lemma 4.1 with ? = 1, the proof is completed. 24

2.3 Let ? ?,?,? and ? ?,?,? be the functions ? and ? of (2.3), where ? and ? are constants and ? ?. The domain where ? and ? are defined is therefore ? = ?,?,? ?0 ? ?, < ? ? < + . Now, we will present the convergence theorem. Theorem 4.1. Let ? ?,?,? and ? ?,?,? belong to ?1? and suppose that the partial derivatives of ? and ? be bounded on ?. Then for arbitrary fixed 0 ? 1 the Modified Simpson approximations of Eq. (3.12) converge to the exact solutions ? ?;? , ? ?;? for ?,? ?3?0,? . Proof. It is sufficient to show lim 0? ?, ;? = ? ?;? , lim By Taylor s theorem, we have ? ??+1;? = ? ?? 1;? + 3 + ? ??+1, ? ??;? + ? ??, ? ??;? ,? ??;? ),? ??;? + ? ??, ? ??;? ,? ??;? + 3 6? 3.12 0? ?, ;? = ? ?;? . ? ?? 1, ? ?? 1;? ,? ?? 1;? + 2? ??, ? ??;? ,? ??;? + 2? ??, ? ??;? ,? ??;? ????+1,?3 5 90?4?1,? ?1 ?2,? ?2 , 25

and ? ??+1;? = ? ?? 1;? + 3 ? ?? 1, ? ?? 1;? ,? ?? 1;? + 2? ??, ? ??;? ,? ??;? + 2? ??, ? ??;? ,? ??;? + 3 6? ?2,? ?2 + ? ??+1, ? ??;? + ? ??, ? ??;? ,? ??;? ),? ??;? + ? ??, ? ??;? ,? ??;? ????+1,?3 5 90? 4 ?1,? ?1 , where ?? 1 ?1 ??+1, ?? ?2 ??+1 and ?3 is in between ? ??;? + ? ??,? ??;? and ? ??;? + ? ??,? ??;? + 2 2 ? ?2,? ?2 . Consequently 26

? ??+1;? ? ??+1;? = ? ?? 1;? ? ?? 1;? + +2 3 +2 3 + 3 3? ?? 1, ? ?? 1;? ,? ?? 1;? ? ?? 1, ? ?? 1;? ,? ?? 1;? ? ??, ? ??;? ,? ??;? ? ??, ? ??;? ,? ??;? ? ??, ? ??;? ,? ??;? ? ??, ? ??;? ,? ??;? ? ??+1,???;? + ? ??, ? ??;? ,? ??;? ,???;? + ? ??, ? ??;? ,? ??;? + 3? ? , ? ??+1,???;? + ? ??, ? ??;? ,? ??;? ,???;? + ? ??, ? ??;? ,? ??;? 27

? ??+1;? ? ??+1;? = ? ?? 1;? ? ?? 1;? + +2 3 +2 3 + 3 3? ?? 1, ? ?? 1;? ,? ?? 1;? ? ?? 1, ? ?? 1;? ,? ?? 1;? ? ??, ? ??;? ,? ??;? ? ??, ? ??;? ,? ??;? ? ??, ? ??;? ,? ??;? ? ??, ? ??;? ,? ??;? ? ??+1,???;? + ? ??, ? ??;? ,? ??;? ,???;? + ? ??, ? ??;? ,? ??;? + 3? ? , ? ??+1,???;? + ? ??, ? ??;? ,? ??;? ,???;? + ? ??, ? ??;? ,? ??;? 28

where ??= ? ? ,? ? ????+1,?3 2 ????+1,?3 2 1 6? ,1 6? ?2,? ?2 4 90?4?1,? ?1 = ?2,? ?2 90? ?1,? ?1 . Set ???;? = ???;? ???;? , ???;? = ? ??;? ? ??;? . Then ??+1?;? +2 ?1 +4 ?2 ?? 1?;? max ?0 ? ? ?? 1?;? , ?? 1?;? max ?0 ? ? ???;? , ???;? 3 3 +4 ?3 +2 ?4 + 3?, max ?0 ? ? ???;? , ???;? max ?0 ? ? ???;? , ???;? 3 3 ??+1?;? +2 ?5 +4 ?6 ?? 1?;? max ?0 ? ? ?? 1?;? , ?? 1?;? max ?0 ? ? ???;? , ???;? 3 3 +4 ?7 +2 ?8 + 3?, max ?0 ? ? ???;? , ???;? max ?0 ? ? ???;? , ???;? 29 3 3

where ? and ? are upper bound for ? ? , ? ? respectively. 3 Set ? = max ?1,?2,?3,?4,?5,?6,?8 < 2 , then ??+1?;? + ??+1?;? +4 ?1 +8 ?2 ?? 1?;? + ?? 1?;? max ?0 ? ? +4 ?4 ?? 1?;? , ?? 1?;? max ?0 ? ? ???;? , ???;? 3 3 +8 ?3 + 3? + ? max ?0 ? ? ???;? , ???;? max ?0 ? ? ???;? , ???;? 3 3 +4 ?1 +8 ?2 ?? 1?;? + ?? 1?;? ?? 1?;? + ?? 1?;? ???;? + ???;? 3 3 +8 ?3 +4 ?4 + 3? + ? . ???;? + ???;? ???;? + ???;? 3 3 30

Ifsetting ???;? + ???;? , we obtain, = ???;? +4 ? +8 ? +8 ? +4 ? + 3? + ? ??+1?;? ?? 1?;? ?? 1?;? ???;? ???;? ???;? 3 3 3 3 = 1 +4 ? +20 ? + 3? + ? . 3) ?? 1?;? ???;? lemma 4.2 3 where ? = max ?,? . By using lemma 4.2 we have: ??+1+ ? ? ?? ???1+ ? ? ?0 + ??? 1 ? 1, ?2+4?+? 2 , ? = 1 +4 ? 3, ? =20 ? 3 and ? = 3?, Because of ?0= ?0= 0 and ?1= ?1= 0, for ? = ? 1, where ? = we have 31

0??1?1+ ? ? ?0 + ???1 1 lim = 0, ? 1 therefore, we have lim 0??+ ? ? ?? 1 = 0 and consequently lim and the proof is completed. 0??= 0, in other words, lim 0??= 0 0??= lim Remark 3.1. Above theorem results that convergence order is ? 2. 32

5. Numerical Results Example 5.1. Consider the initial value problem ? ? = ? ? + ? + 1, ? 0 = 0.96 + 0.04?,1.01 0.01? , ? 0.01 = 0.01 + 0.985 + 0.015? ? 0.01 1 ? 0.025?0.01,0.01 + 0.985 + 0.015? ? 0.01+ 1 ? 0.025?0.01, the exact solution at ? = 0.1 for 0 ? 1 is given by ? 0.1;? = 0.1 + 0.985 + 0.015? ? 0.1 1 ? 0.025?0.1,0.1 + 0.985 + 0.015? ? 0.1+ 1 ? 0.025?0.1. A comparison between the approximate solutions by Modified Simpson rule, ? ?;? , Trapezoidal rule, ??????;? , Midpoint rule, ?????;? , and the exact solution, ? ?;? at ? = 0.1 with ? = 10, is shown in Table 5.1 and Figure 5.1. 33

Table 5.1. 34

Figure 5.1. ()Exact solution, ( ) Simpson, ( ) Trapezoidal and ( ) Midpoint approximated points. 35

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

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