Approximating the Solution of the Linear and Nonlinear Fuzzy Volterra Integrodifferential Equations Using Expansion Method

Abstract

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 [3]. It should be mentioned that the concept of LR fuzzy numbers was expressed by Dubois and Prade [4]. 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 [5]. However, Goetschel, Jr., and Voxman have proposed the Riemann integral-type approach [6]. Some mathematicians have separately worked on the existence and having a unique solution of fuzzy Volterra integrodifferential equation [7–9]. 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, 10–13].

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: with initial condition , and

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

2. Basic Concepts

Here basic definitions of a fuzzy number are given as follows [14–19].

Let be a set of all triangular fuzzy numbers.

Definition 1. An arbitrary fuzzy number in the parametric form is represented by an ordered pair of functions which satisfy the following requirements.(i) is a bounded left-continuous nondecreasing function over .(ii) is a bounded left-continuous nonincreasing function over .(iii).

Definition 2. For arbitrary fuzzy numbers , one uses the distance (Hausdorff metric) [6] and it is shown in [6] that is a complete metric space and the following properties are well known:

Definition 3. A triangular fuzzy number is defined as a fuzzy set in , which is specified by an ordered triple with such that are the endpoints of -level sets for all , where and . Here, , which is denoted by .

Definition 4. A fuzzy number is of type if there exist shape functions (for left), (for right), and scalar with the mean value of , is a real number, and , are called the left and right spreads, respectively. is denoted by .

Definition 5. Let , , and . Then,

Definition 6. The integral of a fuzzy function was defined in [6] by using the Riemann integral concept.
Let , for each partition of and for arbitrary , , and suppose The definite integral of over is provided that this limit exists in the metric .

If the fuzzy function is continuous in the metric , its definite integral exists [17], and also

Definition 7 (see [14]). Consider . If there exists such that , then is called the H-difference of and and is denoted by .

Proposition 8 (see [14]). If is a continuous fuzzy-valued function, then is differentiable, with derivative .

Definition 9 (see [20]). Let be a fuzzy valued function. If, for arbitrary fixed and , a such that is said to be continuous.

Definition 10 (see [21]). Let be and . One says that is differentiable at if
there exists an element such that, for all sufficiently near to , there are ,  , and the limits or
there exists an element such that, for all , sufficiently near to 0, there are , and the limits

Lemma 11 (see [14]). For the fuzzy differential equation where is supposed to be continuous, if equivalent to one of the integral equations: or
On some intervalunder the differentiability condition, (i) or (ii), respectively.

Definition 12 (see [22]). For fuzzy number , one writes (1) , if , (2) , if , (3) , if , and (4) , if .

Theorem 13 (see [14]). Let and . If is differentiable on , then the function , defined by ,, is differentiable on and one has .

Corollary 14 (see [14]). Let and . And define by ,. If is differentiable on and is differentiable on , then is differentiable on and twice differentiable on , with .

Remark 15 (see [14]). In general, if the above is times differentiable on and is differentiable on , then is differentiable of order on and .

Theorem 16 (see [21]). Let be a function and denote , for each . Then one has the following.(i)If is differentiable in the first form (Definition 10), then , are differentiable functions and (ii)If is differentiable in the second form (Definition 10), then , are differentiable functions and

3. Approximation Based on the Expansion Method

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

Suppose the solution of the system of fuzzy integrodifferential equations (1) is as follows: where , for all . By using initial conditions, we have

The coefficients of (18) are computed step by step. Firstly, the solution of problem (1) is considered as

where , and are unknown. With derivative of (20) we have and by substituting , (20) into (1), we have where by integration and sort of terms of above equation we obtain the following system:

where is constant matrix, is fuzzy vector, , and are polynomials of order equal or greater than 1. If by neglecting , we have fuzzy linear equations system of . By solving this system, the coefficient of in (20) can be determined.

In the second step, we assume that

where and are known and is unknown. With derivative of (23) we have case (1),; case (2),,

and by substituting , (23) into (1), we have or where by integration and sort of terms of above equation we obtain the following system: where is constant matrix, is fuzzy vector , and ,, and are polynomials of order greater than unity, where by neglecting , we have again fuzzy system of linear equations of and by solving this system, coefficients of 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 .

Theorem 17. Let be the exact solution of the following fuzzy integrodifferential equation:

Assume that has a power series representation. Then,

Proof. According to the proposed method, we assume that the approximate solution to (27) is as follows:
Hence, it is sufficient that we only prove for .
Note that, for , the initial condition gives
Moreover, for , if we set and in (27), we obtain
On the other hand, from (29) and (31), we have
By substituting (33) into (27) and setting ,we get
For , differentiating (27) with respect to s, we have
Setting in (35), we get
According to (29), (31), and (34), let
By substituting (37) into (35) and setting , we obtain
So, with comparison (36) and (38), we conclude that or By constituting the above procedure, we can easily prove (30) for .

4. Numerical Illustrations

Example 1. Consider the following system of fuzzy linear Volerra integrodifferential equations: with initial conditions
From the initial conditions
Let the solution of (41)be
For obtaining ,, we substitute (44) into (41); then we will have
where are and by neglecting them, we have
where So,
And then
We go to next step. Let Similar to previous step, by substituting (50) into (41), we have So,
By neglecting ,  ,  ,   which are O and solve system , we obtain
and then in a similar way go to next step and we have
And

Example 2. As second example we consider the following nonlinear fuzzy integrodifferential equation:

Typically, we use the power series method for obtaining the solution of problem. From the initial condition,  , let the solution of (56) be the form

For obtaining , we substitute (57) into (56); we will have or

By neglecting ,, which are , we obtain and then

For the next step, we assume that

By substituting (61) into (56), we have

From above relation and by neglecting , we have .

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

Example 3. Consider the following nonlinear fuzzy integrodifferential equation:

Again, we use the power series method for obtaining the solution of the problem. From the initial condition, . Assume that the solution of (63) is the form

By substituting (64) into (63), we obtain By neglecting , we obtain and then For the next step, we assume that

And by substituting it into (63), we have From the above relation and by neglecting , we have 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.