Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.
Fuzzy integral equations of the second kind have attracted the attention of many scientists and researchers in recent years. These equations appear frequently in fuzzy control, fuzzy finance, approximate reasoning, and economic systems [
In recent years, numerous methods have been proposed for solving Volterra integral equations [
The paper is organized as follows: in Section
The proposed methods are implemented using a numerical example with known exact solution by applying the MAPLE software in Section
A standard form of the Volterra integral equation of the second kind is given by [
The second kind fuzzy Volterra integral equations system is of the following form
This method depends on differentiating the fuzzy integral equation of the second kind
From (
To obtain the solution in the form of expression (
Using the Leibniz rule which is dealing with differentiation of product of functions, system (
Substituting (
Consequently, (
One can show that the above numerical method converges to the exact solution of the fuzzy system (
Let the kernel be bounded and belong to
This method provides a sequence of functions, which converges to the exact solution of the problem and is based on the use of Lagrange multipliers for identification of optimal value of a parameter in a functional. Consider the following general nonlinear system [
Next, we take the partial derivative to both sides of the Volterra integral equation (
If we substitute the value of the Lagrange multiplier into (
Now, using the variational iteration method and (
In virtue of (
In this section, in order to examine the accuracy of the proposed methods, we have chosen one example of linear fuzzy integral equation of the second kind. Moreover, the numerical results will be compared with the exact solution.
Consider the following fuzzy linear Volterra integral equations:
Input
Input the Taylor expansion degree
Calculate
Calculate
Calculate
Put
where
Calculate
Put
Put
Denote
Put
Put
Solve the following linear system
Estimate
We get the following results:
Exact and numerical solutions at
Absolute error between the exact and numerical solutions.
Consider the linear fuzzy Volterra integral equation (
In the view of the variational iteration method, we construct a correction functional in the following form:
Start with the initial approximation in (
Input
Input
For
Table
Absolute error between the exact and numerical solutions using the Taylor expansion method and the variational iteration method.

Exact solution 
Taylor numerical solution 
Variation numerical solution 
Exact solution 
Taylor numerical solution 
Variation numerical solution 
Error = 
Error = 

0 








0.1 








0.2 








0.3 








0.4 








0.5 








0.6 








0.7 








0.8 








0.9 








1.0 








Exact and numerical solutions at
Absolute error between the exact and numerical solutions.
In this article, Taylor expansion and variational iteration methods are proposed to solve a fuzzy linear Volterra integral equation of the second kind. The results of the example show that the convergence and accuracy of both methods were in a good agreement with the analytical solution. According to comparison of numerical results, mentioned in tables and figures, we conclude that the variational iteration method provides more accurate results and therefore is more advantageous.
The authors declare that they have no conflicts of interest.