AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 713892 10.1155/2014/713892 713892 Research Article Approximating the Solution of the Linear and Nonlinear Fuzzy Volterra Integrodifferential Equations Using Expansion Method Allahviranloo T. Abbasbandy S. Hashemzehi S. Ezzati Reza 1 Department of Mathematics Science and Research Branch Islamic Azad University Tehran Iran azad.ac.ir 2014 642014 2014 26 01 2014 19 02 2014 6 4 2014 2014 Copyright © 2014 T. Allahviranloo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present research study introduces an innovative method applying power series to solve numerically the linear and nonlinear fuzzy integrodifferential equation systems. Finally, it ends with some examples supporting the idea.

1. Introduction

Fuzzy integrodifferential equations have attracted great interests in recent years since they play a major role in different areas of theory such as control theory. For the first time, Chang and Zadeh have introduced fuzzy numbers as well as the related arithmetic operations [1, 2]. Furthermore, applying the operators on fuzzy numbers has been developed by Mizumoto and Tanaka . It should be mentioned that the concept of LR fuzzy numbers was expressed by Dubois and Prade . In this regard, they made a significant contribution by providing a computational formula for operations on fuzzy numbers. After that, the notation of fuzzy derivative was presented by Seikkala . However, Goetschel, Jr., and Voxman have proposed the Riemann integral-type approach . Some mathematicians have separately worked on the existence and having a unique solution of fuzzy Volterra integrodifferential equation . Recently, numerical methods have been applied to solve the linear as well as nonlinear differential equation fuzzy integral equation and fuzzy integrodifferential equation [7, 8, 1013].

In this paper, we use the power series method of the exact solution of linear or nonlinear fuzzy integrodifferential equations, which is obtained by recursive procedure as follows.

We consider the following system of fuzzy integrodifferential equations: (1) X ~ ( s ) = G ( s , X ~ ( s ) ) 0 s k ( s , t , X ~ ( t ) , X ~ ( t ) ) d t , with initial condition F ~ ( 0 ) = a ~ , and (2) X ~ =    [ x ~ 1 , x ~ 2 , , x ~ n ] T , G ~ = [ g ~ 1 , g ~ 2 , , g ~ n ] T , v v v v v v K ~ = [ k ~ i j ] , i , j = 1,2 , , n , a ~ = [ a ~ 1 , a ~ 2 , , a ~ n ] T .

In (1), G ~ and K ~ are given fuzzy functions and, also, a ~ is fuzzy vector and vector fuzzy function X ~ is solution of (1), which will be determined.

2. Basic Concepts

Here basic definitions of a fuzzy number are given as follows .

Let E be a set of all triangular fuzzy numbers.

Definition 1.

An arbitrary fuzzy number u ~ E in the parametric form is represented by an ordered pair of functions ( u _ , u ¯ ) which satisfy the following requirements.

u ¯ : r u _ ( r ) is a bounded left-continuous nondecreasing function over [ 0,1 ] .

u _ : r u ¯ ( r ) is a bounded left-continuous nonincreasing function over [ 0,1 ] .

u _ u ¯ ,          0 r 1 .

Definition 2.

For arbitrary fuzzy numbers u ~ , v ~ E , one uses the distance (Hausdorff metric)  (3) D ( u ( r ) , v ( r ) ) = max { sup r [ 0,1 ] | u _ ( r ) - v _ ( r ) | , sup r [ 0,1 ] | u ¯ ( r ) - v ¯ ( r ) | } , and it is shown in  that ( E , D ) is a complete metric space and the following properties are well known: (4) D ( u ~ + w ~ , v ~ + w ~ ) = D ( u ~ , v ~ ) , u ~ , v ~ E , D ( k u ~ , k v ~ ) = | k | D ( u ~ , v ~ ) , k , u ~ , v ~ E , D ( u ~ + v ~ , w ~ + e ~ ) D ( u ~ , w ~ ) + D ( v ~ , e ~ ) , u ~ , v ~ , w ~ , e ~ E .

Definition 3.

A triangular fuzzy number is defined as a fuzzy set in E , which is specified by an ordered triple u = ( a , b , c ) 3 with a b c such that [ u ] r = [ u _ ( r ) , u ¯ ( r ) ] are the endpoints of r -level sets for all r [ 0,1 ] , where u _ ( r ) = a + ( b - a ) r and u ¯ ( r ) = c - ( c - b ) r . Here, u _ ( 0 ) = a ,    u ¯ ( 0 ) = c ,    u _ ( 1 ) = u ¯ ( 1 ) = b , which is denoted by u 1 .

Definition 4.

A fuzzy number A ~ is of L R type if there exist shape functions L (for left), R (for right), and scalar α 0 ,    β 0 with (5) μ ~ A ( x ) = { L ( a - x α ) x a , R ( x - a β ) x a ; the mean value of A ~ , a is a real number, and α , β are called the left and right spreads, respectively. A ~ is denoted by ( a , α , β ) .

Definition 5.

Let M ~ = ( m , α , β ) L R , N ~ = ( n , γ , δ ) L R , and λ + . Then, (6) ( 1 ) λ M ~ = ( λ m , λ α , λ β ) L R ( 2 ) λ M ~ = ( - λ m , λ β , λ α ) L R ( 3 ) M ~ N ~ = ( m + n , α + γ , β + δ ) L R ( 4 ) M ~ N ~ { ( m n , m γ + n α , m δ + n β ) L R M ~ , N ~ > 0 ( m n , n α - m δ , n β - m γ ) L R M ~ > 0 , N ~ < 0 ( m n , - n β - m δ , - n α - m γ ) L R M ~ , N ~ < 0 .

Definition 6.

The integral of a fuzzy function was defined in  by using the Riemann integral concept.

Let f : [ a , b ] E 1 , for each partition P = { t 0 , t 1 , , t n } of [ a , b ] and for arbitrary ξ i [ t i - 1 , t i ] , 1 i n , and suppose (7) R p = i = 1 n f ( ξ i ) ( t i - t i - 1 ) , Δ : = max { | t i - t i - 1 | , 1 i n } . The definite integral of f ( t ) over [ a , b ] is (8) a b f ( t ) d t = lim Δ 0 R p , provided that this limit exists in the metric D .

If the fuzzy function f ( t ) is continuous in the metric D , its definite integral exists , and also (9) ( a b f ( t , r ) d t _ ) = a b f _ ( t , r ) d t , ( a b f ( t , r ) d t ¯ ) = a b f ¯ ( t , r ) d t .

Definition 7 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

Consider x ~ , y ~ E . If there exists z ~ E such that x ~ = y ~ + z ~ , then z ~ is called the H-difference of x ~ and y ~ and is denoted by x ~ y ~ .

Proposition 8 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

If f ~ : ( a , b ) E is a continuous fuzzy-valued function, then g ~ ( x ) = a x f ~ ( t ) is differentiable, with derivative g ~ ( x ) = f ~ ( x ) .

Definition 9 (see [<xref ref-type="bibr" rid="B24">20</xref>]).

Let f : R E be a fuzzy valued function. If, for arbitrary fixed t 0 R and ϵ > 0 , a δ > 0 such that (10) | t - t 0 | < δ d ( f ( t ) , f ( t 0 ) ) < ϵ , f is said to be continuous.

Definition 10 (see [<xref ref-type="bibr" rid="B27">21</xref>]).

Let be f ~ : ( a , b ) E and x 0 ( a , b ) . One says that f ~ is differentiable at x 0 if

( 1 )    there exists an element f ~ ( x 0 ) E such that, for all h > 0 sufficiently near to 0 , there are f ~ ( x 0 + h ) f ~ ( x 0 ) ,      f ~ ( x 0 ) f ~ ( x 0 - h ) , and the limits (11) lim h 0 + f ~ ( x 0 + h ) f ~ ( x 0 ) h = lim h 0 + f ~ ( x 0 ) f ~ ( x 0 - h ) h = f ~ ( x 0 ) or

( 2 ) there exists an element f ~ ( x 0 ) E such that, for all h < 0 , sufficiently near to 0, there are f ~ ( x 0 + h ) f ~ ( x 0 ) , f ~ ( x 0 ) f ~ ( x 0 - h ) , and the limits (12) lim h 0 - f ~ ( x 0 + h ) f ~ ( x 0 ) h = lim h 0 - f ~ ( x 0 ) f ~ ( x 0 - h ) h = f ~ ( x 0 ) .

Lemma 11 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

For x ~ 0 R the fuzzy differential equation (13) y ~ = f ~ ( x , y ) , y ~ ( x 0 ) = y ~ 0 E , where f ~ : R × E E is supposed to be continuous, if equivalent to one of the integral equations: (14) y ~ ( x ) = y ~ 0 + x 0 x f ( t , y ~ ( t ) ) , x [ x 0 , x 1 ] or (15) y ~ ( x ) = y ~ 0 + ( - 1 ) x 0 x f ( t , y ~ ( t ) ) , x [ x 0 , x 1 ] .

On some interval ( x 0 , x 1 ) under the differentiability condition, (i) or (ii), respectively.

Definition 12 (see [<xref ref-type="bibr" rid="B23">22</xref>]).

For fuzzy number u ~ ( r ) = ( u _ ( r ) , u ¯ ( r ) ) ,    0 r 1 , one writes (1) u ~ > 0 , if u _ ( r ) > 0 , (2) u ~ 0 , if u _ ( r ) 0 , (3) u ~ < 0 , if u ¯ < 0 , and (4) u ~ 0 , if u ¯ 0 .

Theorem 13 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

Let c E and g : ( a , b ) R . If g is differentiable on x 0 , then the function f : ( a , b ) R , defined by f ( x ) = c g ( x ) , x ( a , b ) , is differentiable on x 0 and one has f ( x 0 ) = c g ( x 0 ) .

Corollary 14 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

Let c E and g : ( a , b ) R . And define f : ( a , b ) R by f ( x ) = c g ( x ) , x ( a , b ) . If g is differentiable on ( a , b ) and g is differentiable on x 0 ( a , b ) , then f is differentiable on ( a , b ) and twice differentiable on x 0 , with f ′′ ( x 0 ) = c g ′′ ( x 0 ) .

Remark 15 (see [<xref ref-type="bibr" rid="B11">14</xref>]).

In general, if the above g is n - 1 times differentiable on ( a , b ) and g n - 1 is differentiable on x 0 , then f ( x ) = c g ( x ) is differentiable of order n on x 0 and f n ( x 0 ) = c g n ( x 0 ) .

Theorem 16 (see [<xref ref-type="bibr" rid="B27">21</xref>]).

Let f ~ : ( a , b ) E be a function and denote [ f ~ ( x ) ] r = [ f _ ( r ) , f ¯ ( r ) ] , for each r [ 0,1 ] . Then one has the following.

If f ~ is differentiable in the first form (Definition 10), then f _ , f ¯ are differentiable functions and (16) [ f ~ ( x ) ] r = [ f _ ( x ) , f ¯ ( x ) ] .

If f ~ is differentiable in the second form (Definition 10), then f _ , f ¯ are differentiable functions and (17) [ f ~ ( x ) ] r = [ f ¯ ( x ) , f _ ( x ) ] .

3. Approximation Based on the Expansion Method

Since s is positive so all derivatives of X ~ ( s ) in (18) are in case (i) in Definition 10.

Suppose the solution of the system of fuzzy integrodifferential equations (1) is as follows: (18) X ~ i ( s ) = j = 0 m e ~ i j s j , i = 1,2 , , n , where e ~ i j E , for all i = 1,2 , , n . By using initial conditions, we have (19) e ~ i 0 = x ~ i ( 0 ) , i = 1,2 , , n .

The coefficients of (18) are computed step by step. Firstly, the solution of problem (1) is considered as (20) X ~ ( s ) = e ~ 0 e ~ 1 s ,

where e ~ i j ,    i = 1,2 , , n , and e ~ 1 are unknown. With derivative of (20) we have X ~ ( s ) = e ~ 1 and by substituting X ~ ( s ) , (20) into (1), we have (21) e ~ 1 = G ( s , e ~ 0 e ~ 1 s ) 0 s k ( s , t , e ~ 0 e ~ 1 s , e ~ 1 ) d t , where by integration and sort of terms of above equation we obtain the following system: (22) ( A 1 e ~ 1 b ~ 1 ) Q ~ 1 ( s ) = 0 ~ ,

where A 1 is n × n constant matrix, b ~ 1 is n × 1 fuzzy vector, Q ~ 1 ( s ) = [ q ~ i 1 ( s ) ] ,    i = 1,2 , , n , and q ~ i 1 ( s ) are polynomials of order equal or greater than 1. If s = 0 by neglecting Q ~ 1 ( s ) , we have fuzzy linear equations system of e ~ 1 . By solving this system, the coefficient of x ~ in (20) can be determined.

In the second step, we assume that (23) X ~ ( s ) = e ~ 0 e ~ 1 s e ~ 2 s 2 ,

where e ~ 0 and e ~ 1 are known and e ~ 2 is unknown. With derivative of (23) we have

case (1), if e ~ 1 > 0 : X ~ ( s ) = e ~ 1 2 e ~ 2 s ;

case (2), if    e ~ 1 < 0 : X ~ ( s ) = e ~ 1 2 e ~ 2 s ,

and by substituting X ~ ( s ) , (23) into (1), we have (24) e ~ 1 2 e ~ 2 s = G ( s , e ~ 0 e ~ 1 s e ~ 2 s 2 ) 0 s k (    s 2 , e ~ 1 2 e ~ 2 s s , t , e ~ 0 e ~ 1 s e ~ 2 v    s 2 , e ~ 1 2 e ~ 2 s ) d t or (25) e ~ 1 2 e ~ 2 s = G ( s , e ~ 0 e ~ 1 s e ~ 2 s 2 ) 0 s k (    s 2 , e ~ 1 2 e ~ 2 s s , t , e ~ 0 e ~ 1 s e ~ 2    s 2 , e ~ 1 2 e ~ 2 s ) d t , where by integration and sort of terms of above equation we obtain the following system: (26) ( A 2 e ~ 2 b ~ 2 ) Q ~ 2 ( s ) = 0 ~ , where A 2 is n × n constant matrix, b ~ 2 is n × 1 fuzzy vector A 1 , and Q ~ 2 ( s ) = [ q ~ i 2 ( s ) ] , i = 1,2 , , n , and q ~ i 2 ( s ) are polynomials of order greater than unity, where by neglecting Q ~ 2 ( s ) , we have again fuzzy system of linear equations of e ~ 2 and by solving this system, coefficients of s 2 in (23) can be determined. This procedure can be repeated till the arbitrary order coefficients of power series of the solution for the problem are obtained.

The following theorem shows convergence of the method. Without loss of generality, we prove it for n = 1 .

Theorem 17.

Let X ~ = F ~ ( s ) be the exact solution of the following fuzzy integrodifferential equation: (27) X ~ ( s ) = G ( s , X ~ ( s ) ) 0 s k ( s , t , X ~ ( t ) , X ~ ( t ) ) d t , X ~ ( 0 ) = a ~ .

Assume that f ~ ( s ) has a power series representation. Then, (28) lim m f ~ m ( s ) = f ~ ( s ) .

Proof.

According to the proposed method, we assume that the approximate solution to (27) is as follows: (29) f ~ m ( s ) = e ~ 0 e ~ 1 s e ~ 2 s 2 e ~ m s m .

Hence, it is sufficient that we only prove (30) if: e ~ m > 0 , e ~ m = f ~ ( m ) ( 0 ) m ! , if: e ~ m < 0 , e ~ m = f ~ ( m ) ( 0 ) m ! for m = 1,2 , 3 , .

Note that, for m = 0 , the initial condition gives (31) e ~ 0 = f ~ ( 0 ) = a ~ .

Moreover, for m = 1 , if we set X ~ = f ~ ( s ) and s = 0 in (27), we obtain (32) f ~ ( 0 ) = g ( 0 , f ~ ( 0 ) ) 0 ~ .

On the other hand, from (29) and (31), we have (33) f ~ 1 ( s ) = e ~ 0 e ~ 1 s .

By substituting (33) into (27) and setting s = 0 ,we get (34) if: e ~ 1 > 0 , e ~ 1 = g ( 0 , f ~ ( 0 ) ) 0 ~ = f ~ ( 0 ) , if: e ~ 1 < 0 , e ~ 1 = g ( 0 , f ~ ( 0 ) ) 0 ~ = f ~ ( 0 ) .

For m = 2 , differentiating (27) with respect to s, we have (35) f ~ ′′ ( s ) = σ σ s g ( s , f ~ ( s ) ) σ σ s g ( s , f ~ ( s ) ) f ~ ( s ) K ( s , f ~ ( s ) , f ~ ( s ) ) 0 s σ σ s K ( s , t , f ~ ( t ) , f ~ ( t ) ) d t .

Setting s = 0 in (35), we get (36) f ~ ′′ ( 0 ) = σ σ s g ( 0 , f ~ ( 0 ) ) σ σ s g ( 0 , f ~ ( 0 ) ) f ~ ( 0 ) K ( 0 , f ~ ( 0 ) , f ~ ( 0 ) ) .

According to (29), (31), and (34), let (37) f ~ 2 ( s ) = f ~ ( 0 ) f ~ ( 0 ) s e ~ 2 s 2 .

By substituting (37) into (35) and setting s = 0 , we obtain (38) if: e ~ 2 > 0 ,    e ~ 1 > 0 , 2 e ~ 2 = σ σ s g ( 0 , e ~ 0 ) σ σ s g ( 0 , e ~ 0 ) e ~ 1 K ( 0 , e ~ 0 , e ~ 1 ) if: e ~ 2 > 0 ,    e ~ 1 < 0 , 2 e ~ 2 = σ σ s g ( 0 , e ~ 0 ) σ σ s g ( 0 , e ~ 0 ) ( e ~ 1 ) K ( 0 , e ~ 0 , e ~ 1 ) if: e ~ 2 < 0 ,    e ~ 1 > 0 , 2 e ~ 2 = σ σ s g ( 0 , e ~ 0 ) σ σ s g ( 0 , e ~ 0 ) e ~ 1 K ( 0 , e ~ 0 , e ~ 1 ) if: e ~ 2 < 0 ,    e ~ 1 < 0 , 2 e ~ 2 = σ σ s g ( 0 , e ~ 0 ) σ σ s g ( 0 , e ~ 0 ) ( e ~ 1 ) K ( 0 , e ~ 0 , e ~ 1 ) .

So, with comparison (36) and (38), we conclude that (39) 2 e ~ 2 = f ~ ′′ ( 0 ) e ~ 2 = f ~ ′′ ( 0 ) 2 ! or (40) 2 e ~ 2 = f ~ ′′ ( 0 ) e ~ 2 = f ~ ′′ ( 0 ) 2 ! . By constituting the above procedure, we can easily prove (30) for m = 3,4 , .

4. Numerical Illustrations Example 1.

Consider the following system of fuzzy linear Volerra integrodifferential equations: (41) x ~ 1 ( s ) = g 1 ( s , x ~ 1 ( s ) , x ~ 2 ( s ) ) 0 s k 1 ( s , t , x ~ 1 ( t ) , x ~ 2 ( t ) ) d t , x ~ 2 ( s ) = g 2 ( s , x ~ 1 ( s ) , x ~ 2 ( s ) ) 0 s k 2 ( s , t , x ~ 1 ( t ) , x ~ 2 ( t ) ) d t , with initial conditions (42) x ~ 1 ( 0 ) = ( 1 + 0.5 ( r - 1 ) , 1 - 0.5 ( r - 1 ) ) , x ~ 2 ( 0 ) = ( - 1 + 0.5 ( r - 1 ) , - 1 - 0.5 ( r - 1 ) ) , g 1 ( s , x ~ 1 ( s ) , x ~ 2 ( s ) ) = 1 + s + s 2 x ~ 2 ( s ) , g 2 ( s , x ~ 1 ( s ) , x ~ 2 ( s ) ) = - 1 - s x ~ 1 ( s ) , k 1 ( s , t , x ~ 1 ( t ) , x ~ 2 ( t ) ) = x ~ 1 ( t ) x ~ 2 ( t ) , k 2 ( s , t , x ~ 1 ( t ) , x ~ 2 ( t ) ) = x ~ 1 ( t ) x ~ 2 ( t ) .

From the initial conditions (43) e ~ 0 = [ 1 + 0.5 ( r - 1 ) , 1 - 0.5 ( r - 1 ) - 1 + 0.5 ( r - 1 ) , - 1 - 0.5 ( r - 1 ) ] T .

Let the solution of (41)be (44) x ~ 1 ( s ) = e ~ 10 e ~ 11 s = [ 1 + 0.5 ( r - 1 ) , 1 - 0.5 ( r - 1 ) ] e ~ 11 s , x ~ 2 ( s ) = e ~ 20 e ~ 21 s = [ - 1 + 0.5 ( r - 1 ) , - 1 - 0.5 ( r - 1 ) ] e ~ 21 s .

For obtaining e ~ 11 , e ~ 21 , we substitute (44) into (41); then we will have (45) ( e _ 11 - 2 - 0.5 ( r - 1 ) ) + ( - s - s 2 + e ¯ 21 s + ( 1 - r ) s - e ¯ 11 s 2 2 - e _ 21 s 2 2 ) = 0 , ( e ¯ 11 - 2 + 0.5 ( r - 1 ) ) + ( - s - s 2 + e _ 21 s + ( 1 - r ) s -    e _ 11 s 2 2 - e ¯ 21 s 2 2 ) = 0 , ( e _ 21 - 0.5 ( r - 1 ) ) + ( 3 s - e _ 11 s + e ¯ 11 s 2 2 + e _ 21 s 2 2 ) = 0 , ( e ¯ 21 + 0.5 ( r - 1 ) ) + ( 3 s - e ¯ 11 s + e _ 11 s 2 2 + e ¯ 21 s 2 2 ) = 0 , v v v v v ( e _ 11 - 2 - 0.5 ( r - 1 ) ) + q _ 11 ( s ) = 0 , v v v v ( e ¯ 11 - 2 + 0.5 ( r - 1 ) ) + q ¯ 11 ( s ) =    0 , ( e _ 21 - 0.5 ( r - 1 ) ) + q _ 21 ( s ) = 0 , ( e ¯ 21 + 0.5 ( r - 1 ) ) + q ¯ 21 ( s ) = 0 ,

where q _ 11 ( s ) ,    q ¯ 11 ( s ) ,    q _ 21 ( s ) ,    q ¯ 21 ( s ) are O ( s ) and by neglecting them, we have (46) A 1 e ~ 1 = b ~ 1 ,

where (47) A 1 = [ 1 0 0 1 ] , b ~ 1 = [ ( 2 + 0.5 ( r - 1 ) , 2 - 0.5 ( r - 1 ) ) ( 0.5 ( r - 1 ) , - 0.5 ( r - 1 ) ) ] , e ~ 1 = [ ( e _ 11 , e ¯ 11 ) ( e _ 21 , e ¯ 21 ) ] . So, (48) e ~ 1 = [ ( 2 + 0.5 ( r - 1 ) , 2 - 0.5 ( r - 1 ) ) ( 0.5 ( r - 1 ) , - 0.5 ( r - 1 ) ) ] .

And then (49) x ~ 1 ( s ) = [ 1 + 0.5 ( r - 1 ) , 1 - 0.5 ( r - 1 ) ] [ 2 + 0.5 ( r - 1 ) , 2 - 0.5 ( r - 1 ) ] s , x ~ 2 ( s ) = [ - 1 + 0.5 ( r - 1 ) , - 1 - 0.5 ( r - 1 ) ] [ 0 . 5 ( r - 1 ) , - 0.5 ( r - 1 ) ] s .

We go to next step. Let (50) v v v v v x ~ 1 ( s ) = [ 1 + 0.5 ( r - 1 ) , 1 - 0.5 ( r - 1 ) ] [ 2 + 0.5 ( r - 1 ) , 2 - 0.5 ( r - 1 ) ] s e ~ 12 s 2 , x ~ 2 ( s ) = [ - 1 + 0.5 ( r - 1 ) , - 1 - 0.5 ( r - 1 ) ] [ 0.5 ( r - 1 ) , - 0.5 ( r - 1 ) ] s e ~ 22 s 2 . Similar to previous step, by substituting (50) into (41), we have (51) ( 2 e _ 12 - 1.5 ( r - 1 ) - 1 ) s + ( e _ 22 s 2 - ( r - 1 ) s 2 2 + e ¯ 12 s 3 3 + e ¯ 22 s 3 3 ) = 0 , ( 2 e ¯ 12 + 1.5 ( r - 1 ) - 1 ) s + ( e _ 22 s 2 + ( r - 1 ) s 2 2 + e _ 12 s 3 3 + e _ 22 s 3 3 ) = 0 , ( 2 e _ 22 - 1.5 ( r - 1 ) + 1 ) s + ( s 2 - e _ 22 s 2 + ( r - 1 ) s 2 2 + e ¯ 12 s 3 3 + e ¯ 22 s 3 3 ) = 0 , ( 2 e ¯ 22 + 1.5 ( r - 1 ) + 1 ) s + ( s 2 + e _ 22 s 2 + ( r - 1 ) s 2 2 + e _ 12 s 3 3 + e _ 22 s 3 3 ) = 0 . So, (52) ( 2 e _ 12 - 1.5 ( r - 1 ) - 1 ) s + q _ 12 ( s ) = 0 , ( 2 e ¯ 12 + 1.5 ( r - 1 ) - 1 ) s + q ¯ 12 ( s ) = 0 , ( 2 e _ 22 - 1.5 ( r - 1 ) + 1 ) s + q _ 22 ( s ) = 0 , ( 2 e ¯ 22 + 1.5 ( r - 1 ) + 1 ) s + q ¯ 22 ( s ) = 0 .

By neglecting q _ 12 ( s ) ,   q ¯ 12 ( s ) ,   q _ 22 ( s ) ,   q ¯ 22 ( s ) which are O ( s 2 ) and solve system A 2 e ~ 2 = b ~ 2 , we obtain (53) e ~ 2 = [ ( 3 4 ( r - 1 ) + 1 2 , - 3 4 ( r - 1 ) + 1 2 ) ( 3 4 ( r - 1 ) - 1 2 , - 3 4 ( r - 1 ) - 1 2 ) ]

and then in a similar way go to next step and we have (54) ( 3 e _ 13 - 5 4 ( r - 1 ) - 0.5 ) s 2 + q _ 13 ( s ) = 0 , ( 3 e ¯ 13 - 5 4 ( r - 1 ) - 0.5 ) s 2 + q ¯ 13 ( s ) = 0 , ( 3 e _ 23 - 5 4 ( r - 1 ) + 0.5 ) s 2 + q _ 23 ( s ) = 0 , ( 3 e ¯ 23 + 5 4 ( r - 1 ) + 0.5 ) s 2 + q ¯ 23 ( s ) = 0 .

And (55) e ~ 3 = [ ( 5 12 ( r - 1 ) + 1 6 , - 5 12 ( r - 1 ) + 1 6 ) ( 5 12 ( r - 1 ) + 1 6 , - 5 12 ( r - 1 ) + 1 6 ) ] .

Example 2.

As second example we consider the following nonlinear fuzzy integrodifferential equation: (56) x ~ ( s ) = ( 1,0.1,0.4 ) 0 s x ~ ( t ) x ~ ( t ) d t , x ~ ( 0 ) = ( 0,0.2,0.6 ) .

Typically, we use the power series method for obtaining the solution of problem. From the initial condition,   e ~ ( 0 ) = ( 0,0.2,0.6 ) , let the solution of (56) be the form (57) x ~ ( s ) = e ~ 0 e ~ 1 s = ( 0,0.2,0.6 ) e ~ 1 s .

For obtaining e ~ 1 = ( m , α , β ) , we substitute (57) into (56); we will have (58) ( m , α , β ) = ( 1 + m s 2 2 , 0.1 + α s 2 2 + 0.2 s , 0.4 + β s 2 2 + 0.3 s ) , or (59) ( m , α , β ) = ( 1 + q 1 ( s ) , 0.1 + q 2 ( s ) , 0.4 + q 3 ( s ) ) .

By neglecting q 1 , q 2 , q 3 which are O ( s ) , we obtain e ~ 1 = ( 1,0.1,0.4 ) and then (60) x ~ ( s ) = ( 0,0.2,0.6 ) ( 1,0.1,0.4 ) s .

For the next step, we assume that (61) x ~ ( s ) = e ~ 0 e ~ 1 s e ~ 2 s 2 .

By substituting (61) into (56), we have (62) ( 1 + 2 m s , 0.1 + α s , 0.4 + β s ) = ( 1 + s 2 2 , 0.1 + 0.2 s + q 1 ( s ) , 0.4 + 0.6 s + q 2 ( s ) ) .

From above relation and by neglecting s 2 / 2 ,    q 1 ( s ) ,    q 2 ( s ) , we have e ~ 2 = ( 0,0.1,0.3 ) .

By repeating this method, we can compute more coefficients of the solution.

Example 3.

Consider the following nonlinear fuzzy integrodifferential equation: (63) x ~ ( s ) = e s - 1 3 e 3 s + ( 1 3 , 0.1,0.3 ) 0 s x ~ 3 ( t ) d t , x ~ ( 0 ) = ( 1,0.4,0.4 ) .

Again, we use the power series method for obtaining the solution of the problem. From the initial condition, e ~ 0 = ( 1 , 0.4 , 0.4 ) . Assume that the solution of (63) is the form (64) x ~ ( s ) = e ~ 0 e ~ 1 s = ( 1,0.4,0.4 ) e ~ 1 s .

By substituting (64) into (63), we obtain (65) ( m , α , β ) = ( 1 + q 1 ( s ) , 0.4 + q 2 ( s ) , 0.4 + q 3 ( s ) ) . By neglecting q 1 ( s ) ,    q 2 ( s ) ,    q 3 ( s ) , we obtain e ~ 1 = ( 1,0.4,0.4 ) and then (66) x ~ ( s ) = ( 1,0.4,0.4 ) ( 1,0.4,0.4 ) s For the next step, we assume that (67) x ~ ( s ) = ( 1,0.4,0.4 ) ( 1,0.4,0.4 ) s e ~ 2 s 2 .

And by substituting it into (63), we have (68) ( 1 + 2 m s , 0.4 + 2 α s , 0.4 + 2 β s ) = ( 1 + s + q 1 ( s ) , 0.4 + 0.6 s + q 2 ( s ) , 0.4 + 0.6 s + q 3 ( s ) ) . From the above relation and by neglecting q 1 ( s ) ,    q 2 ( s ) ,    q 3 ( s ) , we have (69) e ~ 2 = ( 1 2 , 0.3,0.3 ) . By continuing this procedure, more coefficients of the solution can be computed.

5. Conclusion

In summary, this study has exploited power series to find a numerical solution for linear as well as nonlinear fuzzy Volterra integrodifferential equations. In effect, using power series can provide an approximate solution for the mentioned integral equations. Since there are challenging issues to solve the nonlinear integrodifferential equations, the presented method can be simply applied to find an appropriate solution for this kind of equations that is regarded as a considerable benefit of this method undoubtedly.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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