The current research attempts to offer a new method for solving fuzzy linear Volterra integral equations system. This method converts the given fuzzy system into a linear system in crisp case by using the Taylor expansion method. Now the solution of this system yields the unknown Taylor coefficients of the solution functions. The proposed method is illustrated by an example and also results are compared with the exact solution by using computer simulations.
Many mathematical formulations of physical phenomena contain integral equations. These equations appear in physics, biological models, and engineering. Since these equations are usually difficult to solve explicitly, so it is required to obtain approximate solutions. In recent years, numerous methods have been proposed for solving integral equations. For example, Tricomi, in his book [
In this paper, we want to propose a new numerical approach to approximate the solution of a fuzzy linear Volterra integral equations system. This method converts the given fuzzy system that supposedly has a unique fuzzy solution, into crisp linear system. For this scope, first, the Taylor expansions of unknown functions are substituted in parametric form of the given fuzzy system. Then we differentiate both sides of the resulting integral equations of the system
In this section, the most basic used notations in fuzzy calculus and integral equations are briefly introduced. We started by defining the fuzzy number.
A fuzzy number is a fuzzy set there are real numbers
The set of all fuzzy numbers (as given by Definition
A fuzzy number
A popular fuzzy number is the triangular fuzzy number
We briefly mentioned fuzzy number operations that have had been defined by the extension principle [
The above operations on fuzzy numbers are numerically performed on level sets (i.e.,
For arbitrary fuzzy numbers
Let
The basic definition of integral equation is given in [
The Fredholm integral equation of the second kind is
The second kind fuzzy linear Volterra integral equations system is in the form
Now let
Let us first recall the basic principles of the Taylor polynomial method for solving Fredholm’s fuzzy integral equations system (
In this section, we proved that the above numerical method converges to the exact solution of fuzzy system (
Let the kernel be bounded and belong to
Consider the system (
In this section, we present an example of fuzzy linear Volterra integral equations system and results will be compared with the exact solution.
Consider the system of fuzzy linear Volterra integral equations with
Fuzzy integral equations systems, which have a very important place in physics and engineering, are usually difficult to solve analytically. Therefore, it is required to obtain approximate solutions. In this study, mechanization of solving fuzzy linear Volterra integral equations system of the second kind by using the Taylor expansion method have proposed. This Taylor method transforms the given problem to a linear algebraic system in crisp case. The solution of the resulting system is used to compute unknown Taylor coefficients of the solution functions. Consider that to get the best approximating solutions of the given fuzzy equations, the truncation limit