A new efficient method using Bernoulli polynomials to solve systems of linear fuzzy Volterra integral equations

Abstract

In this work, the collocation method to solve fuzzy Volterra integral equations is introduced. Then, solving these systems by collocation method with Bernoulli polynomials is presented. Also, we will discuss the convergence of this method. In addition, the superiority of this method compared to other methods is proved, by solving of some systems of fuzzy Volterra integral equations.

Keywords

Collocation method Bernoulli polynomials systems of fuzzy Volterra integral equations convergence

1 Introduction

The differential equations and the integral equations are very important today due to their natural structure and compatibility. With a collocation method, many problems of applied mathematics, such as differential equations and integral equations, can be solved.

Brunner and Kauthen in [15, 24] have presented the collocation method to solve Fredholm-Volterra integral equations. In [5, 37] an application of Walsh functions for Fredholm-Hammerstein integro-differential equations was introduced. Fuzzy integral equations arise in many fields of electrical engineering, medical, biology, geographic, physics, social sciences, etc. Many practical problems in science and engineering can be transformed into fuzzy Volterra integral equations; thus, their solution is one of the main goals in various areas of applied sciences and engineering. The basic arithmetic structure for fuzzy numbers was later developed by Dubois and Prade [17, 18], Mizumoto and Tanaka [32, 33], and Nahmias [35]. The study of these equations began with the investigations of Kaleva [23] and then continued by Goetschel and Voxman [22], Nanda [36] and Seikkala [47] for the fuzzy Volterra integral equations that is equivalent to the initial value problem for the first order fuzzy differential equations. The collocation method based Bernoulli operational [45, 46] matrix for numerical solution of generalized pantograph equation is realized in [6]. The concept of fuzzy sets, originally introduced by Zadeh [54], led to the definition of the fuzzy number and implementation in approximate reasoning problems [53] and others.

In this work, we study the collocation methods using Bernoulli polynomials. In order to find the numerical solution of the systems of liner fuzzy Volterra integral equations, we used the method of approximation of the unknown function u (x) and v (x) in our collocation point.

This paper is organized as follows. In Section 2, introduce Bernoulli polynomials. In the third section, we present a brief summary of the main concepts of fuzzy. Section 4 is devoted to the study of fuzzy systems of Volterra integral equations. Section 5, explains the collocation method described above. In Section 6, we prove the existence of the solution for systems of linear fuzzy Volterra integral equations. In the seventh section, convergence analysis is proved. The proposed method is applied to several numerical examples in Section 8. Also, a conclusion is given in Section 9.

2 Bernoulli polynomials and some their properties

Bernoulli polynomials play an important role in various expansions and approximation formulas which are useful both in analytic theory of numbers and in classical and numerical analysis. The Bernoulli numbers have been defined here by the exponential generating function [49]: $\frac{z}{e^{z} - 1} = \sum_{k = 0}^{\infty} B_{k} \frac{z^{k}}{k!} .$ (1)

The question considered here is how to introduce a parameter into these numbers. One option is to multiply the left-hand side of by a simple function and to see what happens. Naturally, from the view point of generating functions, the simplest possible function is an exponential one. Therefore, consider the expansion $\frac{z}{e^{z} - 1} \times e^{xz} = \sum_{k = 0}^{\infty} B_{k} (x) \frac{z^{k}}{k!},$ (2) where the coefficients on the right-hand side depend on parameter x.

It follows that $B_{i} (x) = \sum_{n = 0}^{i} i n B_{n} x^{i - n},$ (3) is a polynomial in x, called the Bernoulli polynomial. The first few Bernoulli polynomials are: $\begin{matrix} B_{0} (x) & = & 1, \\ B_{1} (x) & = & x - \frac{1}{2}, \\ B_{2} (x) & = & x^{2} - x + \frac{1}{6}, \\ B_{3} (x) & = & x^{3} - \frac{3}{2} x^{2} + \frac{1}{2} x . \end{matrix}$

If, we define vector A_i+1, as $A_{i + 1} = {[(_{j}^{i}) B_{i} (_{i - 1}^{i}) B_{i - 1} (_{i - 2}^{i}) B_{i - 2} \geq \dots (_{1}^{i}) B_{1} (_{0}^{i}) B_{0} \overset{p - i}{\hat{0, \dots, 0}}]}_{1 \times (p + 1)}$ then B_i (z) = A_i+1X^T (z), for i = 0, 1, ⋯, p, where $X (z) = [\begin{matrix} 1 z z^{2} \dots z^{p} \end{matrix}] .$

Now, we can expand the matrix $B (z) = [\begin{matrix} B_{0} (z) & B_{1} (z) & \dots & B_{p} (z) \end{matrix}]$ as $B (z) = X (z) A^{T},$ (4) where

$\begin{array}{l} A = [\begin{matrix} ⋮ \end{matrix}] \\ 0 & 0 & 0 & \dots & 0 \\ (_{1}^{1}) & (_{1}^{1}) & 0 & 0 & \dots & 0 \\ (_{2}^{2}) & (_{1}^{2}) & (_{0}^{2}) & 0 & \dots & 0 \\ (_{3}^{3}) & (_{2}^{3}) & (_{1}^{3}) & (_{0}^{3}) & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ (_{p}^{p}) & (_{p - 1}^{p}) & (_{p - 2}^{p}) & (_{p - 3}^{p}) & \dots & (_{0}^{p})] (p + 1) \times (p + 1) \end{array}$

3 Preliminaries in fuzzy calculus

The set of all fuzzy numbers is denoted by E¹ and is a convex cone. An equivalent parametric definition of fuzzy numbers is given in [13 , 24–26]].

Definition 3.1. [1 , 38] An arbitrary fuzzy number $\tilde{u}$ in the parametric form is represented by an ordered pair of functions $(\underline{u}, \bar{u})$ performing the following requirements

$\bar{u} \to u_{r}^{-} \in R$ is a bounded left-continuous non-decreasing function over [0, 1].

$\underline{u} : r \to u_{r}^{+} \in R$ is a bounded left-continuous non-increasing function over [0, 1].

$\underline{u} ⩽ \bar{u}$ for all 0 ⩽ r ⩽ 1.

For arbitrary fuzzy numbers $\tilde{u} = (\underline{u} (r), \bar{u} (r))$ and $\tilde{v} = (\underline{v} (r), \bar{v} (r))$ , we define addition, subtraction, scalar multiplication and multiplication as follows

Addition: $\underline{u (r) \oplus v (r)} = \underline{u} (r) \oplus \underline{v} (r)$ and $\bar{u (r) \oplus v (r)} = \bar{u} (r) \oplus \bar{v} (r)$ .

Subtraction: $\underline{u (r) ⊖ v (r)} = \underline{u} (r) ⊖ \underline{v} (r)$ and $\bar{u (r) ⊖ v (r)} = \bar{u} (r) ⊖ \bar{v} (r)$ .

Scalar product: $t ⊙ \tilde{u} = {\begin{matrix} (t \underline{u} (r), t \bar{u} (r)), & t \geq 0, \\ (t \underline{u} (r), t \bar{u} (r)), & t < 0 . \end{matrix}$

Multiplication: $\tilde{u} ⊙ \tilde{v} = {\begin{matrix} (\underline{u . v}) (r) = \max {\underline{u} (r) . \underline{v} (r), \underline{u} (r) . \bar{v} (r), \bar{u} (r) . \underline{v} (r), \bar{u} (r) . \bar{v} (r)}, \\ (\bar{u . v}) (r) = \min {\underline{u} (r) . \underline{v} (r), \underline{u} (r) . \bar{v} (r), \bar{u} (r) . \underline{v} (r), \bar{u} (r) . \bar{v} (r)} . \end{matrix}$

Definition 3.2. [2 , 34] For arbitrary fuzzy numbers $\tilde{u} = (\underline{u} (r), \bar{u} (r))$ and $\tilde{v} = (\underline{v} (r), \bar{v} (r))$ , the quantity $\begin{matrix} D (\tilde{u}, \tilde{v}) & = & max {sup_{r \in [0, 1]} | \underline{u} (r) - \underline{v} (r) |, \\ sup_{r \in [0, 1]} | \bar{u} (r) - \bar{v} (r) |}, \end{matrix}$ is the distance between $\tilde{u}$ and $\tilde{v}$ .

This metric is equivalent to the one used by Puri and Ralescu [43] and Kaleva [23]. It is shown that (E¹, D) is a complete metric space [42, 44]. Now we follow Goestschel and Voxman [22] and define the integral of a fuzzy function using the Riemann integral concept.

Definition 3.3. [48]. A fuzzy function $\tilde{h} : [a, b] \to E^{1}$ is said to be continuous if for arbitrary fixed x₀ ∈ [a, b] and γ > 0 there exists σ > 0 such that if |x - x₀| < σ, then $D (\tilde{h} (x), \tilde{h} (x_{0})) < γ$ .

Definition 3.4. Take $\tilde{h} : [a, b] \to E^{1}$ .

Definite integral of h (x) over [a, b] is $\int_{a}^{b} h (x) d x = \lim_{A \to 0} C_{q} = Σ_{i = 0}^{n} \tilde{h} (ξ) (x_{i} - x_{i - 1})$ for each partition q = {x₀, x₁, ⋯, x_n} of [a, b] and arbitrary ξ_i : x_i-1 ⩽ ξ_i ⩽ x_i, i = 1, 2, ⋯, n and $A = \max_{1 \leq i \leq n} | x_{i} - x_{i - 1} |$ is continuous in metric D, its definite integral exists [11, 16] and also, $\begin{matrix} \underline{\int_{a}^{b} h (x; r) dx} & = & \int_{a}^{b} \underline{h} (x; r) dx, \\ \bar{\int_{a}^{b} h (x; r) dx} & = & \int_{a}^{b} \bar{h} (x; r) dx . \end{matrix}$

It should be noted that the fuzzy integral also can be defined by using the Lebesgue-type approach [10].

Definition 3.5. [3 , 52]. $\tilde{h} : [a, b] \to E^{1}$ is fuzzy-Riemann integrable to $L (\tilde{h}) \in E^{1}$ if for any δ > 0, there exists σ > 0 such that for any division W = {[f, g]; η} of [a, b] with the norms Δ (W) < σ, we have $D (\sum_{w}^{*} (g - f) ⊙ \tilde{h} (η), L (\tilde{h})) < δ,$ where $\sum_{w}^{*}$ denotes the fuzzy summation.

Lemma 3.6.[4, 41] If $\tilde{h}, \tilde{l} : [a, b] \subseteq R \to E^{1}$ is fuzzy continuous functions, then the function H : [a, b] → R₊ by

$H (x) = D (\tilde{h} (x), \tilde{l} (x))$ is continuous on [a, b], and $D (\int_{a}^{b} \tilde{h} (x) dx, \int_{a}^{b} \tilde{l} (x) dx) ⩽ \int_{a}^{b} D (\tilde{h} (x), \tilde{l} (x)) dx .$

Theorem 3.7. [14 , 31]

The pair (E¹, ⊕) is a commutative semigroup with $\tilde{0} = χ {0}$ zero element.

For fuzzy numbers which are not crisp, there is no opposite element (that is, (E¹, ⊕) cannot be a group).

The function of ||.||_F : E¹ → R by $‖ u ‖_{F} = D (\tilde{u}, \tilde{0})$ has the usual properties of norm, that is, ||u||_F = 0 if and only if $\tilde{u} = \tilde{0},$ $‖ β ⊙ \tilde{u} ‖ = | β | ‖ \tilde{u} ‖_{F}$ and $‖ \tilde{u} \oplus \tilde{v} ‖_{F} ⩽ ‖ \tilde{u} ‖_{F} + ‖ \tilde{v} ‖_{F} .$

$| ‖ \tilde{u} ‖_{F} - ‖ \tilde{v} ‖_{F} | ⩽ D (\tilde{u}, \tilde{v})$ and $D (\tilde{u}, \tilde{v}) ⩽ ‖ \tilde{u} ‖_{F} + ‖ \tilde{v} ‖_{F}$ for any $\tilde{u}, \tilde{v} \in E^{1}$ .

Definition 3.8. [28]. The fuzzy linear system $A ⊙ \tilde{x} = B ⊙ \tilde{x} \oplus \tilde{y},$ (5) where A = (a_ij), B = (b_ij), 1 ⩽ i, j ⩽ n are crisp coefficient matrices and $\tilde{y}$ a fuzzy number vector, is called dual fuzzy linear system.

Definition 3.9. [19, 20]. A fuzzy number vector $({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ given by ${\tilde{x}}_{i} = (\underline{{\tilde{x}}_{i}} (r), \bar{{\tilde{x}}_{i}} (r)),$ i = 1, 2, ⋯, n, 0 ⩽ r ⩽ 1, is called a solution of (3.5) if for all i = 1, 2, ⋯, n, we have $\begin{array}{l} \int_{a}^{b} h (x) d x = \lim_{A \to 0} C_{q} = Σ_{i = 0}^{n} \tilde{h} (ξ) (x_{i} - x_{i - 1}) & \int_{a}^{b} h (x) d x = \lim_{A \to 0} C_{q} = Σ_{i = 0}^{n} \tilde{h} (ξ) (x_{i} - x_{i - 1}) \\ {\underline{Σ_{j = 1}^{n} * a_{i j} ⊙ {\tilde{x}}_{j}} \\ = Σ_{j = 1}^{n} \underline{* a_{i j} ⊙ {\tilde{x}}_{j}} = Σ_{j = 1}^{n} \underline{* b_{i j} ⊙ {\tilde{x}}_{j}} + \underline{\tilde{y} i}, \\ \bar{Σ_{j = 1}^{n} * a_{i j} ⊙ {\tilde{x}}_{j}} \\ = Σ_{j = 1}^{n} \bar{* a_{i j} ⊙ {\tilde{x}}_{j}} = Σ_{j = 1}^{n} \bar{* b_{i j} ⊙ {\tilde{x}}_{j}} + \bar{{\tilde{y}}_{i}} . \end{array}$

4 Fuzzy systems of Volterra integral equations

The general fuzzy Volterra integral equation systems of the second kind are fuzzy integral equations for x ∈ [a, b] of the from [39, 40]: $\begin{matrix} \tilde{U} (x) = \tilde{F} (x) \oplus λ \int_{a}^{x} (K_{1} (x, t) ⊙ \tilde{u} (t) \oplus \tilde{K_{1}} (x, t) \\ ⊙ \tilde{v} (t) + \dots) dt, \\ \tilde{V} (x) = \tilde{G} (x) \oplus λ \int_{a}^{x} (K_{2} (x, t) ⊙ \tilde{u} (t) \oplus \tilde{K_{2}} (x, t) \\ ⊙ \tilde{v} (t) + \dots) dt, \\ ⋮ \end{matrix}$ (6) where $\begin{matrix} \tilde{U} (x) = (\underline{u}, \bar{u}) (x) \\ \approx (\sum_{i = 1}^{n} {\underline{u}}_{i} p_{i} (x, r), \sum_{i = 1}^{n} {\bar{u}}_{i} p_{i} (x, r)), 0 ⩽ r ⩽ 1, \\ \tilde{V} (x) = (\underline{v}, \bar{v}) (x) \\ \approx (\sum_{i = 1}^{n} {\underline{v}}_{i} p_{i} (x, r), \sum_{i = 1}^{n} {\bar{v}}_{i} p_{i} (x, r)), 0 ⩽ r ⩽ 1, \end{matrix}$ $\tilde{U} (x) = [{\tilde{u}}_{1} (x), {\tilde{u}}_{2} (x), \dots, {\tilde{u}}_{n} (x)]^{T},$ $\tilde{V} (x) = [{\tilde{v}}_{1} (x), {\tilde{v}}_{2} (x), \dots, {\tilde{v}}_{n} (x)]^{T},$ are unknown functions. Also, $\begin{matrix} {\tilde{F}}_{i} (x) & = & ({\underline{f}}_{i} (x, r), {\bar{f}}_{i} (x, r)), i = 1, 2, \dots, n, \\ \tilde{f} (x) & = & [{\tilde{f}}_{1} (x), {\tilde{f}}_{2} (x), \dots, {\tilde{f}}_{n} (x)]^{T} \\ {\tilde{G}}_{i} (x) & = & ({\underline{g}}_{i} (x, r), {\bar{g}}_{i} (x, r)), i = 1, 2, \dots, n, \\ \tilde{g} (x) & = & [{\tilde{g}}_{1} (x), {\tilde{g}}_{2} (x), \dots, {\tilde{g}}_{n} (x)]^{T} \end{matrix}$ and $\begin{matrix} K (x, t) = [k_{i, j} (x, t)], i, j = 1, 2, \dots, n \\ k_{i, j} (x, t) ⊙ {\underline{u}}_{j} (t, r) \\ = {k_{i, j} (x, t) {\underline{u}}_{j} (t, r) & k_{i, j} (x, t) ⩾ 0 \\ k_{i, j} (x, t) {\bar{u}}_{j} (t, r) & k_{i, j} (x, t) < 0 \end{matrix}$ and $\begin{matrix} k_{i, j} (x, t) ⊙ {\bar{u}}_{j} (t, r) \\ = {\begin{matrix} k_{i, j} (x, t) {\bar{u}}_{j} (t, r) & k_{i, j} (x, t) ⩾ 0 \\ k_{i, j} (x, t) {\underline{u}}_{j} (t, r) & k_{i, j} (x, t) < 0 . \end{matrix} \end{matrix}$

5 Collocation method based on Bernoulli polynomials

In this section, we use the collocation method based on Bernoulli polynomials (BP) to solve Fuzzy Systems of Volterra Integral Equations (FSVIE).

Let P (x) = [p₁ (x), p₂ (x), ⋯, p_n (x)] be a BP on x ∈ D. Then, we have

$\begin{matrix} \sum_{i = 1}^{n} a_{i} p_{i} (x) \\ = f_{i} (x) \oplus \sum_{i = 1}^{n} (λ \int_{a}^{x} (k_{1} (x_{i}, t) ⊙ a_{i} p_{i} (x) \\ \oplus {\tilde{k}}_{1} (x_{i}, t) ⊙ b_{i} p_{i} (x)) dt, \\ \sum_{i = 1}^{n} b_{i} p_{i} (x) \\ = g_{i} (x) \oplus \sum_{i = 1}^{n} (λ \int_{a}^{x} (k_{2} (x_{i}, t) ⊙ a_{i} p_{i} (x) \\ \oplus {\tilde{k}}_{2} (x_{i}, t) ⊙ b_{i} p_{i} (x)) dt . \end{matrix}$ (7)

Let us consider $\begin{matrix} u_{n} (x) & = & \sum_{i = 1}^{n} a_{i} B_{i} (x), \\ v_{n} (x) & = & \sum_{i = 1}^{n} b_{i} B_{i} (x) . \end{matrix}$

We can write the system 6 in the matrix form

$(\begin{matrix} ({\underline{f}}_{1} (x, r), {\bar{f}}_{1} (x, r)) \\ ({\underline{f}}_{2} (x, r), {\bar{f}}_{2} (x, r)) \\ ⋮ \\ ({\underline{f}}_{n} (x, r), {\bar{f}}_{n} (x, r)) \end{matrix}) = (\begin{matrix} ({\underline{u}}_{1} (x, r), {\bar{u}}_{1} (x, r)) \\ ({\underline{u}}_{2} (x, r), {\bar{u}}_{2} (x, r)) \\ ⋮ \\ ({\underline{u}}_{n} (x, r), {\bar{u}}_{n} (x, r)) \end{matrix}) - (\begin{matrix} ({\underline{L}}_{1}, {\bar{L}}_{1}) \\ ({\underline{L}}_{2}, {\bar{L}}_{2}) \\ ⋮ \\ ({\underline{L}}_{n}, {\bar{L}}_{n}) \end{matrix})$ (8) and

$\begin{matrix} (\begin{matrix} ({\underline{g}}_{1} (x, r), {\bar{g}}_{1} (x, r)) \\ ({\underline{g}}_{2} (x, r), {\bar{g}}_{2} (x, r)) \\ ⋮ \\ ({\underline{g}}_{n} (x, r), {\bar{g}}_{n} (x, r)) \end{matrix}) \\ = (\begin{matrix} ({\underline{v}}_{1} (x, r), {\bar{v}}_{1} (x, r)) \\ ({\underline{v}}_{2} (x, r), {\bar{v}}_{2} (x, r)) \\ ⋮ \\ ({\underline{v}}_{n} (x, r), {\bar{v}}_{n} (x, r)) \end{matrix}) - (\begin{matrix} ({\underline{L}}_{1}, {\bar{L}}_{1}) \\ ({\underline{L}}_{2}, {\bar{L}}_{2}) \\ ⋮ \\ ({\underline{L}}_{n}, {\bar{L}}_{n}) \end{matrix}) \end{matrix}$ (9) where for all i = 1, 2, ⋯, n, we have $\begin{matrix} ({\underline{L}}_{i}, {\bar{L}}_{i}) & = & (\sum_{j = 1}^{n} (\int_{a}^{x} (k_{i, j} (x, t) ⊙ {\underline{u}}_{j} (t, r) \oplus {\tilde{k}}_{i, j} (x, t) {\underline{v}}_{j} (t, r)) dt, \\ \sum_{j = 1}^{n} (\int_{a}^{x} (k_{i, j} (x, t) ⊙ {\bar{u}}_{j} (t, r) \oplus {\tilde{k}}_{i, j} (x, t) {\bar{v}}_{j} (t, r)) dt) . \end{matrix}$

6 Existence of the solution

In this section, we will study the solvability of the linear fuzzy Volterra integral equation systems 6. First, we introduce some simplifying notations. Given an n × m matrix (or, in particular, a vector) C with (i, j) entry c_i,j, we define |C|, the absolute value of C, to be the n × m matrix whose (i, j) entry is |c_ij|. Also, given two n × m matrices C and F, we will write C ⩽ F if and only if c_ij ⩽ f_ij for all i and j. With these definitions, we can show [[7 –9, 90]

If |C| ⩽ |F| (elementwise), then ||C||_∞ ⩽ ||F||_∞.

||C||_∞ = || |C| ||_∞.

Theorem 6.1.Let ${\tilde{f}}_{i} (x)$ and ${\tilde{g}}_{i} (x) i = 1, 2, \dots, n$ be the fuzzy continuous functions and k_i,j (x, t), i, j = 1, 2, ⋯, n be continuous for a ⩽ x, t ⩽ b, and $k_{i, j} = max_{a ⩽ x, t ⩽ b} | k_{i, j} (x, t) |$ .

Let $A_{1} = max {k_{i, j}$ , i, j = 1, 2, ⋯, n} and $Y = {\tilde{f} : [0, 1] \to E^{1} : \tilde{f} is continuous}$ be a space of fuzzy continuous functions with the metric $D^{*} (\tilde{f}, \tilde{g}) = sup_{a ⩽ x ⩽ b} D (\tilde{f} (x), \tilde{g} (x))$ (that is called the uniform distance between fuzzy number value functions). If $Θ = n (b - a) A_{1} < 1$ then the fuzzy system 6 has an unique solution ${\tilde{U}}^{*}, {\tilde{V}}^{*}$ in Yⁿ which can be obtained by the following successive approximations method $\begin{matrix} {\tilde{U}}_{0} (x) & = & \tilde{F} (x), {\tilde{V}}_{0} (x) = \tilde{G} (x) \\ {\tilde{U}}_{m} (x) & = & \tilde{F} (x) \oplus \int_{a}^{x} k_{1} (x, t) ⊙ {\tilde{U}}_{m - 1} (t) \\ \oplus {\tilde{k}}_{1} (x, t) ⊙ {\tilde{V}}_{m - 1} dt, m \geq 1, \\ {\tilde{V}}_{m} (x) & = & \tilde{G} (x) \oplus \int_{a}^{x} k_{2} (x, t) ⊙ {\tilde{U}}_{m - 1} (t) \\ \oplus {\tilde{k}}_{2} (x, t) ⊙ {\tilde{V}}_{m - 1} dt, m \geq 1 . \end{matrix}$

Moreover, the sequence of successive approximations, ${\tilde{U}}_{m}$ and ${\tilde{V}}_{m}$ converges to the solution ${\tilde{U}}^{*}$ and ${\tilde{V}}^{*}$ . Furthermore, the following error bound holds

$\begin{matrix} ‖ D ({\tilde{U}}^{*} (t), {\tilde{U}}_{m} (t)) ‖_{\infty} ⩽ \frac{Θ^{m + 1}}{1 - Θ} A_{0}, \\ ‖ D ({\tilde{V}}^{*} (t), {\tilde{V}}_{m} (t)) ‖_{\infty} ⩽ \frac{Θ^{m + 1}}{1 - Θ} B_{0}, \end{matrix}$ where $A_{0} = sup_{x \in [a, b]} ‖ D (\tilde{0}, {\tilde{f}}_{i} (x)) ‖_{\infty}, i = 1, 2, \dots, n$ and $B_{0} = sup_{x \in [a, b]} ‖ D (\tilde{0}, {\tilde{g}}_{i} (x)) ‖_{\infty}, i = 1, 2, \dots, n .$

Proof. First, note that since k_i,j (x, t), i, j = 1, 2, ⋯, n, x ∈ [a, b] is continuous for a ⩽ x, t ⩽ b and uniformly continuous with respect to t, there exists k_ij > 0 such that $k_{ij} = max_{a ⩽ x, t ⩽ b} | k_{i, j} (x, t) |$ . To prove this theorem, we investigate the conditions of the Banach fixed point principle. First, we define the operator Q : Yⁿ → Yⁿ by $\begin{matrix} Q (\tilde{U} (x)) & = & \tilde{F} (x) \oplus \int_{a}^{x} (K_{1} (x, t) ⊙ \tilde{U} (t) \\ \oplus {\tilde{K}}_{1} (x, t) ⊙ \tilde{V} (t)) dt, \end{matrix}$ for each $\tilde{U}, \tilde{V} \in Y^{n}$ . We show that Q maps Yⁿ into Yⁿ (i.e.Q (Yⁿ) ⊂ Yⁿ). To the end, we show that the operator Q is uniformly continuous. Since ${\tilde{f}}_{i} (x), i = 1, 2, \dots, n,$ is continuous on the compact set of [a, b], we deduce that it is uniformly continuous and hence for δ_i > 0, i = 1, 2, ⋯, n.

There exists σ_i > 0 such that $D ({\tilde{f}}_{i} (x_{1}), {\tilde{f}}_{i} (x_{2})) < δ_{i}, whenever | x_{1} - x_{2} | < σ_{i},$ x₁, x₂ ∈ [a, b].As described above, k_i,j (x, t), i, j = 1, 2, ⋯, n, also is uniformly continuous, thus for γ_ij > 0 there exists σ_ij > 0 such that |k_i,j (x₁, t) - k_i,j (x₂, t) | < γ_ij, whenever |x₁ - x₂| < σ_i,j, x₁, x₂ ∈ [a, b]. Take $\begin{matrix} δ^{*} & = & max {δ_{i}, i = 1, 2, \dots, n}, \\ γ^{*} & = & max {γ_{ij}, i, j = 1, 2, \dots, n}, \\ σ_{1}^{*} & = & min {σ_{i}, i = 1, 2, \dots, n} \end{matrix}$ and $σ_{2}^{*} = min {σ_{ij}, i, j = 1, 2, \dots, n} .$

Therefore $\begin{matrix} | K (x_{1}, t) - K (x_{2}, t) | \\ = (\begin{matrix} | k_{1, 1} (x_{1}, t) - k_{1, 1} (x_{2}, t) | & \dots & | k_{1, n} (x_{1}, t) - k_{1, n} (x_{2}, t) | \\ | k_{2, 1} (x_{1}, t) - k_{2, 1} (x_{2}, t) | & \dots & | k_{2, n} (x_{1}, t) - k_{2, n} (x_{2}, t) | \\ ⋮ & ⋱ & ⋮ \\ | k_{n, 1} (x_{1}, t) - k_{n, 1} (x_{2}, t) | & \dots & | k_{n, n} (x_{1}, t) - k_{n, n} (x_{2}, t) | \end{matrix}) \\ ⩽ {(\begin{matrix} γ^{*} & \dots & γ^{*} \\ γ^{*} & \dots & γ^{*} \\ ⋮ & ⋱ & ⋮ \\ γ^{*} & \dots & γ^{*} \end{matrix})}_{n \times n}, \end{matrix}$

and $\begin{matrix} D (\tilde{F} (x_{1}), \tilde{F} (x_{2})) \\ = (\begin{matrix} D ({\tilde{f}}_{1} (x_{1}), {\tilde{f}}_{1} (x_{2})) \\ D ({\tilde{f}}_{2} (x_{1}), {\tilde{f}}_{2} (x_{2})) \\ ⋮ \\ D ({\tilde{f}}_{n} (x_{1}), {\tilde{f}}_{n} (x_{2})) \end{matrix}) ⩽ {(\begin{matrix} δ^{*} \\ δ^{*} \\ ⋮ \\ δ^{*} \end{matrix})}_{n \times 1} . \end{matrix}$

So, we have $\begin{matrix} D (Q (\tilde{U}) (x_{1}), Q (\tilde{U}) (x_{2})) ⩽ D (\tilde{F} (x_{1}), \tilde{F} (x_{2})) \\ \oplus D (\int_{a}^{x} K (x_{1}, t) ⊙ \tilde{U} (t) dt, \int_{a}^{x} K (x_{2}, t) \\ ⊙ \tilde{U} (t) dt) ⩽ D (\tilde{F} (x_{1}), \tilde{F} (x_{2})) \\ + | K (x_{1}, t) - K (x_{2}, t) | \int_{a}^{x} D (\tilde{0}, \tilde{U} (t)) dt . \end{matrix}$

Take $\begin{matrix} D (\tilde{0}, \tilde{U} (t)) = {(\begin{matrix} D (\tilde{0}, {\tilde{u}}_{1} (t)) \\ D (\tilde{0}, {\tilde{u}}_{2} (t)) \\ ⋮ \\ D (\tilde{0}, {\tilde{u}}_{n} (t)) \end{matrix})}_{n \times 1}, \\ A_{2} = sup_{a ⩽ t ⩽ b} ‖ D (\tilde{0}, {\tilde{u}}_{i} (t)) ‖_{\infty}, i = 1, 2, \dots, n . \end{matrix}$

Therefore, $‖ D (Q (\tilde{U}) (x_{1}), Q (\tilde{U}) (x_{2})) ‖_{\infty} ⩽ δ^{*} + n (b - a) A_{2} γ^{*} .$

By choosing $δ^{*} = \frac{δ}{2}$ and $γ^{*} = \frac{δ}{2 n (b - a) A_{2}}$ , we derive $D (Q (\tilde{U}) (x_{1}), Q (\tilde{U}) (x_{2})) ⩽ δ .$

This shows that Q is uniformly continuous for any $\tilde{U} \in Y^{n}$ , and continuous on [a, b], and hence Q (Yⁿ) ⊂ Yⁿ. Now, we prove that the operator Q is a continuous map. So, for $\tilde{U}, \tilde{W} \subset Y^{n}$ and x ∈ [a, b], we have $\begin{matrix} D (Q (\tilde{U} (x), Q (\tilde{W} (x)) ⩽ D (\tilde{F} (x), \tilde{F} (x)) \\ + D (\int_{a}^{x} K (x, t) ⊙ \tilde{U} (t) dt, \int_{a}^{x} K (x, t) ⊙ \tilde{W} (t) dt) \\ ⩽ \int_{a}^{x} D (K (x, t) ⊙ \tilde{U} (t) dt, K (x, t) ⊙ \tilde{W} (t)) dt \\ ⩽ Z (b - a) D^{*} (\tilde{U}, \tilde{W}), \end{matrix}$ where $\begin{matrix} Z = {(\begin{matrix} A_{1} & A_{1} & \dots & A_{1} \\ A_{1} & A_{1} & \dots & A_{1} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ A_{1} & A_{1} & \dots & A_{1} \end{matrix})}_{n \times n}, \\ D^{*} (\tilde{U} (t), \tilde{W} (t)) = {(\begin{matrix} D^{*} ({\tilde{u}}_{1} (t), {\tilde{w}}_{1} (t)) \\ D^{*} ({\tilde{u}}_{2} (t), {\tilde{w}}_{2} (t)) \\ ⋮ \\ D^{*} ({\tilde{u}}_{n} (t), {\tilde{w}}_{n} (t)) \end{matrix})}_{n \times 1}, \end{matrix}$ and thus, $\begin{matrix} ‖ D^{*} (Q (\tilde{U}), Q (\tilde{W})) ‖_{\infty} \\ ⩽ n (b - a) A_{1} ‖ D^{*} (\tilde{U}, \tilde{W}) ‖_{\infty} . \end{matrix}$

Since Θ < 1, the operator Q is a contraction on the Banach space (Yⁿ, D^*). Consequently, the Banach fixed point principle implies that 6 has a unique solution ${\tilde{U}}^{*}$ in Yⁿ and the following inequality holds

$\begin{matrix} D (\tilde{U} (x), {\tilde{U}}_{m} (x)) & ⩽ & D^{*} ({\tilde{U}}^{*}, {\tilde{U}}_{m}) \\ ⩽ & \frac{Θ^{m}}{1 - Θ} D^{*} ({\tilde{U}}_{0}, {\tilde{U}}_{1}), \end{matrix}$ (10) since $\begin{matrix} D ({\tilde{U}}_{0} (x), {\tilde{U}}_{1} (x)) \\ = D (\tilde{F} (x), \tilde{F} (x) \oplus \int_{a}^{x} K (x, t) ⊙ {\tilde{U}}_{0} (t) dt) \\ ⩽ D (\tilde{F} (x), \tilde{F} (x)) \oplus D (\tilde{0}, \int_{a}^{x} K (x, t) ⊙ \tilde{F} (t) dt) \\ ⩽ \int_{a}^{x} D (\tilde{0}, K (x, t) ⊙ \tilde{F} (t)) dt \\ ⩽ | K (x, t) | \int_{a}^{x} D (\tilde{0}, \tilde{F} (t)) dt \\ ⩽ Z (\begin{matrix} \int_{a}^{x} D (\tilde{0}, {\tilde{f}}_{1} (t)) \\ \int_{a}^{x} D (\tilde{0}, {\tilde{f}}_{2} (t)) \\ ⋮ \\ \int_{a}^{x} D (\tilde{0}, {\tilde{f}}_{n} (t)) \end{matrix}) . \end{matrix}$

We have $‖ D ({\tilde{U}}_{0} (x), {\tilde{U}}_{1} (x)) ‖_{\infty} ⩽ n (b - a) A_{1} A_{0} = Θ A_{0} .$

Therefore, from 11 we have $\begin{matrix} ‖ D ({\tilde{U}}^{*} (x), {\tilde{U}}_{m} (x)) ‖_{\infty} ⩽ \frac{Θ^{m}}{1 - Θ} . \\ L A_{0} = \frac{Θ^{m} + 1}{1 - Θ} A_{0} . \end{matrix}$ □

7 Convergence analysis

In this section, we prove that the present numerical method converges to the exact solution.

Theorem 7.1.Suppose that $\tilde{u} (x)_{i, P}, \tilde{v} (x)_{i, P}$ and ${\tilde{u}}_{i} (x), i = 1, 2, \dots, n, {\tilde{v}}_{i} (x), i = 1, 2, \dots, n$ are the approximate and exact solution of system 6, respectivly. In linear system 6, if k_i,j (x, t), i, j = 1, 2, ⋯, n and a ⩽ x, t ⩽ b are bounded and continuous, then

$\tilde{u} (x)_{i, P} \to {\tilde{u}}_{i} (x), \tilde{v} (x)_{i, P} \to {\tilde{v}}_{i} (x)$ as P → ∞.

Table 1
The approximate solution and absolute error of $(\underline{u} (x, r), \bar{u} (x, r))$ and $(\underline{v} (x, r), \bar{v} (x, r))$ of Example 8.1 for N = 4

r Approximate Solution Approximate Solution Absolute Error Absolute Error

$({\underline{u}}_{4}, {\bar{u}}_{4})$ $({\underline{v}}_{4}, {\bar{v}}_{4})$ $(| \underline{u} - {\underline{u}}_{4} |, | \bar{u} - {\bar{u}}_{4} |)$ $(| \underline{v} - {\underline{v}}_{4} |, | \bar{v} - {\bar{v}}_{4} |)$

0. (-0.877584, 1.755168) (0.479424, 1.438272) (1.850488E^-6, 3.700978E^-6) (1.403340E^-6, 4.209990E^-6)

0.1 (-0.702068, 1.667410) (0.431961, 1.428683) (1.480403E^-6, 3.515930E^-6) (1.264414E^-6, 4.181920E^-6)

0.2 (-0.526551, 1.579652) (0.387375, 1.399920) (1.110320E^-6, 3.330881E^-6) (1.133897E^-6, 4.097721E^-6)

0.3 (-0.351034, 1.491893) (0.348541, 1.351980) (7.402350E^-7, 3.145834E^-6) (1.020224E^-6, 3.957383E^-6)

0.4 (-0.175517, 1.404135) (0.318338, 1.284860) (3.701506E^-6, 2.960790E^-6) (9.318133E^-7, 3.760913E^-6)

0.5 (-6.61E^-11, 1.31638) (0.299640, 1.198560) (6.612821E^-11, 2.775742E^-6) (8.770820E^-7, 3.508310E^-6)

0.6 (0.175517, 1.228618) (0.295325, 1.093090) (3.700183E^-7, 2.590690E^-6) (8.644510E^-7, 3.199573E^-6)

0.7 (0.351034, 1.140860) (0.308270, 0.968437) (7.401027E^-7, 2.405642E^-6) (9.023404E^-7, 2.834701E^-6)

0.8 (0.526551, 1.053101) (0.341350, 0.824610) (1.110187E^-6, 2.220601E^-6) (9.991703E^-7, 2.413700E^-6)

0.9 (0.702068, 0.965343) (0.397443, 0.661605) (1.480272E^-6, 2.035575E^-6) (1.163360E^-6, 1.936560E^-6)

1. (0.877584, 0.877584) (0.479424, 0.479424) (1.850356E^-6, 1.85057E^-6) (1.403331E^-6, 1.403282E^-6)

r	Approximate Solution	Approximate Solution	Absolute Error	Absolute Error
0.	(-0.877584, 1.755168)	(0.479424, 1.438272)	(1.850488E^-6, 3.700978E^-6)	(1.403340E^-6, 4.209990E^-6)
0.1	(-0.702068, 1.667410)	(0.431961, 1.428683)	(1.480403E^-6, 3.515930E^-6)	(1.264414E^-6, 4.181920E^-6)
0.2	(-0.526551, 1.579652)	(0.387375, 1.399920)	(1.110320E^-6, 3.330881E^-6)	(1.133897E^-6, 4.097721E^-6)
0.3	(-0.351034, 1.491893)	(0.348541, 1.351980)	(7.402350E^-7, 3.145834E^-6)	(1.020224E^-6, 3.957383E^-6)
0.4	(-0.175517, 1.404135)	(0.318338, 1.284860)	(3.701506E^-6, 2.960790E^-6)	(9.318133E^-7, 3.760913E^-6)
0.5	(-6.61E^-11, 1.31638)	(0.299640, 1.198560)	(6.612821E^-11, 2.775742E^-6)	(8.770820E^-7, 3.508310E^-6)
0.6	(0.175517, 1.228618)	(0.295325, 1.093090)	(3.700183E^-7, 2.590690E^-6)	(8.644510E^-7, 3.199573E^-6)
0.7	(0.351034, 1.140860)	(0.308270, 0.968437)	(7.401027E^-7, 2.405642E^-6)	(9.023404E^-7, 2.834701E^-6)
0.8	(0.526551, 1.053101)	(0.341350, 0.824610)	(1.110187E^-6, 2.220601E^-6)	(9.991703E^-7, 2.413700E^-6)
0.9	(0.702068, 0.965343)	(0.397443, 0.661605)	(1.480272E^-6, 2.035575E^-6)	(1.163360E^-6, 1.936560E^-6)
1.	(0.877584, 0.877584)	(0.479424, 0.479424)	(1.850356E^-6, 1.85057E^-6)	(1.403331E^-6, 1.403282E^-6)

Table 2

The approximate solution and absolute error of $(\underline{u} (x, r), \bar{u} (x, r))$ and $(\underline{v} (x, r), \bar{v} (x, r))$ of Example 8.2 for N = 8

r	Approximate Solution	Approximate Solution	Absolute Error	Absolute Error
	$({\underline{u}}_{8}, {\bar{u}}_{8})$	$({\underline{v}}_{8}, {\bar{v}}_{8})$	$(\| \underline{u} - {\underline{u}}_{8} \|, \| \bar{u} - {\bar{u}}_{8} \|)$	$(\| \underline{v} - {\underline{v}}_{8} \|, \| \bar{v} - {\bar{v}}_{8} \|)$
0.	(0.000000, 6.594890)	(-0.606531, 1.819591)	(0.000000, 1.932765E^-11)	(1.950110E^-12, 5.851100E^-12)
0.1	(0.131898, 6.589940)	(0.539812, 1.758332)	(0.263795, 1.931344E^-11)	(1.735612E^-12, 5.654144E^-12)
0.2	(0.197847, 6.555320)	(0.460963, 1.693434)	(0.395693, 1.921219E^-11)	(1.482092E^-12, 5.445640E^-12)
0.3	(0.197847, 6.461340)	(-0.369984, 1.621260)	(0.395693, 1.893596E^-11)	(1.189604E^-12, 5.213390E^-12)
0.4	(0.131898, 6.278330)	(-0.266873, 1.538162)	(0.263795, 1.839950E^-11)	(8.850360E^-13, 4.946270E^-12)
0.5	(0.000000, 5.976614)	(-0.151633, 1.440510)	(0.000000, 1.751577E^-11)	(4.875544E^-13, 4.632294E^-12)
0.6	(-0.197847, 5.526513)	(-0.024261, 1.324663)	(0.395693, 1.619682E^-11)	(7.796541E^-14, 4.259930E^-12)
0.7	(-0.461642, 4.898350)	(0.115241, 1.186980)	(0.923284, 1.435563E^-11)	(3.705092E^-13, 3.817170E^-12)
0.8	(-0.791386, 4.062450)	(0.266873, 1.023824)	(1.582772, 1.190514E^-11)	(8.580360E^-13, 3.292700E^-12)
0.9	(-1.187079, 2.989131)	(0.430637, 0.831554)	(2.374160, 8.760104E^-12)	(1.384615E^-12, 2.674310E^-12)
1.	(1.648721, 1.648721)	(0.606531, 0.606531)	(3.297442, 4.831690E^-12)	(1.950110E^-12, 1.950884E^-12)

Proof. We have $\begin{matrix} D ({\tilde{u}}_{i} (x), {\tilde{u}}_{i, P}) \\ = D (\sum_{j = 1}^{n} \int_{a}^{x} k_{i, j} (x, t) ⊙ {\tilde{u}}_{j} (t) dt, \\ \sum_{j = 1}^{n} \int_{a}^{x} k_{i, j} (x, t) ⊙ (\sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t)) dt) \\ ⩽ α D (\int_{a}^{x} \sum_{j = 1}^{n} {\tilde{u}}_{j} (t) dt, \\ \int_{a}^{x} \sum_{j = 1}^{n} (\sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t)) dt) \\ ⩽ α \int_{a}^{x} D (\sum_{j = 1}^{n} {\tilde{u}}_{j} (t), \sum_{j = 1}^{n} (\sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t))) dt, \end{matrix}$ where $α = max_{a ⩽ x, t ⩽ b} | k_{i, j} (x, t) | < \infty, i, j = 1, 2, \dots, n .$

Therefore $\begin{matrix} D ({\tilde{u}}_{i} (x), {\tilde{u}}_{i, M} (x)) \\ ⩽ α \sum_{j = 1}^{n} \int_{a}^{x} D ({\tilde{u}}_{j} (x), \sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t)) dt . \end{matrix}$

So $\begin{matrix} lim_{P \to \infty} D ({\tilde{u}}_{i} (x), {\tilde{u}}_{i, M} (x)) \\ ⩽ α lim_{P \to \infty} \sum_{j = 1}^{n} \int_{a}^{x} D ({\tilde{u}}_{j} (x), \sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t)) dt, \end{matrix}$ thus, we have $\begin{matrix} lim_{P \to \infty} D ({\tilde{u}}_{i} (x), {\tilde{u}}_{i, M} (x)) \\ ⩽ α \sum_{j = 1}^{n} \int_{a}^{x} lim_{P \to \infty} D ({\tilde{u}}_{j} (x), \sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t)) dt . \end{matrix}$

By using 4, we have $\begin{matrix} {\tilde{u}}_{i} (x) & = & lim_{P \to \infty} \sum_{m = 1}^{M}^{*} a_{j, m} ⊙ B_{m} (t) \\ \Rightarrow & lim_{P \to \infty} D ({\tilde{u}}_{i} (x), \sum_{m = 1}^{M}^{*} a_{i, m} ⊙ B_{m} (t)) \to 0, \end{matrix}$ for i = 1, 2, ⋯, n. Finally, since α is bounded, we conclude that $lim_{P \to \infty} D ({\tilde{u}}_{i} (x), {\tilde{u}}_{i, M} (x)) \to 0,$ and the proof is completed. □

Fig.1

Exact and Approximate Solution and Absolute Error functions obtained by the present method for N = 4 of Example 8.1.

Fig.2

Exact and Approximate Solution and Absolute Error functions obtained by the present method for N = 8 of Example 8.2.

8 Numerical examples

In order to illustrate the accuracy of the method we present below some numerical examples and the accuracy of the suggested method. As can be viewed in Tables 1 and 2, for both these cases, the absolute error of $(\underline{u} (x, r), \bar{u} (x, r))$ and $(\underline{v} (x, r), \bar{v} (x, r))$ by the presented method for N = 4 and N = 8, respectively. Figures 1 and 2 display exact and approximate solution and absolute error functions obtained by the present method for N = 4 and N = 8. All the numerical computations have been done using mathematic.

Example 8.1. Consider the system of fuzzy Volterra integral equations with $\begin{matrix} \underline{f} (x, r) \\ = (2 r - 1) cos x + \frac{1}{2} (r^{3} - r + 1) (x cos x - sin x) \\ - \frac{1}{2} (2 r - 1) (x cos x + sin x), \\ \bar{f} (x, r) \\ = (2 - r) cos x - \frac{1}{2} (2 r^{2} - 3) (x cos x - sin x) \\ + \frac{1}{2} (r - 2) (x cos x + sin x), \\ \underline{g} (x, r) \\ = \frac{1}{2} (r^{3} - r + 1) (x cos x - sin x) \\ + (r^{3} - r + 1) sin x \\ - \frac{1}{2} (2 r - 1) (x cos x + sin x), \\ \bar{g} (x, r) \\ = \frac{- 1}{2} (2 r^{2} - 3) (x cos x - sin x) + (3 - 2 r^{2}) sin x \\ + \frac{1}{2} (r - 2) (x cos x + sin x), \end{matrix}$ and kernel functions $\begin{matrix} k_{1} (x, t) = cos (x - t), {\tilde{k}}_{1} (x, t) = sin (x - t), \\ k_{2} (x, t) = sin (x - t), {\tilde{k}}_{2} (x, t) = cos (x - t) . \end{matrix}$

The exact solution in this case is given by $\begin{matrix} U (x, r) & = & (\underline{u} (x, r), \bar{u} (x, r)) \\ = & ((2 r - 1) cos x, (2 - r) cos x), \\ V (x, r) & = & (\underline{v} (x, r), \bar{v} (x, r)) \\ = & ((r^{3} - r + 1) sin x, (3 - 2 r^{2}) sin x) . \end{matrix}$

Example 8.2. Consider the system of fuzzy Volterra integral equations with $\begin{matrix} \underline{f} (x, r) & = & e^{- x} (1 - r - r^{2}) + x (1 + r^{2}) \\ + (3 r^{2} + 2 r - 1), \\ \bar{f} (x, r) & = & e^{- x} (r^{3} + r - 3) - x (2 r^{3} - r - 1) \\ - (4 r^{3} + r - 7), \\ \underline{g} (x, r) & = & x (r^{2} + 1) - e^{x} (2 r^{2} + r) \\ - r (e^{- x} - 3 r - 1), \\ \bar{g} (x, r) & = & e^{x} (3 r^{3} - 4) + x (1 + r - 2 r^{3}) \\ - (4 r^{3} + r - 7), \end{matrix}$ and kernel functions K (x, t) = (x - t). We get $\begin{matrix} U (x, r) & = & (\underline{u} (x, r), \bar{u} (x, r)) \\ = & ((2 r^{2} - r) e^{x}, (4 - 3 r^{3}) e^{x}), \\ V (x, r) & = & (\underline{v} (x, r), \bar{v} (x, r)) \\ = & ((r^{2} + r - 1) e^{- x}, (3 - r^{3} - r) e^{- x}) \end{matrix}$ that is the exact solution.

9 Conclusions

In this paper, we used the collocation method based on Bernoulli polynomials to approximate the solution of the systems of linear fuzzy Volterra integral equations. Collocation method is easily implemented and is easy and efficient to approximate the solution of systems of integral Equation 6. In the above presented numerical examples, we see that the proposed method well is carried out for the system of linear fuzzy integral equations and the existence and convergence results are approved. In addition, the presented method is simple and quick to compute.

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