Iterative method for approximate solution of fuzzy

ARTICLE IN PRESS beni-suef university journal of basic and applied sciences ■■ (2016) ■■–■■

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Iterative method for approximate solution of fuzzy integro-differential equations

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Mahmood Otadi *, Maryam Mosleh Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

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A R T I C L E

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I N F O

A B S T R A C T

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Article history:

In this paper, we interpret a nonlinear fuzzy Fredholm integro-differential equations by using

Received 4 July 2016

the strongly generalized differentiability concept. Based on the parametric form of a fuzzy

Received in revised form 23 October

number, a fuzzy integro-differential equation converts to two systems of integro-differential

2016

equations in the crisp case. Also, we use the parametric form of fuzzy number, and an it-

Accepted 4 November 2016

erative approach for obtaining approximate solution for a class of nonlinear fuzzy Fredholm

Available online

integro-differential equation of the second kind is proposed. This paper presents a method

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based on Newton–Cotes methods with positive coefficient. Then we obtain approximate so-

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Q3 Keywords: Nonlinear fuzzy integro-differential


equations


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© 2016 Production and hosting by Elsevier B.V. on behalf of Beni-Suef University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

Newton–Cotes methods


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lution of the nonlinear fuzzy integro-differential equations by an iterative approach.

licenses/by-nc-nd/4.0/).

Parametric form of a fuzzy number

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1.

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Introduction

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The study of fuzzy integral equations (FIE) forms a suitable setting for mathematical modeling of real-world problems in which uncertainties or vagueness pervade. The topics of fuzzy integral equations which is a growing interest for some time, in particular in relation to fuzzy control, have been rapidly developed in recent years. The fuzzy mapping function was introduced by Chang and Zadeh (1972). Later, Dubois and Prade (1982) presented an elementary fuzzy calculus based on the extension principle also the concept of integration of fuzzy functions. Since some of the fuzzy integral equations cannot be solved explicitly, it is often necessary to resort to numerical tech-

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niques which are appropriate combinations of numerical integration and interpolation. There are several numerical methods for solving fuzzy linear Fredholm integral equation (Abbasbandy et al., 2007; Babolian et al., 2005; Friedman et al., 1999) and fuzzy differential equation (Mosleh and Otadi, 2012a, 2012b). Hosseini Fadravi et al. (2014) considered solutions of fuzzy Fredholm integral equations using neural networks. Based on the parametric form of a fuzzy number, a Fredholm fuzzy integral equation converts to two systems of integral equations of the second kind in the crisp case. In their approach a Q4 growing neural network as the approximate solution of the integral equations, for which the activation functions are logsigmoid and linear functions. Recently Otadi and Mosleh (2016) introduced new generalized interval-valued fuzzy linear Fredholm integral equation concepts. The work of this paper

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* Corresponding author. Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran. E-mail address: [email protected], [email protected] (M. Otadi). http://dx.doi.org/10.1016/j.bjbas.2016.11.008 2314-8535/© 2016 Production and hosting by Elsevier B.V. on behalf of Beni-Suef University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Mahmood Otadi, Maryam Mosleh, Iterative method for approximate solution of fuzzy integro-differential equations, Beni-Suef University Journal of Basic and Applied Sciences (2016), doi: 10.1016/j.bjbas.2016.11.008

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ARTICLE IN PRESS 2

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beni-suef university journal of basic and applied sciences ■■ (2016) ■■–■■

is an expansion of the research of real fuzzy linear Fredholm integral equations. In this paper interval-valued fuzzy neural network (IVFNN) can be trained with crisp and intervalvalued fuzzy data. In this paper, a novel hybrid method based on IVFNN and Newton–Cotes methods with positive coefficient for the solution of interval-valued fuzzy linear Fredholm integral equations (IVFLFIEs) of the second kind is presented. Within this paper the fuzzy neural network model is used to obtain an estimate for the fuzzy parameters in a statistical sense. Also Mosleh and Otadi (2016) proved some results concerning the existence of solution for a class of nonlinear fuzzy Fredholm integro-differential equations. Also an iterative approach to obtain approximate solution for a class of nonlinear fuzzy Fredholm integro-differential equation of the second kind is proposed. But they considered the H-derivative of fuzzy numbers. While finding an approximate solution for the nonlinear fuzzy integro-differential equations

A crisp number r is simply represented by u (α ) = u (α ) = r, 0 ≤ α ≤ 1 . The set of all the fuzzy numbers is denoted by E1. This fuzzy number space as shown in Congxin and Ming (1991) can be embedded into the Banach space B = C [0, 1] × C [0, 1] . For arbitrary u = (u (r ) , u (r )) , v = ( v (r ) , v (r )) and k ∈ ℝ we define addition and multiplication by k as

(u + v)(r ) = (u (r ) + v (r )) , (u + v)(r ) = (u (r ) + v (r )) , ku ( r ) = ku ( r ) , ku ( r ) = ku ( r ) , if k ≥ 0, ku (r ) = ku (r ) , ku (r ) = ku (r ) , if k < 0. Definition 3. For arbitrary fuzzy numbers u,v, we use the distance (Goetschel and Vaxman, 1986)

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ii. u (r ) is a bounded monotonically decreasing, left continuous function on (0,1] and right continuous at 0; iii. u (r ) ≤ u (r ) , 0 ≤ r ≤ 1 .

b

X′ ( s) = y ( s) + ∫ k ( s, t, X ( t )) dt, a

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is difficult. In this paper, using strongly generalized differentiability, we present a novel and very simple numerical method based on iterative methods for solving nonlinear fuzzy Fredholm integrodifferential equations of the second kind. The strongly generalized differentiability was introduced in Bede and Gal (2004) and studied in Bede and Gal (2005). This concept allows us to resolve the above-mentioned shortcoming. Indeed, the strongly generalized derivative is defined for a larger class of fuzzy-number-valued functions than the Hukuhara derivative. Hence, we use this differentiability concept in the present paper.

D (u, v) = sup0 ≤ r ≤1max { u ( r ) − v ( r ) , u ( r ) − v ( r ) }

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and it is shown that (E,D) is a complete metric space (Puri and Ralescu, 1986). Definition 4. Let f : [a, b] → E , for each partition P = {t0, t1, … , tn } of [a,b] and for arbitrary ξi ∈[ti−1, ti ], 1 ≤ i ≤ n suppose

R p = ∑ i = 1 f (ξi ) ( ti − ti − 1 ), n

Δ := max { ti − ti − 1 , i = 1, 2, … , n} .

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The definite integral of f(t) over [a,b] is bs_bs_query

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2.

Preliminaries

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In this section the basic notations used in fuzzy operations are introduced. We start by defining the fuzzy number.

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Definition 1. A fuzzy number is a fuzzy set u : → I = [0, 1] such that 1

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i. u is upper semi-continuous; ii. u(x) = 0 outside some interval [a,d]; iii. there are real numbers b and c, a ≤ b ≤ c ≤ d, for which 1. u(x) is monotonically increasing on [a,b], 2. u(x) is monotonically decreasing on [c,d], 3. u ( x ) = 1, b ≤ x ≤ c .

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The set of all the fuzzy numbers (as given in Definition 1) is denoted by E (Klir et al., 1997). An alternative definition which yields the same E1 is given by Kaleva (1987).

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Definition 2. A fuzzy number u is a pair (u, u ) of functions u (r ) and u (r ), 0 ≤ r ≤ 1, which satisfy the following requirements:

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f ( t ) dt = limΔ → 0R p

a

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b

i. u (r ) is a bounded monotonically increasing, left continuous function on (0,1] and right continuous at 0;

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provided that this limit exists in the metric D (Friedman et al., 1999). If the fuzzy function f(t) is continuous in the metric D, its definite integral exists (Goetschel and Vaxman, 1986) and also, Q5

(∫

b

a

)

b

f ( t; r ) dt = ∫ f ( t; r ) dt,

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a

b ⎛ b ⎞ ⎜⎝ ∫a f ( t; r ) dt⎟⎠ = ∫a f ( t; r ) dt.

Definition 5. Let u,v ∈ E. If there exists w ∈ E, such that u = v + w, then w is called the H-difference of u,v and it is denoted u ⊖ v. In this paper the sign ⊖ always stands for the H-difference, and let us remark that u v ≠ u + ( −1) v . Usually we denote u + (−1)v by u − v, while u ⊖ v stands for the H-difference. In what follows, we fix T = (a,b), for a,b ∈ ℝ. Definition 6. Let F:T → E be a fuzzy function. We say F is differentiable at t0 ∈ T if there exists an element F′(t0) ∈ E such that the limits

Please cite this article in press as: Mahmood Otadi, Maryam Mosleh, Iterative method for approximate solution of fuzzy integro-differential equations, Beni-Suef University Journal of Basic and Applied Sciences (2016), doi: 10.1016/j.bjbas.2016.11.008

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limh→ 0+

ferentiable functions and D2F (t ) = ( F′ (t; r ) , F ′ (t;, r )).

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exist and are equal to F′(t0).

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Proof. See Chalco-Cano and Roman-Flores (2008). □


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The above definition is a straightforward generalization of the Hukuhara differentiability of a set-value function. From proposition 4.2.8 in Diamond and Kloeden (1994), it follows that a Hukuhara differentiable function has increasing length of support. Note that this definition of a derivative is very restrictive (Bede and Gal, 2005). The authors of Bede and Gal (2005) introduced a more general definition of a derivative for a fuzzynumber-valued function. In this paper we consider the following Q6 definition (Chalco-Cano and Roman-Flores, 2008):

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3.

Fuzzy integro-differential equation

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In this section, we consider the nonlinear Fredholm integrodifferential equations of the second kind b

X′ ( s) = y ( s) + ∫ k ( s, t, X ( t )) dt, a

X ( s0 ) = X0,

(1)

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Definition 7. Let F:T→E. Fix t0 ∈ T. We say F is differentiable at t0, if there exists an element F′(t0) ∈ E such that

where k is an arbitrary given kernel function and y(s) is a given function of s ∈ [a,b] and X0 is a fuzzy number.

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(1) for all h > 0 sufficiently close to 0, there exist F (t0 + h) F (t0 ) , F (t0 ) F (t0 − h) and the limits (in the metric D)

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limh→ 0+

F ( t0 + h) F ( t0 ) F ( t0 ) F ( t0 − h) = limh→ 0+ = F′ ( t0 ) , h h

Definition 9. Let X:T→E be a fuzzy function such that D1X or D2X exists. If X and D1X satisfy problem (1), we say X is a (1)-solution of problem (1). Similarly, if X and D2X satisfy problem (1), we say X is a (2)-solution of problem (1).

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or (2) for all h > 0 sufficiently close to 0, there exist F (t0 + h) F (t0 ) , F (t0 ) F (t0 − h) and the limits (in the metric D)

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limh→ 0−

F ( t0 + h) F ( t0 ) F ( t0 ) F ( t0 − h) = limh→ 0− = F′ ( t0 ) . h h

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Remark 1. In the previous definition, case (1) corresponds to the H-derivative introduced in Puri and Ralescu (1983), so this differentiability concept is a generalization of the H-derivative.

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Remark 2. In Bede and Gal (2005), the authors consider four cases for derivatives. Here we only consider the two first cases of Definition 5 in Bede and Gal (2005). In the other cases, the derivative is trivial because it is reduced to a crisp element (more precisely, F′ ∈ ℝ; for details see Theorem 7 in Bede and Gal (2005)).

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Definition 8. Let F:T→E. We say F is (1)-differentiable on T if F is differentiable in the sense (1) of Definition 7 and its derivative is denoted D1F, and similarly for (2)-differentiability we have D2F.

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The principal properties of defined derivatives are well known and can be found in Bede and Gal (2005), Chalco-Cano and Roman-Flores (2008), and Kaleva (2006). In this paper, we make use of the following Theorem (Chalco-Cano and Roman-Flores, 2008).

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Theorem 1. Let F:T→E and put F (t ) = ( F (t; r ) , F (t; r )) for each 0 ≤ r ≤ 1.

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(i) If F is (1)-differentiable then F (t; r ) and F (t; r ) are differentiable functions and D1F (t ) = ( F ′ (t; r ) , F′ (t; r )).

For solving Eq. (1) we may replace Eq. (1) by a system of integral equations. Let X (s) = ( X (s, r ) , X (s, r )). If X(s) is (1)-differentiable then Eq. (1) translates into the following system:

⎧X′ ( s) = y ( s) + b k ( s, t, X ( t )) dt = y ( s) + b F ( s, t, X, X ) dt, ∫a ∫a ⎪ ⎪⎪ X ( s0 ) = X0, ⎨ b b ⎪X′ ( s) = y ( s) + ∫ k ( s, t, X ( t )) dt = y ( s) + ∫ G ( s, t, X, X ) dt, a a ⎪ ⎪⎩ X ( s0 ) = X0.

⎧X′ ( s) = y ( s) + k ( s, t, X ( t )) dt = y ( s) + F ( s, t, X, X ) dt, ∫a ∫a ⎪ ⎪⎪ X ( s0 ) = X0, ⎨ b b ⎪X′ ( s) = y ( s) + ∫ k ( s, t, X ( t )) dt = y ( s) + ∫ G ( s, t, X, X ) dt, a a ⎪ ⎪⎩ X ( s0 ) = X0.

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(2) 263

Also, if X(s) is (2)-differentiable then Eq. (1) translates into the following system: b

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(ii) If F is (2)-differentiable then F (t; r ) and F (t; r ) are dif-

F ( t0 + h) F ( t0 ) F ( t0 ) F ( t0 − h) , and limh→ 0+ h h

b

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(3)

The parametric forms of (2) and (3) are given by

⎧X′ ( s; r ) = y ( s; r ) + b F ( s, t, X ( t; r ) , X ( t; r )) dt, X ( s0; r ) = X0 ( r ) , ∫a ⎪ (4) ⎨ b ⎪X′ ( s; r ) = y ( s; r ) + ∫ G ( s, t, X ( t; r ) , X ( t; r )) dt, X ( s0; r ) = X0 ( r ) , ⎩ a

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⎧X′ ( s; r ) = y ( s; r ) + b F ( s, t, X ( t; r ) , X ( t; r )) dt, X ( s0; r ) = X0 ( r ) , ∫a ⎪ (5) ⎨ b ⎪X′ ( s; r ) = y ( s; r ) + ∫ G ( s, t, X ( t; r ) , X ( t; r )) dt, X ( s0; r ) = X0 ( r ) ⎩ a

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for r ∈ [0,1]. In most cases, however, analytical solution to (4) and (5) may not be found and a numerical approach must be considered.

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4.

The numerical approach

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We replace the interval [a,b] by a set of discrete equally spaced grid points

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a = s0 < s1 < … < sN = b

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at

which

the

exact

solutions

( X2 (s; r ), X2 (s; r ))

( X1 (s; r ), X1 (s; r ))

and

are approximated by some ( x1 (s; r ) , x1 (s; r )) and ( x2 (s; r ) , x2 (s; r )), respectively. The exact and approximate solutions at si, 0 ≤ i ≤ N are denoted by X1i (r ) = ( X1i (r ) , X1i (r )) , , X2i (r ) = ( X2i (r ) , X2i (r )) , x1i (r ) = ( x1i (r ) , x1i (r )) and x2i (r ) = ( x2i (r ) , x2i (r )) , respectively. The grid points at which the solution is calculated are

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si = s0 + ih, h = ( b − a) N ; 1 ≤ i ≤ N.

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The first-order approximation of X′ (s; r ) and X′ (s; r ) is given

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by

(

)

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(11) 320

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Z ( s + h; r ) − Z ( s; r ) Z′ ( s; r ) ≈ h

(6)

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where Z(s;r) is X (s; r ) and X (s; r ) alternatively. By virtue of Eq. (6) and Eqs. (4) and Eqs. (5) we obtain

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( (

) )

⎧X1 ( r ) = X1 ( r ) + h ⎡ yi ( r ) + b F si , t, X1 ( t; r ) , X1 (t; r ) dt ⎤ + O ( h2 ) , ∫a i ⎣⎢ ⎦⎥ ⎪ i +1 b ⎪ 2 ⎡ ⎨X1 i + 1 ( r ) = X1 i ( r ) + h ⎣⎢ yi ( r ) + ∫a G si, t, X1 ( t; r ) , X1 ( t; r ) dt ⎤⎦⎥ + O ( h ) , ⎪ ⎪X1 ( s0; r ) = X0 ( r ) , X1 ( s0; r ) = X0 ( r ) , i = 0, 1, … , N − 1, ⎩ (7)

( (

) )

⎧X2 ( r ) = X2 ( r ) + h ⎡ yi ( r ) + F si, t, X2 ( t; r ) , X2 ( t; r ) dt ⎤ + O ( h2 ) , i ∫a ⎣⎢ ⎦⎥ ⎪ i +1 b ⎪ ⎡ = + + X r X r h y r G s , t , X t ; r , X t ; r dt ) 2 ( ) ⎤⎥⎦ + O (h2 ) , 2i( ) 2( ⎨ 2 i +1 ( ) i ⎢⎣ i ( ) ∫a ⎪ ⎪X2 ( s0; r ) = X0 ( r ) , X2 ( s0; r ) = X0 ( r ) , i = 0, 1, … , N − 1, ⎩ (8) b

where X0 is an initial value. The Newton–Cotes method (Atkinson, 1987) is given by

∫

b

a

N

Z ( t ) dt = ∑ w jZ ( t j ) + O ( h j =0

ν

)

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324

)) )

( (

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⎧x2 i + 1 ( r ) = x2 i ( r ) + h ⎡ y ( r ) + ∑ N w jF si, t j, x2 ( r ) , x2 j ( r ) ⎤ , j j=0 ⎣ i ⎦ ⎪ N ⎪ ⎡ ⎨x2 i + 1 ( r ) = x2 i ( r ) + h ⎣ yi ( r ) + ∑ j = 0 w jG si, t j, x2 j ( r ) , x2 j ( r ) ⎤⎦ , ⎪ ⎪x2 ( s0; r ) = x0 ( r ) , X2 ( s0; r ) = x0 ( r ) , i = 0, 1, … , N − 1. ⎩

}

}

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{

}

}

x1 ( s; h; r ) ⎡⎣s0, x1 0 ( r )⎤⎦ , ⎡⎣s1, x1 1 ( r )⎤⎦ , , … , ⎡⎣sN, x1 N ( r )⎤⎦ x2 ( s; h; r ) {⎡⎣s0, x2 0 ( r )⎤⎦ , ⎡⎣s1, x2 1 ( r )⎤⎦ ,} , … , ⎡⎣sN, x2 N ( r )⎤⎦} , x2 ( s; h; r ) ⎡⎣s0, x2 0 ( r )⎤⎦ , ⎡⎣s1, x2 1 ( r )⎤⎦ , , … , ⎡⎣sN, x2 N ( r )⎤⎦

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(13)

x1 ( s; h; r ) {⎡⎣s0, x1 0 ( r )⎤⎦ , ⎡⎣s1 , x1 1 ( r )⎤⎦ ,} , … , ⎡⎣sN, x1 N ( r )⎤⎦} ,

{

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(12)

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(14) 330

are the approximates to X1 (s; r ) , X1 (s; r ) , X2 (s; r ) and X2 (s; r ) , respectively, over the interval s0 ≤ s ≤ sN. Let F (s, t, u, v) and G (s, t, u, v) be the functions F and G of Eq. (2) where u and v are constants and u ≤ v. In other words F (s, t, u, v) and G (s, t, u, v) are obtained by substituting X = (u,v) in Eq. (2). The domain where F and G are defined is therefore

B = {( s, t, u, v) a ≤ s, t ≤ b, − ∞ < v < +∞, − ∞ < u ≤ v} .

331 332 333 334 335 336 337 338 339

Theorem 1. Let F (s, t, u, v) and G (s, t, u, v) belong to C1(B), let Q7 the partial derivatives of F,G be bounded over B and D ( X1 p, x1 p ) = max0≤i≤ N {D ( X1i, x1i )} , D ( X2q, x2q ) = max0≤i≤ N {D ( X2i, x2i )} . Then, for arbitrary fixed r: 0 ≤ r ≤ 1,

limh→ 0 x1 p ( r ) = X1 p ( r ) , limh→ 0 x1 p ( r ) = X1 p ( r ) .

340 341 342 343 344 345 346

Proof. Let

where Z is F and G alternatively and ν depends on the used method of Newton–Cotes with positive coefficient for estimating the integral in Eq. (9). By virtue of Eq. (9) and Eqs. (7) and Eqs. (8) we obtain

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) )

⎧x1 ( r ) = x1 ( r ) + h ⎡ y ( r ) + ∑ N w jF si, t j, x1 ( r ) , x1 j ( r ) ⎤ , i j j=0 ⎣ i ⎦ ⎪ i +1 N ⎪ ⎡ ⎨x1 i + 1 ( r ) = x1 i ( r ) + h ⎣ yi ( r ) + ∑ j = 0 w jG si, t j, x1 j ( r ) , x1 j ( r ) ⎤⎦ , ⎪ ⎪x1 ( s0; r ) = x0 ( r ) , x1 ( s0; r ) = x0 ( r ) , i = 0, 1, … , N − 1. ⎩

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(9)

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( (

321 322 323

limh→ 0 x2 q ( r ) = X2 q ( r ) , limh→ 0 x2 q ( r ) = X2 q ( r ) .

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Following Eqs. (10) and Eqs. (11) we define

The polygon curves

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313 314 315 316 317 318

)

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308 309 310 311 312

(

⎧X2 i + 1 ( r ) = X2 i ( r ) + h ⎡ y ( r ) + ∑ N w jF si, t j, X2 ( r ) , X2 j ( r ) ⎤ j j=0 ⎣ i ⎦ ⎪ ⎪+ O ( h2 ) + O ( hν + 1 ) , ⎪⎪ N ⎨X2 i + 1 ( r ) = X2 i ( r ) + h ⎡⎣ yi ( r ) + ∑ j = 0 w jG si, t j, X2 j ( r ) , X2 j ( r ) ⎤⎦ ⎪ ν +1 2 ⎪+ O (h ) + O (h ) , ⎪ ⎪⎩X2 ( s0; r ) = X0 ( r ) , X2 ( s0; r ) = X0 ( r ) , i = 0, 1, … , N − 1.

(

)

⎧X1 ( r ) = X1 ( r ) + h ⎡ y ( r ) + ∑ N w jF si, t j, X1 ( r ) , X1 j ( r ) ⎤ i j j=0 ⎣ i ⎦ ⎪ i +1 ⎪+ O ( h2 ) + O ( hν + 1 ) , ⎪⎪ N ⎨X1 i + 1 ( r ) = X1 i ( r ) + h ⎡⎣ yi ( r ) + ∑ j = 0 w jG si, t j, X1 j ( r ) , X1 j ( r ) ⎤⎦ ⎪ 2 ν +1 ⎪+ O (h ) + O (h ) , ⎪ ⎪⎩X1 ( s0; r ) = X0 ( r ) , X1 ( s0; r ) = X0 ( r ) , i = 0, 1, … , N − 1.

(

)

(

)

(

)

N X1 p ( r ) = X1 p − 1 ( r ) + h ⎡ y ( r ) + ∑ j = 0 w jF sp − 1, t j, X1 j ( r ) , X1 j ( r ) ⎤ ⎦ ⎣ p−1 + O ( h2 ) + O ( hν + 1 ) , N X1 p ( r ) = X1 p − 1 ( r ) + h ⎡ y p − 1 ( r ) + ∑ j = 0 w jG sp − 1, t j, X1 j ( r ) , X1 j ( r ) ⎤ ⎣ ⎦ + O ( h2 ) + O ( hν + 1 )

347 348 349

( (

) )

x1 p ( r ) = x1 p − 1 ( r ) + h ⎡ yp − 1 ( r ) + ∑ j = 0 w jF sp − 1, t j, x1 j ( r ) , x1 j ( r ) ⎤ , ⎦ ⎣ (16) N x1 p ( r ) = x1 p − 1 ( r ) + h ⎡ yp − 1 ( r ) + ∑ j = 0 w jG sp − 1, t j, x1 j ( r ) , x1 j ( r ) ⎤ . ⎣ ⎦ N

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(15)

and (10)

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354 355 356

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However, we should first study the conditions that guarantee the convergence of the approximate solutions.

Consequently

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X1 p ( r ) − x1 p ( r ) = X1 p − 1 ( r ) − x1 p − 1 ( r ) +

((

) (

))

h ⎡∑ j = 0 w j F sp − 1, t j, X1 j ( r ) , X1 j ( r ) − F sp − 1, t j, x1 j ( r ) , x1 j ( r ) ⎤ + ⎣ ⎦ O ( h2 ) + O ( hν + 1 ) , N

357

358 359 360

X1 p ( r ) − x1 p ( r ) = X1 p − 1 ( r ) − x1 p − 1 ( r ) +

( (

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) (

))

N h ⎡∑ j = 0 w j G sp − 1, t j, X1 j ( r ) , X1 j ( r ) − G sp − 1, t j, x1 j ( r ) , x1 j ( r ) ⎤ + ⎣ ⎦ O ( h2 ) + O ( hν + 1 ) .

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Denote W1 p = X1 p (r ) − x1 p (r ) , V1 p = X1 p (r ) − x1 p (r ). Then

W1 p ≤ W1 p − 1 + 2Lh ( b − a ) D ( X1 p, x1 p ) + O ( h ) + O ( h

ν +1

2

V1 p ≤ V1 p − 1 + 2Lh ( b − a) D ( X1 p, x1 p ) + O ( h ) + O ( h

ν +1

2

361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391

392

393

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),

D( x1(kp), x1*p ) = max0≤i≤ N {D( x1(ki ), x1*i )} , D( x2(kq), x2*q ) = max0≤i≤ N {D( x2(ki ), x2*i )} ,

398

the produced sequences x1(k) and x2(k) from the iteration process (17) and (18) tend to the exact solutions of (12) and

399 400

(13) respectively, say x1* and x2* , for any arbitrary fuzzy initial

401

vectors x1(0) and x2(0) with x1(k) (s0; r ) = x0 (r ) and x2(k) (s0; r ) = x0 (r ) for all k.

402 403 404 405 406 407

Proof. By (12) and (17) we have,

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Theorem 2. Considering assumptions of Theorem 1 and

394 395 396 397

),

(k + 1)

x1 p

where L > 0 is a bound for the partial derivatives of F,G. Thus, we have

(k )

(r ) − x1 *p (r ) ≤ x1 p − 1 (r ) − x1 *p − 1 (r ) +

(

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W1 p ≤ W10 + p2Lh ( b − a) D ( X1 p, x1 p ) + O ( h ) + O ( h

ν +1

V1 p ≤ V10 + p2Lh ( b − a) D ( X1 p, x1 p ) + O ( h2 ) + O ( hν + 1 ) .

),

(k + 1)

x1 p

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(k )

(

408

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)

(k )

h∑ j = 0 w j G sp − 1, t j, x1 j ( r ) , x1 j ( r ) − G( sp − 1, t j, x1 *j ( r ) , x1 *j ( r )) (k )

also, by (13) and (18) we have,

Since W0 = V0 = 0 we obtain

409 410 411

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(k + 1)

x2 p

W1 p ≤ p2Lh ( b − a) D ( X1 p, x1 p ) + O ( h2 ) + O ( hν + 1 ) ,

(k )

(r ) − x2 *p (r ) ≤ x2 p − 1 (r ) − x2 *p − 1 (r ) +

(

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)

V1 p ≤ p2Lh ( b − a) D ( X1 p, x1 p ) + O ( h2 ) + O ( hν + 1 )

(k ) n h∑ j = 0 w j F sp − 1, t j, x2 (jk) ( r ) , x2 j ( r ) − F( sp − 1, t j, x2 *j ( r ) , x2 *j ( r )) ,

and if h→0 we get W1 p → 0 , V1 p → 0 and also, we have

x2 (pk + 1) ( r ) − x2 *p ( r ) ≤ x2 (pk−)1 ( r ) − x2 *p − 1 ( r ) +

(

412

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)

(k )

h∑ j = 0 w j G sp − 1, t j, x2 j ( r ) , x2 j ( r ) − G( sp − 1, t j, x2 *j ( r ) , x2 *j ( r )) n

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)

(r ) − x1 *p (r ) ≤ x1 p − 1 (r ) − x1 *p − 1 (r ) +

n

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(k ) n h∑ j = 0 w j F sp − 1, t j, x1 (jk) ( r ) , x1 j ( r ) − F( sp − 1, t j, x1 *j ( r ) , x1 *j ( r )) ,

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2

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limh→ 0 x2 q ( r ) = X2 q ( r ) , limh→ 0 x2 q ( r ) = X2 q ( r ) ,

(k )

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and according to the conditions of Theorem 1,

similarity which concludes the proof. □

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So far, we came to the nonlinear equation systems (12) and (13) with a special form that lets us offer a numerical approach for obtaining the approximate solution. Iterative methods are widely used for finding approximate solution of nonlinear equation systems (Stoer and Bulirsch, 1993). The nonlinear equation systems (12) and (13) also have a structure that permits to approximate its solution by an iterative method. For this purpose, we apply a successive substitution, similar to Jacobi method of solving linear equation systems and thereby define an iterative process leading to the sequence of vectors x1 (k) , x1

(k )

(k )

x2 (k) and x2 ,

xj

(k + 1) (k + 1)

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( (

) )

(k + 1)

(k )

( (

(k )

) )

⎧x2 i + 1 ( r ) = x2 i ( r ) + h ⎡ yi ( r ) + ∑ w jF si, t j, x2 ( r ) , x2 j ( r ) ⎤ , j j=0 ⎣ ⎦ ⎪ (k ) N ⎪ (k + 1) (k ) (k ) ⎡ ⎨x2 i + 1 ( r ) = x2 i ( r ) + h ⎣ yi ( r ) + ∑ j = 0 w jG si, t j, x2 j ( r ) , x2 j ( r ) ⎤⎦ , (18) ⎪ ⎪x2 ( s0; r ) = x0 ( r ) , x2 ( s0; r ) = x0 ( r ) , i = 0, 1, … , N − 1, k = 0, 1, … , K. ⎩ N

(k )

(r ) − x j * (r ) + 2Lh (b − a) D( x jp , x*jp ),

(k )

416

Denote Wjp(k+1) = x(jpk+1) (r ) − x *jp (r ), Vjp(k+1) = x (jpk+1) (r ) − x jp (r ) , j = 1, 2.

417

(k + 1)

Wjp

(k )

(k )

≤ Wjp − 1 + 2Lh ( b − a) D( x jp , x*jp ),

Vjp(k + 1) ≤ Vjp(k−) 1 + 2Lh ( b − a ) D( x(jpk), x*jp ), j = 1, 2. Thus, we have (k + 1)

Wjp

(k )

(k )

≤ Wj 0 + 2Lph ( b − a ) D( x jp , x*jp ),

Vjp(k + 1) ≤ Vj(0k) + 2 pLh ( b − a) D( x(jpk), x*jp ), j = 1, 2. Since Wj(0k) = Vj(0k) = 0 for all k we obtain (k + 1) jp

W

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p−1

Then

where the components of the vectors satisfy the iteration formulas,

⎧x1 (k + 1) ( r ) = x1 (k) ( r ) + h ⎡ yi ( r ) + ∑ N w jF si, t j, x1 (k) ( r ) , x1 (jk) ( r ) ⎤ , i j j=0 ⎣ ⎦ ⎪ i +1 (k ) (k ) N ⎪ (k + 1) (k ) ⎡ ⎨x1 i + 1 ( r ) = x1 i ( r ) + h ⎣ yi ( r ) + ∑ j = 0 w jG si, t j, x1 j ( r ) , x1 j ( r ) ⎤⎦ , (17) ⎪ ⎪x1 ( s0; r ) = x0 ( r ) , x1 ( s0; r ) = x0 ( r ) , i = 0, 1, … , N − 1, k = 0, 1, … , K, ⎩

p−1

(k )

(r ) − x j *p (r ) ≤ x j p − 1 (r ) − x j *p − 1 (r ) + 2Lh (b − a) D( x(jpk), x*jp ), j = 1, 2.

xj p

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(r ) − x j * (r ) ≤ x j p

p

(k )

413 414 415

(k ) jp

≤ 2Lph ( b − a) D( x , x*jp ),

418 419 420 421 422 423 424 425 426 427 428

Vjp(k + 1) ≤ 2 pLh ( b − a ) D( x(jpk), x*jp ), j = 1, 2.

429

and if h→0 we get Wjp(k+1) → 0 , Vjp(k+1) → 0 for all k which con-

430

cludes the proof. □

431


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ARTICLE IN PRESS 6


1 Exact solution Approximate solution 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452

453

454

0

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1.4

1.2

1

0.8

Fig. 1 – Comparison of the exact solution and obtained solutions at s = 0.5 of example 5.1.

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5.

Numerical examples

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To illustrate the technique proposed in this paper, consider the following examples.

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Example 5.1. Consider the following fuzzy integro-differential equation

⎧x1 (i k++11) ( r ) = x1 (i k) ( r ) ⎪ tj 2 r2 ⎞ N ⎡⎛ ⎤ ⎞ ⎪ min ⎡⎣x1 2j ; r , x1 j ; r⎟ , x1 j x1 j ⎥ , + h ⎢⎜ r − ⎟ + ∑ j=0 wj ⎪ ⎠ 40 ⎠ 10 ⎣⎝ ⎦ ⎪⎪ (k + 1) (k ) ⎨ x1 i + 1 ( r ) = x1 i ( r ) ⎪ tj 2 N ⎡ ⎤ ⎪ + h ⎢ yi ( r ) + ∑ j = 0 w j max ⎡⎣x1 2j ; r , x1 j ; r , x1 j x1 j ⎥ , ⎪ 10 ⎣ ⎦ ⎪ 0 0 = 0 ; r = 0 . x ; r , x ) ) ⎪⎩ 1 ( 1(

455

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)

)

)

456

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1 t r 2 76 − 36r − r 2 ⎞ ⎛ X ′ ( s) = ⎜ r − X2 ( t ) dt, , ⎟ + ∫0 ⎝ ⎠ 40 40 10 X ( 0 ) = 0; 0 ≤ s, t ≤ 1, 0 ≤ r ≤ 1.

Using the formulation (2) we get the exact solution

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X1 ( s; r ) = rs, X1 ( s; r ) = (2 − r ) s, that is a (1)-differentiable solution of the problem (1). The parametric equations are

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1 t r2 ⎞ ⎧ ⎛ 2 2 ⎪X′ ( s; r ) = ⎜⎝ r − 40 ⎟⎠ + ∫0 10 min ⎡⎣ X ( t; r ) , X ( t; r ) , X ( t; r ) X (t; r )⎤⎦ dt, ⎪ ⎪ ⎛ 76 − 36r − r 2 ⎞ ⎪X′ ( s; r ) = ⎜ ⎟⎠ ⎝ 40 ⎨ ⎪ 1 t ⎪ +∫ max ⎡⎣ X2 (( t; r )) , X2 ( t; r ) , X (t; r ) X (t; r )⎤⎦ dt, 0 10 ⎪ ⎪⎩ X ( 0; r ) = 0, X ( 0; r ) = 0, 0 ≤ s, t ≤ 1, 0 ≤ r ≤ 1.

Following Eqs. (17) we obtain

In this example we take max0≤i≤n {D ( x1(ki +1), x1(ki ) )} < 10 −2 and a 1 . The initial partition with the discretization parameter h = 100 (0 ) vector x1 = 0 is considered for starting. The exact and obtained solution of fuzzy Fredholm integrodifferential equation in this example at s = 0.5 is shown in Fig. 1. Example 5.2. Consider the following fuzzy integro-differential equation

1 X′ ( s) = y ( s) + ∫ − X ( t ) dt, 0 2 X ( 0 ) = [ r − 1, 1 − r ]; 0 ≤ s, t ≤ 1, 0 ≤ r ≤ 1, 1

y ( s) = ( r − 1 ) e − s + ( r − 1 )

where

(e

( e −1 − 1 ) 2

and

y ( s ) = (1 − r ) e − s

−1

X2 ( s; r ) = [( r − 1) e , (1 − r ) e −s

−s

458 459 460 461 462 463 464 465 466 467

− 1) . 2 Using the formulation (3) we get the exact solution

+ (1 − r )

457

]


468 469 470 471 472 473

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ARTICLE IN PRESS beni-suef university journal of basic and applied sciences ■■ (2016) ■■–■■

7

1 Exact solution Approximate solution 0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 −0.8

474 475 476 477 478 479

480

481 482 483

484

485 486 487 488 489

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−0.6

−0.4

−0.2

0.2

0

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0.8

Fig. 2 – Comparison of the exact solution and obtained solutions at s = 0.5 of example 5.2.

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that is a (2)-differentiable solution of the problem (1). The parametric equations are

490

6.

Summary and conclusions

491 492

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(e−1 − 1) + 1 − 1 Xdt, ⎧ −s ⎪X′ ( s; r ) = ( r − 1) e + ( r − 1) ∫0 2 2 ⎪ −1 1 − e 1 1 ( ) ⎪ −s + ∫ − Xdt, ⎨X′ ( s; r ) = (1 − r ) e + (1 − r ) 0 2 2 ⎪ ⎪ X ( 0; r ) = r − 1, X ( 0; r ) = 1 − r, 0 ≤ s, t ≤ 1, 0 ≤ r ≤ 1. ⎪ ⎩ Following Eqs. (18) we obtain

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⎧ (k + 1) ⎡ (k ) (e−1 − 1) − 1 N w x , − si ∑ j 2j ⎪x2 i+1 (r ) = x1 i (r ) + h ⎢(r − 1) e + (r − 1) 2 2 j=0 ⎣ ⎪ ⎪ ⎪ ⎪ ⎪ ( e −1 − 1 ) − 1 N w x , − s)i ⎪ (k + 1) (k ) ⎨x2 i+1 (r ) = x2 i (r ) + h[(1 − r ) e + (1 − r ) ∑ j 2j 2 j=0 2 ⎪ ⎪x (0; r ) = r − 1, x (0; r ) = 1 − r, 0 ≤ r ≤ 1. 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪⎩ In this example we take max0≤i≤n {D ( x1(ki +1), x1(ki ) )} < 10 −2 and a 1 . partition with the discretization parameter h = 100 The exact and obtained solution of fuzzy Fredholm integro-differential equation in this example at s = 0.5 is shown in Fig. 2.

We proposed a general numerical procedure for treating nonlinear fuzzy Fredholm integro-differential equations of the second kind by using the strongly generalized differentiability concept. The original nonlinear fuzzy Fredholm integrodifferential equation is replaced by two parametric nonlinear Fredholm integro-differential equations which are then solved numerically using classical algorithm. In this paper the standard Newton–Cotes method is designed for approximating integral. Also we can execute this method in a computer simply.

493 494 495 496 497 498 499 500 501 502

Acknowledgements

503 504

We would like to present our sincere thanks to the Editor in Chief, Managing Editor and referees for their valuable suggestions.

505 506 507

Uncited references

Q9

508 509

Nieto et al, 2009

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REFERENCES

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514 Abbasbandy S, Babolian E, Alavi M. Numerical method for solving linear Fredholm fuzzy integral equations of the second kind. Chaos Solitons Fractals 2007;31:138–46.


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