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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.

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 [

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:

In (

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

Let

An arbitrary fuzzy number

For arbitrary fuzzy numbers

A triangular fuzzy number is defined as a fuzzy set in

A fuzzy number

Let

The integral of a fuzzy function was defined in [

Let

If the fuzzy function

Consider

If

Let

Let be

For

On some interval

For fuzzy number

Let

Let

In general, if the above

Let

If

If

Since s is positive so all derivatives of

Suppose the solution of the system of fuzzy integrodifferential equations (

The coefficients of (

where

where

In the second step, we assume that

where

case (1),

case (2),

and by substituting

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

Let

Assume that

According to the proposed method, we assume that the approximate solution to (

Hence, it is sufficient that we only prove

Note that, for

Moreover, for

On the other hand, from (

By substituting (

For

Setting

According to (

By substituting (

So, with comparison (

Consider the following system of fuzzy linear Volerra integrodifferential equations:

From the initial conditions

Let the solution of (

For obtaining

where

where

And then

We go to next step. Let

By neglecting

and then in a similar way go to next step and we have

And

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,

For obtaining

By neglecting

For the next step, we assume that

By substituting (

From above relation and by neglecting

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

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,

By substituting (

And by substituting it into (

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.

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