An efficient method to solve fuzzy Volterra integral equations using Fibonacci polynomials

Abstract

According to a huge interest in implementation of the fuzzy Volterra integral equations, especially the second kind, researchers have been investigating to solve such equations using numerical methods since analytical ones might not be accessible usually. In this research paper, we introduce a new approach based on Fibonacci polynomials collocation method to numerically solve them. Several properties of such polynomials were considered to implement in the collocation method due to approximate the solution of the second kind of fuzzy Volterra integral equations. We approved the existence, uniqueness of the solution, convergence and the error analysis of the proposed method in detail. In order to show the authenticity and applicability of the proposed method, we employed several illustrative examples. The numerical results show that the convergence and precision of the recent method were in a good settlement with the exact solution. Also, the calculations of the suggested method are simple and low computational complexity in respect to other methods as an advantage feature of the presented approach.

Keywords

Fuzzy Volterra integral equation Fibonacci polynomial collocation method

1 Introduction

Fuzzy systems have been employed in a several of problems ranging from fuzzy fundamental metric spaces [1], fuzzy topological spaces [2], control chaotic systems [3, 4], fuzzy differential equations and fuzzy fractional differential equations [5, 9], granular differentiability and granular fractional differentiability [10, 11], neutrosophic theory [12] and particle physics [13 –16].

The study of fuzzy integral equations (FIE) has attracted a growing attention for various times in relation with fuzzy modelling, specifically in recent decades. The significance of integration of fuzzy functions was primarily introduced by Dubois and Prade [17]. Alternative schemes were later proposed by Goetschel and Voxman [18], Kaleva [21], Seikkala [22].

Recently, several numerical computational have been suggested for solving FIE of the second kind in one-dimensional space (FIE-2). For instance, Babolian et al. [23] used the Adomian decomposition method (ADM) to solve the second kind of the fuzzy Fredholm integral equations (FFIE-2). Also, Allahviranloo et al. [24] applied the homotopy perturbation technique for solving fuzzy Volterra integral equations (FVIE). After, Ghanbari [25] applied Homotopy analysis method to solve fuzzy linear Volterra integral equations of the second kind. In addition, Allahviranloo et al. [26] approximated the numerical solution of FFIE-2 by modified trapezoidal method. Later, Allahviranloo and Behzadi [27] used airfoil and Chebyshev functions to solve fuzzy Fredholm integro-differential equations with Cauchy kernel. Likewise, Ali Hussain and Waly Ali [28] applied modified trapezoidal method to solve linear Volterra fuzzy integral equations. Mosleh and Otadi [29] solved fuzzy Volterra integral equations using Bernstein polynomial basis. Afterwards, Mirzaee and Hoseini [30] solved systems of linear Fredholm integro-differential equations with Fibonacci polynomials. Besides, Behzadi et al. [31] used fuzzy collocation methods for solving second-order fuzzy Abel-Volterra integro-differential equations. Mirzaee and Hoseini [32] solved a class of Fredholm-Volterra integral equations in two-dimensional spaces by Fibonacci collocation method. Narayanamoorthy and Sathiyapriya [33] used homotopy perturbation method to evaluate linear and nonlinear fuzzy Volterra integral equations. Furthermore, Hamaydi and Qatanani [34] used a computational methods for solving linear fuzzy Volterra integral equation. Moreover, Alijani and Kangro [35] applied collocation method for fuzzy Volterra integral equations.

Now, in this work, we use the Fibonacci polynomials collocation method in order to find a numerical solution for the fuzzy Volterra integral equations of the second kind. The existence and uniqueness of approximation solution are proved to remove hesitation about the method. Also, the convergence and error analysis of the presented method are examined thoroughly. The results show that the calculations of the method respect to others are simple and low cost indeed.

The rest of the paper is organized as follows: The essential notations and concepts of fuzzy numbers, fuzzy functions and fuzzy integrals have been illustrated in Section 2. In Section 3, provides a brief exposition of representation of fuzzy integral equations. In Section 4, we point out several properties of the Fibonacci polynomials and collocation scheme which is applied for solving fuzzy Volterra integral equations of the second kind. In Section 5, the existence and uniqueness of the approximate solution, convergence and error analysis are discussed. In Section 6, deals with numerical implementations and comparisons for a few numerical examples are provided. Section 7 is given a brief conclusion.

2 Basic fuzzy concepts

In this section we provide several basic definitions of a fuzzy calculus as follows

Definition 2.1. [23] An arbitrary fuzzy number $\tilde{u}$ in parametric form is represented by an ordered pair of functions $(\underline{u} (r), \bar{u} (r))$ , 0 ≤ r ≤ 1, such that satisfy the following necessities:

$\underline{u} (r)$ is a bounded monotonic increasing left continuous function,

$\bar{u} (r)$ is a bounded monotonic decreasing left continuous function,

$\underline{u} (r) \leq \bar{u} (r), 0 \leq r \leq 1,$

and

(\underline{u} (r), \bar{u} (r))

, 0 ≤ r ≤ 1 are called the r-cut sets of

\tilde{u}

. The set of all such fuzzy numbers is shown in the form of E¹.

Lemma 2.1. [27] Suppose $(\underline{u} (r), \bar{u} (r))$ , 0 ≤ r ≤ 1 is a given family of non-empty intervals. If

$(\underline{u} (r_{1}), \bar{u} (r_{1})) \supseteq (\underline{u} (r_{2}), \bar{u} (r_{2}))$ for 0 ≤ r₁ ≤ r₂ ≤ 1,

$(lim_{k \to \infty} \underline{u} (r_{k}), lim_{k \to \infty} \bar{u} (r_{k})) = (\underline{u} (r), \bar{u} (r))$ , whenever {r_k} is a non-decreasing converging sequence converges to 0 ≤ r ≤ 1,

then the family

(\underline{u} (r), \bar{u} (r))

, 0 ≤ r ≤ 1, represent the r-cut sets of a fuzzy number

\tilde{u} \in E^{1}

. On the contrary, suppose

(\underline{u} (r), \bar{u} (r))

, 0 ≤ r ≤ 1, are the r-cut sets of a fuzzy number

\tilde{u} \in E^{1}

, then the conditions (1) and (2) hold.

Definition 2.2. [24] For arbitrary $\tilde{u} = (\underline{u} (r), \bar{u} (r))$ , $\tilde{v} = (\underline{v} (r), \bar{v} (r))$ and k ∈ R, we define addition and multiplication by k as follows:

$\begin{matrix} (\underline{u + v}) (r) = (\underline{u} (r) + \underline{v} (r)), \\ (\bar{u + v}) (r) = (\bar{u} (r) + \bar{v} (r)), \\ (\underline{ku}) (r) = k \underline{u} (r), (\bar{ku}) (r) = k \bar{u} (r) if k \geq 0, \\ (\bar{ku}) (r) = k \underline{u} (r), (\underline{ku}) (r) = k \bar{u} (r) if k < 0 . \end{matrix}$

Definition 2.3. [18] For arbitrary $\tilde{u} = (\underline{u} (r), \bar{u} (r))$ , $\tilde{v} = (\underline{v} (r), \bar{v} (r))$ the distance between $\tilde{u}, \tilde{v}$ is define as follows (Hausdorff metric): $D (\tilde{u}, \tilde{v}) = max {sup_{0 \leq r \leq 1} | \underline{u} (r) - \underline{v} (r) |, sup_{0 \leq r \leq 1} | \bar{u} (r) - \bar{v} (r) |}$ (2.1)

Theorem 2.2. [19] For any $\tilde{u}, \tilde{v}, \tilde{w}, \tilde{e} \in E^{1}$ , we have

(E¹, D) is a complete metric space,

$D (\tilde{u} + \tilde{w}, \tilde{v} + \tilde{w}) = D (\tilde{u}, \tilde{v}),$

$D (k \tilde{u}, k \tilde{v}) = | k | D (\tilde{u}, \tilde{v}), \forall k \in R$ ,

$D (\tilde{u} + \tilde{v}, \tilde{w} + \tilde{e}) \leq D (\tilde{u}, \tilde{w}) + D (\tilde{v}, \tilde{e}) .$

Definition 2.4. [31] The mapping $\tilde{f} : [a, b] \to E^{1}$ for some interval [a, b] is called a fuzzy process. Therefore, its r-level set can be written as follows:

$\tilde{f} (s) = (\underline{f} (s, r), \bar{f} (s, r)), s \in [a, b], r \in [0, 1] .$

Definition 2.5. [25] A function $\tilde{f} : [a, b] \to E^{1}$ is said to be continuous if for arbitrary fixed s₀ ∈ [a, b] and ε > 0 there exists ξ > 0 such that

$∣ s - s_{0} ∣ < ξ \Rightarrow D (\tilde{f} (s), \tilde{f} (s_{0})) < ε .$

Definition 2.6. [18] Let $\tilde{f} : [a, b] \to E^{1}$ . For each partition $S = {s_{0}, s_{1}, \dots, s_{n}}$ of [a, b] and for arbitrary ξ_i : s_i-1 ≤ ξ_i ≤ s_i, 1 ≤ i ≤ n, let $λ = max_{1 \leq i \leq n} {∣ s_{i} - s_{i - 1} ∣}$ and

$R_{S} = \sum_{i = 1}^{n} \tilde{f} (ξ_{i}) (s_{i} - s_{i - 1}) .$

The definite integral of $\tilde{f} (s)$ over [a, b] is

$\int_{a}^{b} \tilde{f} (s) ds = lim_{λ \to 0} R_{S},$ provided that this limit exists in the metric D.

If the fuzzy function $\tilde{f} (s)$ is continuous in the metric D, its definite integral exists [18], and

$(\underline{\int_{a}^{b} f (s, r) ds}) = \int_{a}^{b} \underline{f} (s, r) ds,$ $(\bar{\int_{a}^{b} f (s, r) ds}) = \int_{a}^{b} \bar{f} (s, r) ds,$ where $(\underline{f} (s, r), \bar{f} (s, r))$ is the parametric form of $\tilde{f} (s)$ .

Now, we introduce the parametric form of a fuzzy Volterra integral equation of the second kind. Presume $(\underline{f} (s, r), \bar{f} (s, r))$ and $(\underline{u} (s, r), \bar{u} (s, r))$ , 0 ≤ r ≤ 1, a ≤ s ≤ b be parametric forms of $\tilde{f} (s)$ and $\tilde{u} (s)$ , respectively. Then, the parametric form of FVIE-2 is as follows:

$\underline{u} (s, r) = \underline{f} (s, r) + λ \int_{a}^{s} \underline{K (s, t) u (t, r)} dt,$ $\bar{u} (s, r) = \bar{f} (s, r) + λ \int_{a}^{s} \bar{K (s, t) u (t, r)} dt,$

where

$\underline{K (s, t) u (t, r)} = {\begin{matrix} K (s, t) \underline{u} (t, r), K (s, t) \geq 0, \\ K (s, t) \bar{u} (t, r), K (s, t) \leq 0, \end{matrix}$

and

$\bar{K (s, t) u (t, r)} = {\begin{matrix} K (s, t) \bar{u} (t, r), K (s, t) \geq 0, \\ K (s, t) \underline{u} (t, r), K (s, t) \leq 0, \end{matrix}$ for each 0 ≤ r ≤ 1 [25].

Definition 2.7. [20] For L ≥ 0, a function $\tilde{f} : [a, b] \to E^{1}$ is L-Lipschitz if

$D (\tilde{f} (s), \tilde{f} (t)) \leq L ∣ s - t ∣,$ for any s, t ∈ [a, b] .

3 Fuzzy integral equations

The Fredholm integral equation of the second kind [38] is given by

$u (s) = f (s) + λ \int_{a}^{b} K (s, t) u (t) dt,$ (3.1) where λ > 0, a and b are constant, K (s, t) is an arbitrary kernel function over the square a ≤ s, t ≤ b and f (s) is a function of a ≤ s ≤ b. If the kernel satisfies K (s, t) =0, s > t, we obtain the Volterra integral equation of the second kind and is given by

$u (s) = f (s) + λ \int_{a}^{s} K (s, t) u (t) dt .$ (3.2)

In the above equation a refers to constant and s is a variable. If $\tilde{f} (s)$ is a fuzzy function then we have the following equations.

The fuzzy Fredholm integral equation of the second kind is displayed as follows:

$\tilde{u} (s) = \tilde{f} (s) + λ \int_{a}^{b} K (s, t) \tilde{u} (t) dt, a \leq s \leq b,$ (3.3) if for the kernel function we have

$K (s, t) = 0, s > t,$ result in the fuzzy Volterra integral equation as

$\tilde{u} (s) = \tilde{f} (s) + λ \int_{a}^{s} K (s, t) \tilde{u} (t) dt, s \geq a,$ (3.4) where $\tilde{u} (s)$ and $\tilde{f} (s)$ are fuzzy functions on [a, b] and K (s, t) is an arbitrary kernel function over [a, b] × [a, b], λ is a crisp constant and $\tilde{u}$ is unknown on [a, b] [39].

Theorem 3.1. [39] Let K (s, t) be continuous for a ≤ s, t ≤ b, λ > 0 and $\tilde{f} (s)$ a fuzzy continuous function of s, a ≤ s ≤ b. If $λ < \frac{1}{P (b - a)},$ where $P = max_{a \leq s, t \leq b} ∣ K (s, t) ∣,$ then the iterative scheme $\begin{matrix} {\tilde{u}}_{0} (s) = \tilde{f} (s), \\ {\tilde{u}}_{k} (s) = \tilde{f} (s) + λ \int_{a}^{b} K (s, t) {\tilde{u}}_{k - 1} (t) dt, k \geq 1, \end{matrix}$ converges to the unique solution of Eq. (3.3). Specially, $sup_{a \leq s \leq b} D (\tilde{u}, {\tilde{u}}_{k}) \leq \frac{Q^{k}}{1 - Q} sup_{a \leq s \leq b} D ({\tilde{u}}_{0}, {\tilde{u}}_{1}),$ where $Q = λ P (b - a) < 1$ . This deduces that ${\tilde{u}}_{k}$ converges uniformly in s to $\tilde{u}$ , in other words, given arbitrary ε > 0 we can find N such that $D (\tilde{u}, {\tilde{u}}_{k}) < ε, a \leq s \leq b, k > N .$

The proof this theorem can be easily extended for fuzzy Volterra integral equations of the second kind.

4 The Fibonacci polynomials and collocation method

The collocation technique has produced an acceptable numerical results with sufficient stability for integral equations [40, 41]. On the other hand, applying this technique using series expansion based on different polynomials, plays an important role in the convergence of the method to the accurate solution. In this section, we use the non-orthogonal Fibonacci polynomial that is less considered in fuzzy integral equations, and in the following we give a brief explanation of this polynomial.

4.1 The fibonacci polynomials and their properties

A sequence of polynomials is a Fibonacci polynomials, {F_n (x)}, are defined by the recursion

$F_{n + 2} (x) = {xF}_{n + 1} (x) + F_{n} (x), n \geq 1,$ with initial values F₁ (x) =1 and F₂ (x) = x.

Definition 4.1. [30] Assume that k, the k is an arbitrary positive real number, the k-Fibonacci sequence, {F_k,n} _n∈N is represented recurrently by

$F_{k, n + 1} = {kF}_{k, n} + F_{k, n - 1}, n \geq 1,$ with initial conditions

$F_{k, 0} = 0, F_{k, 1} = 1 .$

The Fibonacci polynomials are given by the explicit formula

$F_{n + 1} (x) = \sum_{i = 0}^{⌊ \frac{n}{2} ⌋} (\begin{matrix} n - i \\ i \end{matrix}) x^{n - 2 i}, n \geq 0,$ (4.1) where $⌊ \frac{n}{2} ⌋$ denotes the greatest integer in $\frac{n}{2}$ . One should bear in mind that F_2n (0) =0 and x = 0 is the unique real root, while F_2n+1 (0) =1 has no real roots. In addition, for x = k ∈ N, we can find the elements of the k-Fibonacci sequence [42].

The Fibonacci polynomials have generating function [48].

$\begin{matrix} G (x, t) = \frac{t}{1 - t^{2} - tx} = \sum_{n = 1}^{\infty} F_{n} (x) t^{n} \\ = t + {xt}^{2} + (x^{2} + 1) t^{3} + (x^{3} + 2 x) t^{4} + \dots . \end{matrix}$

The Fibonacci polynomials are normalized so that F_n (1) = F_n, in which the F_n is the nth Fibonacci number. The equation for the Fibonacci polynomials can be written in matrix form as

$F (x) = BX (x),$

where F (x) = [F₁ (x) , F₂ (x) , …, F_N+1 (x)] ^T, X (x) = [1, x, x², …, x^N] ^T, and B is the lower triangular matrix with entrances the coefficients appearing in expansion of Fibonacci polynomials in increasing powers of x. For instance, for N = 6 we have $[\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 3 & 0 & 1 & 0 & 0 \\ 0 & 3 & 0 & 4 & 0 & 1 & 0 \\ 1 & 0 & 6 & 0 & 5 & 0 & 1 \end{matrix}] .$ Remind that in matrix B the non–zero entrances construct precisely the diagonals of the pascal triangle and the sum of the elements in the same row delivers the classical Fibonacci sequence. In addition, matrix B is invertible. So xⁿ may be written as linear Fibonacci polynomials [42]. These expansions are given in closed form in theorem as follows

Theorem 4.1. [42] For every integer n ≥ 1, x^n-1 is possible to rewrite in a special way as linear mixture of the n first Fibonacci polynomials as

$\begin{matrix} x^{n - 1} = \sum_{i = 0}^{⌊ \frac{n}{2} ⌋} (- 1)^{i} [(\begin{matrix} n \\ i \end{matrix}) - (\begin{matrix} n \\ i - 1 \end{matrix})] \\ F_{n - 2 i} (x), where (\begin{matrix} n \\ - 1 \end{matrix}) = 0 . \end{matrix}$

The Fibonacci polynomials can be also represented using some orthogonal polynomials, such as the Chebyshev polynomials of the second kind [43]. The Fibonacci polynomials can be written as follows

$F_{n} (s) = z^{n - 1} c_{n - 1} (\frac{s}{2 z}), z^{2} = - 1, n \geq 1 .$

Therefore, the expansion of $\tilde{u} (s)$ is the approximated form of Chebyshev polynomials can be shown that

$\tilde{u} (s) ≃ {\tilde{u}}_{N} (s) = \sum_{n = 1}^{N} {\tilde{u}}_{n} z^{n - 1} c_{n - 1} (\frac{s}{2 z}),$ by considering $C_{n} (s) = \sqrt{\frac{2}{π}} c_{n} (s)$ , then $C_{n} (s)$ , for n = 0, 1, …, N - 1, form of an orthonormal polynomial basis in [-1, 1] with regard to weight function $ω (s) = \sqrt{1 - s^{2}}$ , as defined as

$\int_{- 1}^{1} C_{p} (s) C_{q} (s) \sqrt{1 - s^{2}} ds = {\begin{matrix} 0, p \neq q, \\ 1, p = q, \end{matrix}$ and can be mapped in to [0, 1]. At last, it can be shown that

$\tilde{u} (s) ≃ \sum_{j = 1}^{N} {\tilde{t}}_{j} C_{j - 1} (s),$ where ${\tilde{t}}_{j}$ can be written in phrases of ${\tilde{u}}_{n}$ , n = 1, 2, …, N.

Theorem 4.2. [44] If $\tilde{u} (s)$ be a continuous fuzzy function defined on [0, 1] and $D (\tilde{u} (s), \tilde{0}) \leq P, \forall s \in [0, 1]$ , then the expansion of Chebyshev functions defined in $\tilde{u} (s) ≃ \sum_{n = 1}^{N} {\tilde{t}}_{n} C_{n - 1} (s)$ converges uniformly and also

$D ({\tilde{t}}_{n}, \tilde{0}) \leq P \sqrt{2 π} .$

4.2 Explanation of the method

In this section, we solved the Eq. (3.4) using the Fibonacci polynomials and collocation method.

To find the approximation solution of Eq. (3.4), we can write

$\tilde{u} (s) = \sum_{n = 1}^{\infty} {\tilde{U}}_{n} F_{n} (s), 0 \leq a \leq s \leq b,$ (4.2) where ${\tilde{U}}_{n}, n = 1, 2, \dots, N + 1$ , are unknown Fibonacci coefficients, N is an arbitrary positive integer and F_n (x) , n = 1, 2, …, N + 1 are Fibonacci polynomials. The purpose of the method is to get solution as Fibonacci series represented by

$\tilde{u} (s) ≃ {\tilde{u}}_{N} (s) = \sum_{n = 1}^{N + 1} {\tilde{U}}_{n} F_{n} (s) = F (s) U^{T},$ (4.3) where ${\tilde{U}}_{n}, n = 1, 2, \dots, N + 1$ are unknown Fibonacci coefficients,

$\begin{matrix} U = [{\tilde{U}}_{1}, {\tilde{U}}_{2}, \dots, {\tilde{U}}_{N + 1}]^{T}, \\ F (s) = [F_{1} (s), F_{2} (s), \dots, F_{N + 1} (s)]^{T}, \end{matrix}$ where N is an arbitrary positive integer.

Assume without loss of generality that K (s, t) =1, now $\sum_{n = 1}^{N + 1} {\tilde{U}}_{n} F_{n} (s) = \tilde{f} (s) + λ \int_{a}^{s} \sum_{n = 1}^{N + 1} {\tilde{U}}_{n} F_{n} (t) dt,$ (4.4) then

$\begin{matrix} \sum_{n = 1}^{N + 1} {\underline{U}}_{n} (r) F_{n} (s) = \underline{f} (s, r) + λ \int_{a}^{s} \sum_{n = 1}^{N + 1} {\underline{U}}_{n} (r) F_{n} (t) dt, \\ \sum_{n = 1}^{N + 1} {\bar{U}}_{n} (r) F_{n} (s) = \bar{f} (s, r) + λ \int_{a}^{s} \sum_{n = 1}^{N + 1} {\bar{U}}_{n} (r) F_{n} (t) dt, \end{matrix}$ and we have

$\begin{matrix} \sum_{n = 1}^{N + 1} (F_{n} (s) - λ \int_{a}^{s} F_{n} (t) dt) {\underline{U}}_{n} (r) = \underline{f} (s, r), \\ \sum_{n = 1}^{N + 1} (F_{n} (s) - λ \int_{a}^{s} F_{n} (t) dt) {\bar{U}}_{n} (r) = \bar{f} (s, r) . \end{matrix}$

Consider the collocation points as

$s_{i} = a + \frac{b - a}{N + 1} i, i = 1, 2, \dots, N + 1,$

$\tilde{f} (s_{i}) = \sum_{n = 1}^{N + 1} (F_{n} (s_{i}) - λ \int_{a}^{s_{i}} F_{n} (t) dt) {\tilde{U}}_{n} .$

In the other hand, we get

$\begin{matrix} \underline{f} (s_{i}, r) = \sum_{n = 1}^{N + 1} (F_{n} (s_{i}) - λ \int_{a}^{s_{i}} F_{n} (t) dt) {\underline{U}}_{n} (r), \\ \bar{f} (s_{i}, r) = \sum_{n = 1}^{N + 1} (F_{n} (s_{i}) - λ \int_{a}^{s_{i}} F_{n} (t) dt) {\bar{U}}_{n} (r), \end{matrix}$ and we can write

$\begin{matrix} \sum_{n = 1}^{N + 1} (F_{n} (s_{i}) - λ \int_{a}^{s_{i}} F_{n} (t) dt) ({\underline{U}}_{n} (r), {\bar{U}}_{n} (r)) \\ = (\underline{f} (s_{i}, r), \bar{f} (s_{i}, r)) . \end{matrix}$ (4.5)

Hence, the matrix form of Eq. (4.5) is

$FU = M,$ (4.6)

where

$\begin{matrix} F = (F_{ij})_{(N + 1) \times (N + 1)}, \\ M = [{\tilde{M}}_{1}, {\tilde{M}}_{2}, \dots, {\tilde{M}}_{N + 1}]^{T}, \end{matrix}$

with

$\begin{matrix} F_{ij} = F_{ij} (s_{i}) - λ \int_{a}^{s_{i}} F_{ij} (t) dt, i, j = 1, 2, \dots, N + 1, \\ {\tilde{M}}_{i} = ({\underline{M}}_{i} (r), {\bar{M}}_{i} (r)) = (\underline{f} (s_{i}, r), \bar{f} (s_{i}, r)), 1 \leq i \leq N + 1, \\ {\tilde{U}}_{i} = ({\underline{U}}_{i} (r), {\bar{U}}_{i} (r)), 1 \leq i \leq N + 1 . \end{matrix}$

To find fuzzy coefficients of U and obtain the approximate solution, we have to calculate F^-1M with |F|≠0 and use in truncated series ${\tilde{u}}_{N}$ .

Definition 4.2. [45] Consider the (N + 1) × (N + 1) fuzzy linear system of equations as follows

${\begin{matrix} F_{11} {\tilde{U}}_{1} + F_{12} {\tilde{U}}_{2} + \dots + F_{1 (N + 1)} {\tilde{U}}_{N + 1} = {\tilde{M}}_{1}, \\ F_{21} {\tilde{U}}_{1} + F_{22} {\tilde{U}}_{2} + \dots + F_{2 (N + 1)} {\tilde{U}}_{N + 1} = {\tilde{M}}_{2}, \\ ⋮ \\ F_{(N + 1) 1} {\tilde{U}}_{1} + F_{(N + 1) 2} {\tilde{U}}_{2} + \dots + F_{(N + 1) (N + 1)} {\tilde{U}}_{N + 1} = {\tilde{M}}_{N + 1}, \end{matrix}$ (4.7) where the coefficient matrix F = (F_ij) , 1 ≤ i, j ≤ N + 1, is a crisp matrix and ${\tilde{M}}_{i} \in E^{1}, 1 \leq i \leq N + 1$ .

Definition 4.3. [45] A fuzzy number vector $U = [{\tilde{U}}_{1}, {\tilde{U}}_{2}, \dots, {\tilde{U}}_{N + 1}]^{T}$ given by parametric form ${\tilde{U}}_{i} = ({\underline{U}}_{i} (r), {\bar{U}}_{i} (r)), 1 \leq i \leq N + 1, r \in [0, 1]$ , is called a solution of the fuzzy linear system (4.8) if $\begin{matrix} \underline{\sum_{j = 1}^{N + 1} F_{ij} {\tilde{U}}_{j}} = \sum_{j = 1}^{N + 1} \underline{F_{ij} {\tilde{U}}_{j}} = {\underline{M}}_{i} (r), i = 1, 2, \dots, N + 1, \\ \bar{\sum_{j = 1}^{N + 1} F_{ij} {\tilde{U}}_{j}} = \sum_{j = 1}^{N + 1} \bar{F_{ij} {\tilde{U}}_{j}} = {\bar{M}}_{i} (r), i = 1, 2, \dots, N + 1 . \end{matrix}$ (4.8)

In order to solve the system (4.8), we can write crisp linear system as follows [45].

$SU = M,$ (4.9) where S = (s_ij) , 1 ≤ i, j ≤ 2 (N + 1), and

$\begin{matrix} F_{ij} \geq 0 \to s_{ij} = F_{ij}, s_{i, j + N + 1} = F_{ij}, \\ F_{ij} < 0 \to s_{i, j + N + 1} = - F_{ij}, s_{i + N + 1, j} = - F_{ij}, \end{matrix}$ (4.10) and any s_ij is not determined by Eq. (4.10) is zero, then according to [45] we have

$\begin{matrix} U = ({\underline{U}}_{1} (r), {\underline{U}}_{2} (r), \dots, {\underline{U}}_{N + 1} (r), - {\bar{U}}_{1} (r), \\ - {\bar{U}}_{2} (r), \dots, - {\bar{U}}_{N + 1} (r)) \\ M = ({\underline{M}}_{1} (r), {\underline{M}}_{2} (r), \dots, {\underline{M}}_{N + 1} (r), - {\bar{M}}_{1} (r), \\ - {\bar{M}}_{2} (r), \dots, - {\bar{M}}_{N + 1} (r)), \end{matrix}$ (4.11)

The formation of S implies that s_ij ≥ 0, 1 ≤ i, j ≤ 2 (N + 1) and $S = [\begin{matrix} S_{1} & S_{2} \\ S_{2} & S_{1} \end{matrix}]$ (4.12) where S₁ contains the positive entries of F and S₂ the absolute values of the entries of the negative entries of F and F = S₁ - S₂.

Theorem 4.3. [45] The matrix S is nonsingular if and only if the matrices F and S₁ + S₂ are both nonsingular.

Theorem 4.4. [45] Suppose S be nonsingular, then the unique solution U is always a fuzzy vector for arbitrary vector M if and only if S^-1 is nonnegative.

Theorem 4.5. [46] Assume the inverse of matrix F in Eq. (4.8) exists and $U = [{\tilde{U}}_{1}, {\tilde{U}}_{2}, \dots, {\tilde{U}}_{N + 1}]^{T}$ is a fuzzy solution of this equation. Then $\underline{U} (r) + \bar{U} (r) = ({\underline{U}}_{1} (r) + {\bar{U}}_{1} (r), {\underline{U}}_{2} (r) + {\bar{U}}_{2} (r), \dots, {\underline{U}}_{N + 1} (r) + {\bar{U}}_{N + 1} (r))^{T}$ is the solution of the following system $F (\underline{U} (r) + \bar{U} (r)) = (\underline{M} (r) + \bar{M} (r)),$ (4.13) where $\underline{M} (r) + \bar{M} (r) = ({\underline{M}}_{1} (r) + {\bar{M}}_{1} (r), {\underline{M}}_{2} (r) + {\bar{M}}_{2} (r), \dots, {\underline{M}}_{N + 1} (r) + {\bar{M}}_{N + 1} (r))^{T}$ .

Theorem 4.6. [47] For every positive integer i, the following connection formulas between the Fibonacci polynomials and the Chebyshev polynomials hold

$\begin{matrix} C_{i - 1} (s) = 2^{i - 1} \sum_{j = 0}^{⌊ \frac{i - 1}{2} ⌋} (- 1)^{j + 1} (\begin{matrix} i - 1 \\ j \end{matrix}) \\ \frac{- i + 2 j}{i - j} \sum_{k = 0}^{j} \frac{(- j)_{k} (- i + j)_{k}}{(- 4)^{k} (- i + 1)_{k} k!} \sqrt{\frac{2}{π}} F_{i - 2 j} (s), \end{matrix}$

and

$\begin{matrix} F_{i} (s) = \frac{1}{2^{i - 1}} \sum_{j = 0}^{⌊ \frac{i - 1}{2} ⌋} (\begin{matrix} i - 1 \\ j \end{matrix}) \frac{i - 2 j}{i - j} \sum_{k = 0}^{j} \\ \frac{(- j)_{k} (- i + j)_{k} (- 4)^{k}}{(- i + 1)_{k} k!} \sqrt{\frac{π}{2}} C_{i - 2 j - 1} (s), \end{matrix}$

where

$(- j)_{k} = \frac{(- 1)^{k} j!}{(j - k)!} .$

5 Existence and uniqueness of the solution along with error analysis

In this section, uniqueness and existence of the solution are proved. Then error in the method is illustrated.

Let $X = {\tilde{f} : [a, b] \to E^{1}, \tilde{f} is contiuous}$ be the space of fuzzy continuous functions with the metric $D (\tilde{u}, \tilde{v}) = sup_{a \leq s \leq b} D (\tilde{u} (s), \tilde{v} (s))$ . Remember the fact that $(X, D)$ is complete metric space.

The operator $Γ : X \to X$ define by

$Γ (\tilde{u} (s)) = \tilde{f} (s) + \int_{a}^{s} K (s, t) \tilde{u} (t) dt, \forall s \in [a, b], \forall \tilde{u} \in X .$ (5.1)

Theorem 5.1. In the Eq. (3.4), assume that $\tilde{f} (s)$ , s ∈ [a, b] is a fuzzy continuous function and K (s, t) is uniformly continuous for s ∈ [a, b]. Moreover, suppose that $Q > 0$ , $P > 0$ such that $P = max_{a \leq s, t \leq b} ∣ K (s, t) ∣,$

and

$D (K (s_{1}, t), K (s_{2}, t)) \leq Q D (s_{1}, s_{2}), \forall s_{1}, s_{2} \in [a, b] .$

If $Q (b - a) < 1$ , then the integral Eq. (3.4) has a unique solution $\tilde{u} \in X$ , which can be derived through the method of successive approximations starting by any element of $X$ . Moreover, in the approximation of the solution by the terms of the sequence of successive approximations, ${\tilde{u}}_{m}, m \in N$ and ${\tilde{u}}_{0} (s) = \tilde{f} (s)$ .

$\begin{matrix} {\tilde{u}}_{m + 1} (s) = \tilde{f} (s) + \int_{a}^{s} K (s, t) {\tilde{u}}_{m} (t) dt, \\ \forall s \in [a, b], \forall m \in N, \end{matrix}$ the error estimate is

$\begin{matrix} D (\tilde{u} (s), {\tilde{u}}_{m} (s)) \leq \frac{[Q (b - a)]^{m}}{1 - Q (b - a)} P (b - a), \\ \forall s \in [a, b], \forall m \in N, \end{matrix}$

Proof. we first examine the conditions of the Banach fixed point principle and prove $Γ (X) \subset X$ . Let the operator $Γ : X \to X$ defined as

$Γ (\tilde{u} (s)) = \tilde{f} (s) + \int_{a}^{s} K (s, t) \tilde{u} (t) dt, s \in [a, b] .$

In this aim, we see that for all δ > 0, there exists δ₁, δ₂ such that δ₁ + (b - a) δ₂ < δ.

Since $\tilde{f}$ is continuous on the compact interval [a, b], we conclude that it is uniformly continuous and δ₁ > 0 exists γ₁ > 0 so that

$D (\tilde{f} (s_{1}), \tilde{f} (s_{2})) < δ_{1}, \forall s_{1}, s_{2} \in [a, b],$ with ∣s₁ - s₂ ∣ < γ₁. Beside, from the uniform continuity of k, for δ₂, exists γ₂ > 0, such that

$D (K (s_{1}, t), K (s_{2}, t)) < δ_{2}, \forall s_{1}, s_{2} \in [a, b],$ with ∣s₁ - s₂ ∣ < γ₂. Let γ = min(γ₁, γ₂) and s₁, s₂ ∈ [a, b] with ∣s₁ - s₂ ∣ < γ.

According to Theorem (2.2) and Definition (2.3), we have

$\begin{matrix} D (Γ (\tilde{u} (s_{1})), Γ (\tilde{u} (s_{2}))) \\ = D (\tilde{f} (s_{1}) + \int_{a}^{s} K (s_{1}, t) \tilde{u} (t) dt, \tilde{f} (s_{2}) + \int_{a}^{s} K (s_{2}, t) \tilde{u} (t) dt) \\ \leq D (\tilde{f} (s_{1}), \tilde{f} (s_{2})) + D (\int_{a}^{s} K (s_{1}, t) \tilde{u} (t) dt, \int_{a}^{s} K (s_{2}, t) \tilde{u} (t) dt) \\ \leq D (\tilde{f} (s_{1}), \tilde{f} (s_{2})) + Q \int_{a}^{s} D (K (s_{1}, t) \tilde{u} (t) dt, K (s_{2}, t) \tilde{u} (t) dt) \\ \leq δ_{1} + Q \int_{a}^{s} δ_{2} ds < δ_{1} + Q (b - a) δ_{2} < δ . \end{matrix}$

Therefore, $Γ (\tilde{u})$ is uniformly continuous for any $\tilde{u} \in X$ , and consequently continuous on [a, b], then $Γ (X) \subset X$ . For $\tilde{u}, \tilde{v} \in X$ and s ∈ [a, b] follows

$\begin{matrix} D (Γ (\tilde{u} (s)), Γ (\tilde{v} (s))) \\ = D (\tilde{f} (s) + \int_{a}^{s} K (s, t) \tilde{u} (t) dt, \tilde{f} (s) + \int_{a}^{s} K (s, t) \tilde{v} (t) dt) \\ \leq D (\tilde{f} (s), \tilde{f} (s)) + D (\int_{a}^{s} K (s, t) \tilde{u} (t) dt, \int_{a}^{s} K (s, t) \tilde{v} (t) dt) \\ \leq Q \int_{a}^{s} D (K (s, t) \tilde{u} (t) dt, K (s, t) \tilde{v} (t) dt) \leq Q \int_{a}^{s} LD (\tilde{u} (t), \tilde{v} (t)) dt \\ \leq Q \int_{a}^{s} Q D (\tilde{u}, \tilde{v}) dt \leq Q \int_{a}^{b} Q D (\tilde{u}, \tilde{v}) dt = Q D (\tilde{u}, \tilde{v}) Q \int_{a}^{b} dt \\ = Q (b - a) D (\tilde{u}, \tilde{v}), \forall s \in [a, b], \end{matrix}$ then, we have

$D (Γ (\tilde{u}), Γ (\tilde{v})) \leq Q (b - a) D (\tilde{u}, \tilde{v}), \forall \tilde{u}, \tilde{v} \in X,$ since $Q (b - a) < 1$ , the operator Γ is a contraction. Applying the Banach’s fixed point principle we derive that Eq. (3.4) has a unique solution $\tilde{u}$ in $X$ and the following inequality holds $\begin{matrix} D (\tilde{u} (s), {\tilde{u}}_{m} (s)) \leq D (\tilde{u}, {\tilde{u}}_{m}) \\ \leq \frac{(Q (b - a))^{m}}{1 - Q (b - a)} D ({\tilde{u}}_{0}, {\tilde{u}}_{1}), \forall m \in N, \end{matrix}$ for any s ∈ [a, b]. Also,

$\begin{matrix} D ({\tilde{u}}_{0}, {\tilde{u}}_{1}) \\ = sup_{a \leq s \leq b} D (\tilde{f} (s), \tilde{f} (s) + \int_{a}^{s} K (s, t) \tilde{f} (t) dt) \\ \leq sup_{a \leq s \leq b} D (0, \int_{a}^{s} K (s, t) \tilde{f} (t) dt) \leq sup_{a \leq s \leq b} Q \int_{a}^{s} D (0, K (s, t) \tilde{f} (t)) dt \\ = sup_{a \leq s \leq b} \int_{a}^{s} ∣ K (s, t) ∣ dt \leq \sup_{a \leq s \leq b} Q \int_{a}^{s} P dt \leq P (b - a) . \end{matrix}$

Hereupon we obtain the inequality of error estimate. □

Theorem 5.2. In the Eq. (3.4), assume that $\tilde{f} (s)$ , s ∈ [a, b] is a fuzzy continuous function and K (s, t) is continuous for s ∈ [a, b]. Moreover, assume that $\begin{matrix} p = max_{a \leq s \leq b} D (\tilde{u} (s), \tilde{0}), \\ P = max_{a \leq s, t \leq b} ∣ K (s, t) ∣ . \end{matrix}$

If p (b - a) <1, the series solution of the Eq. (4.3) using Fibonacci polynomials converge.

Proof. Let the ${\tilde{u}}_{m} (s)$ and ${\tilde{u}}_{n} (s)$ , are the approximate of Eq. (4.2). By using Eq. (4.3) we can write

$\begin{matrix} {\tilde{u}}_{m} (s) = \sum_{i = 1}^{m} {\tilde{U}}_{i} F_{i} (s), \\ {\tilde{u}}_{n} (s) = \sum_{i = 1}^{n} {\tilde{U}}_{i} F_{i} (s) . \end{matrix}$

Assume ${\tilde{u}}_{m} (s)$ and ${\tilde{u}}_{n} (s)$ are two arbitrary partial sums with m > n. Now, we show that ${\tilde{u}}_{m} (s)$ is a Cauchy sequence in Banach space $X$ .

$\begin{matrix} D ({\tilde{u}}_{m} (s), {\tilde{u}}_{n} (s)) \\ = D ((\tilde{f} (s) + \int_{a}^{s} K (s, t) {\tilde{u}}_{m} (t) dt), (\tilde{f} (s) + \int_{a}^{s} K (s, t) {\tilde{u}}_{n} (t) dt)) \\ \leq D (\tilde{f} (s) + \tilde{f} (s)) + D ((\int_{a}^{s} K (s, t) {\tilde{u}}_{m} (t) dt), \\ (\int_{a}^{s} K (s, t) {\tilde{u}}_{n} (t) dt)) . \end{matrix}$ Then $\begin{matrix} D ({\tilde{u}}_{m} (s), {\tilde{u}}_{n} (s)) & \leq D ((\int_{a}^{s} K (s, t) \sum_{i = 1}^{m} {\tilde{U}}_{i} F_{i} (t) dt), (\int_{a}^{s} K (s, t) \sum_{i = 1}^{n} {\tilde{U}}_{i} F_{i} (t) dt)) \\ = max_{s \in [a, b]} {sup_{0 \leq t \leq 1} | \int K (s, t) \sum_{i = 1}^{m} {\underline{U}}_{i} F_{i} (t) dt - \int K (s, t) \sum_{i = 1}^{n} {\underline{U}}_{i} F_{i} (t) dt |, \\ sup_{0 \leq t \leq 1} | \int K (s, t) \sum_{i = 1}^{m} {\bar{U}}_{i} F_{i} (t) dt - \int K (s, t) \sum_{i = 1}^{n} {\bar{U}}_{i} F_{i} (t) dt |} \\ \leq max_{s \in [a, b]} {sup_{0 \leq t \leq 1} | (\int_{a}^{s} ∣ K (s, t) ∣ ∣ \sum_{i = 1}^{m} {\underline{U}}_{i} F_{i} (t) - 0 ∣ dt) |, sup_{0 \leq t \leq 1} | (\int_{a}^{s} ∣ K (s, t) ∣ ∣ \sum_{i = 1}^{n} {\underline{U}}_{i} F_{i} (t) - 0 ∣ dt) |} \\ + max_{s \in [a, b]} {sup_{0 \leq t \leq 1} | (\int_{a}^{s} ∣ K (s, t) ∣ ∣ \sum_{i = 1}^{m} {\bar{U}}_{i} F_{i} (t) - 0 ∣ dt) |, sup_{0 \leq t \leq 1} | (\int_{a}^{s} ∣ K (s, t) ∣ ∣ \sum_{i = 1}^{n} {\bar{U}}_{i} F_{i} (t) - 0 ∣ dt) |} \\ \leq P (D ({\tilde{u}}_{m} (t), 0)) + P (D ({\tilde{u}}_{n} (t), 0)), \end{matrix}$ with using Theorem (3.1) it is clear that

$\begin{matrix} D ({\tilde{u}}_{m} (t), 0) \leq (p (b - a))^{m} \to 0 and \\ D ({\tilde{u}}_{n} (t), 0) \leq (p (b - a))^{n} \to 0, \end{matrix}$ now, we have

$D ({\tilde{u}}_{m} (s), {\tilde{u}}_{n} (s)) \to 0 .$

By choosing m = n + 1, we have

$D ({\tilde{u}}_{n + 1} (s), {\tilde{u}}_{n} (s)) = 0 .$ □

Now, we obtain error analysis for the Eq. (3.4), by the following theorem.

Theorem 5.3. Let $\tilde{u} (s)$ be the exact solution and ${\tilde{u}}_{N} (s) = F (s) U^{T}$ be the approximation solution of Eq. (3.4), based on Fibonacci polynomials, and also assume that

|K (s, t) | ≤ η,

1 - λη > 0,

$‖ \tilde{f} (s) ‖_{\infty} = sup_{s \in [a, b]} | \tilde{f} (s) | .$

Then we have

$‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty} ⩽ \frac{e (\tilde{f} (s))}{1 - λ η},$ and η is a constant.

Proof. Due to the Eq. (3.4), we can write

$\begin{matrix} | \tilde{u} (s) - {\tilde{u}}_{N} (s) | \\ = | (\tilde{f} (s) + λ \int_{a}^{s} K (s, t) \tilde{u} (t) dt) - ({\tilde{f}}_{N} (s) + λ \int_{a}^{s} K (s, t) {\tilde{u}}_{N} (t) dt) | \\ = | \tilde{f} (s) - {\tilde{f}}_{N} (s) + λ \int_{a}^{s} (K (s, t) \tilde{u} (t) - K (s, t) {\tilde{u}}_{N} (t)) dt | \\ ⩽ | \tilde{f} (s) - {\tilde{f}}_{N} (s) | + λ \int_{a}^{s} | K (s, t) \tilde{u} (t) - K (s, t) {\tilde{u}}_{N} (t) | dt, \\ ⩽ | \tilde{f} (s) - {\tilde{f}}_{N} (s) | + λ \int_{a}^{s} | K (s, t) | | \tilde{u} (t) - {\tilde{u}}_{N} (t) | dt, \end{matrix}$ (5.2) using assumption (i), we have

$\begin{matrix} | K (s, t) \tilde{u} (t) - K (s, t) {\tilde{u}}_{N} (t)) | & = | K (s, t) (\tilde{u} (t) - {\tilde{u}}_{N} (t)) | \\ ⩽ | K (s, t) | | \tilde{u} (t) - {\tilde{u}}_{N} (t) | \\ ⩽ η ‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty}, \end{matrix}$ (5.3) combining Eqs. (5.2), (5.3) and putting $e (\tilde{f} (s)) = ‖ \tilde{f} (s) - {\tilde{f}}_{N} (s) ‖_{\infty}$ , we have

$‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty} ⩽ e (\tilde{f} (s)) + λ η ‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty},$ and

$‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty} - λ η ‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty} ⩽ e (\tilde{f} (s)),$ as a result

$‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{\infty} ⩽ \frac{e (\tilde{f} (s))}{1 - λ η},$ □

Theorem 5.4. Let ${\tilde{u}}_{N} (s) = \sum_{i = 1}^{N} {\tilde{u}}_{i} F_{i} (s)$ be the truncated series, then the truncated error can be represented by $‖ \tilde{u} - {\tilde{u}}_{N} (s) ‖_{2}^{2} \leq P^{2} π \sum_{i = N + 1}^{\infty} \sum_{j = 0}^{⌊ \frac{i - 1}{2} ⌋} \sum_{k = 0}^{j} ▵^{2},$ and $▵ = \frac{4^{k} (i - 2 j) (i - k - 1)!}{2^{i - 1} k! (j - k)! (i - j - k)!} .$

Proof. For any $\tilde{u} \in ([0, 1] \times [0, 1])$ can be expressed by the orthonormal Chebyshev polynomials to the weight function $ω (s) = \sqrt{1 - s^{2}}$ and the Fibonacci polynomials as $\tilde{u} (s) = \sum_{i = 1}^{\infty} {\tilde{u}}_{i} F_{i} (s)$ , if ${\tilde{u}}_{N} (s)$ be the truncated series by Fibonacci polynomials, then

$\sum_{i = 1}^{\infty} {\tilde{u}}_{i} F_{i} (s) = \sum_{i = 1}^{N} {\tilde{u}}_{i} F_{i} (s) + \sum_{i = N + 1}^{\infty} {\tilde{u}}_{i} F_{i} (s),$ and

$\sum_{i = 1}^{\infty} {\tilde{u}}_{i} F_{i} (s) - \sum_{i = 1}^{N} {\tilde{u}}_{i} F_{i} (s) = \sum_{i = N + 1}^{\infty} {\tilde{u}}_{i} F_{i} (s),$ the truncated error can be written as

$\tilde{u} (s) - {\tilde{u}}_{N} (s) = \sum_{i = N + 1}^{\infty} {\tilde{u}}_{i} F_{i} (s),$ now

$\begin{matrix} ‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{2}^{2} = ‖ \sum_{i = N + 1}^{\infty} {\tilde{u}}_{i} F_{i} (s) ‖_{2}^{2} \\ = \int_{0}^{1} | \sum_{i = N + 1}^{\infty} {\tilde{u}}_{i} F_{i} (s) |^{2} \sqrt{1 - s^{2}} ds, \end{matrix}$ (5.4) by using Theorem (4.6) in the Eq. (5.4) we have

$\begin{matrix} ‖ \tilde{u} (s) - {\tilde{u}}_{N} (s) ‖_{2}^{2} \\ \leq \int_{0}^{1} (D (\sum_{i = N + 1}^{\infty} \frac{{\tilde{u}}_{i}}{2^{i - 1}} \sum_{j = 0}^{⌊ \frac{i - 1}{2} ⌋} (\begin{matrix} i - 1 \\ j \end{matrix}) \frac{i - 2 j}{i - j} \\ \times \sum_{k = 0}^{j} \frac{(- j)_{k} (- i + j)_{k} (- 4)^{k}}{(- i + 1)_{k} k!} \sqrt{\frac{π}{2}} C_{i - 2 j - 1 (s), \tilde{0}}))^{2} \sqrt{1 - s^{2}} ds \\ \leq \frac{P^{2} π}{4} \sum_{i = N + 1}^{\infty} \sum_{j = 0}^{⌊ \frac{i - 1}{2} ⌋} \sum_{k = 0}^{j} (\frac{4^{k} (i - 2 j) (i - k - 1)!}{2^{i - 1} k! (j - k)! (i - j - k)!})^{2} \\ = \frac{P^{2} π}{4} \sum_{i = N + 1}^{\infty} \sum_{j = 0}^{⌊ \frac{i - 1}{2} ⌋} \sum_{k = 0}^{j} ▵^{2} . \end{matrix}$ □

6 Numerical examples

In this section, in order to demonstrate the efficiency of the present method, by providing some examples of FVIE-2, we examine and compare the obtained numerical solutions with the exact solution and VIM in [49].

Example 6.1. [49] Consider the fuzzy Volterra integral equation

$\begin{matrix} \underline{f} (s, r) = 2 s (r^{5} + 2 r) [3 - 3 cos (s) - s^{2}], \\ \bar{f} (s, r) = 6 s (2 - r^{3}) [3 - 3 cos (s) - s^{2}], \end{matrix}$ and kernel

$K (s, t) = t cos (s - t), 0 \leq s \leq t, 0 \leq t \leq \frac{π}{2},$ and a = 0, $b = \frac{π}{2}$ , λ = 1, the exact solution is

$\begin{matrix} \underline{u} (s, r) = s^{3} (r^{5} + 2 r), \\ \bar{u} (s, r) = s^{3} (6 - 3 r^{3}) . \end{matrix}$

In this example, K (s, t) ≥0 for each 0 ≤ s ≤ t, we have

$\begin{matrix} \underline{f} (s, r) = \underline{u} (s, r) - λ \int_{0}^{s} t cos (s - t) \underline{u} (t, r) dt, \\ \bar{f} (s, r) = \bar{u} (s, r) - λ \int_{0}^{s} t cos (s - t) \bar{u} (t, r) dt, \end{matrix}$ and

$\begin{matrix} \underline{u} (s, r) = \sum_{i = 0}^{n} {\underline{U}}_{i} (r) F_{i} (s) - \underline{f} (s, r) \\ - \sum_{i = 0}^{n} {\underline{U}}_{i} (r) \int_{0}^{s} t cos (s - t) F_{i} (t) dt, \\ \bar{u} (s, r) = \sum_{i = 0}^{n} {\bar{U}}_{i} (r) F_{i} (s) - \bar{f} (s, r) \\ - \sum_{i = 0}^{n} {\bar{U}}_{i} (r) \int_{0}^{s} t cos (s - t) F_{i} (t) dt . \end{matrix}$

Example 6.1 has been solved by assuming N = 5 and $s = \frac{π}{4}$ . In Fig. 1, a comparison is made between the approximate solution and the exact solution. The absolute error of the approximate solution is clearly shown in Figs. 2 and 3. Table 1 illustrate the behavior of absolute errors of the current technique for N = 5 and the VIM in [49].

Fig. 1

The plot shows a comparison between the approximate solution and the exact solution of the Example 6.1 with N = 5.

Fig. 2

Absolute error $| \underline{u} (s, r) - {\underline{u}}_{5} (s, r) |$ .

Fig. 3

Absolute error $| \bar{u} (s, r) - {\bar{u}}_{5} (s, r) |$ .

Table 1

Numerical results of Example 6.1

	Exact solution	Approximate solution	Absolute error for presented method	Absolute error for VIM in [49]
r	$(\underline{u} (s, r), \bar{u} (s, r))$	$s = \frac{π}{4}$ and N = 5	$s = \frac{π}{4}$ and N = 5	$s = \frac{π}{4}$ and N = 7
0.0	(0.000, 2.906)	(0.000, 2.906)	(0.000e-00, 1.776e-15)	(0.000e-00, 6.373e-10)
0.1	(0.096, 2.905)	(0.096, 2.905)	(4.163e-17, 1.332e-15)	(2.124e-11, 6.370e-10)
0.2	(0.193, 2.895)	(0.193, 2.895)	(1.110e-16, 8.881e-16)	(4.252e-11, 6.347e-10)
0.3	(0.291, 2.768)	(0.291, 2.867)	(2.775e-16, 4.440e-16)	(6.399e-11, 6.287e-10)
0.4	(0.392, 2.813)	(0.392, 2.813)	(5.551e-17, 2.220e-15)	(8.606e-11, 6.169e-10)
0.5	(0.499, 2.725)	(0.499, 2.725)	(2.775e-16, 8.881e-16)	(1.095e-10, 5.974e-10)
0.6	(0.619, 2.592)	(0.619, 2.592)	(1.110e-16, 1.776e-15)	(1.357e-10, 5.685e-10)
0.7	(0.759, 2.408)	(0.759, 2.408)	(3.330e-16, 8,881e-16)	(1.665e-10, 5.280e-10)
0.8	(0.933, 2.162)	(0.933, 2.162)	(3.330e-16, 1.332e-15)	(2.047e-10, 4.741e-10)
0.9	(1.158, 1.847)	(1.158, 1.847)	(2.220e-16, 1.332e-15)	(2.539e-10, 4.050e-10)

Example 6.2. Let the fuzzy Volterra integral equation

$\begin{matrix} \underline{f} (s, r) = r + r^{2} - s (r + r^{2}) sinh (s), \\ \bar{f} (s, r) = 4 - r - r^{3} - s (4 - r - r^{3}) sinh (s), \end{matrix}$

and kernel

$K (s, t) = sinh (s), 0 \leq s \leq t \leq 1,$

and a = 0, b = 1, λ = 1, the exact solution is

$\begin{matrix} \underline{u} (s, r) = r^{2} + r, \\ \bar{u} (s, r) = 4 - r^{3} - r, \end{matrix}$

we have

$\begin{matrix} \underline{f} (s, r) = \underline{u} (s, r) - λ \int_{0}^{s} sinh (s) \underline{u} (t, r) dt, \\ \bar{f} (s, r) = \bar{u} (s, r) - λ \int_{0}^{s} sinh (s) \bar{u} (t, r) dt, \end{matrix}$

and

$\begin{matrix} \underline{u} (s, r) = \sum_{i = 0}^{n} {\underline{U}}_{i} (r) F_{i} (s) - \underline{f} (s, r) \\ - \sum_{i = 0}^{n} {\underline{U}}_{i} (r) \int_{0}^{s} sinh (s) F_{i} (t) dt, \\ \bar{u} (s, r) = \sum_{i = 0}^{n} {\bar{U}}_{i} (r) F_{i} (s) - \bar{f} (s, r) \\ - \sum_{i = 0}^{n} {\bar{U}}_{i} (r) \int_{0}^{s} sinh (s) F_{i} (t) dt . \end{matrix}$

Example 6.2 has been solved by assuming N = 5 and s = 0.3.

As in the previous example, a comparison is made between the approximate solution and the exact solution and the absolute error of the approximate solution is shown in Figs. 4-6, respectively. In Table 2 illustrate the behavior of absolute errors of the presented method for N = 5.

Fig. 4

Comparison between the approximate solution and the exact solution of the Example 6.2 with N = 5.

Fig. 5

Absolute error $| \underline{u} (s, r) - {\underline{u}}_{5} (s, r) |$ .

Fig. 6

Absolute error $| \bar{u} (s, r) - {\bar{u}}_{5} (s, r) |$ .

Table 2

Numerical results of Example 6.2

	Exact solution	Approximate solution	Absolute error for presented method
r	$(\underline{u} (s, r), \bar{u} (s, r))$	s = 0.3 and N = 5	s = 0.3 and N = 5
0.0	(0.000, 4.000)	(0.000, 4.000)	(0.000e-00, 0.000e-00)
0.1	(0.110, 3.899)	(0.110, 3.899)	(0.000e-00, 0.000e-00)
0.2	(0.240, 3.792)	(0.240, 3.792)	(2.775e-17, 0.000e-00)
0.3	(0.390, 3.673)	(0.390, 3.673)	(0.000e-00, 0.000e-00)
0.4	(0.560, 3.536)	(0.560, 3.536)	(1.110e-16, 0.000e-00)
0.5	(0.750, 3.375)	(0.750, 3.375)	(0.000e-00, 0.000e-00)
0.6	(0.960, 3.184)	(0.960, 3.184)	(0.000e-00, 0.000e-00)
0.7	(1.190, 2.957)	(1.190, 2.957)	(0.000e-00, 0.000e-00)
0.8	(1.440, 2.688)	(1.440, 2.688)	(0.000e-00, 0.000e-00)
0.9	(1.710, 2.371)	(1.710, 2.371)	(0.000e-00, 0.000e-00)

Example 6.3. Consider the fuzzy Volterra integral equation of the secon kind

$\begin{matrix} \underline{f} (s, r) = r, \\ \bar{f} (s, r) = 2 - r, \end{matrix}$

and kernel

$K (s, t) = 1, 0 \leq s \leq t \leq 1,$ and a = 0, b = 1 and λ = 1, the exact solution is

$\begin{matrix} \underline{u} (s, r) = {re}^{s}, \\ \bar{u} (s, r) = (2 - r) e^{s}, \end{matrix}$

we have

$\begin{matrix} \underline{f} (s, r) = \underline{u} (s, r) - λ \int_{0}^{s} \underline{u} (t, r) dt, \\ \bar{f} (s, r) = \bar{u} (s, r) - λ \int_{0}^{s} \bar{u} (t, r) dt, \end{matrix}$

and

$\begin{matrix} \underline{u} (s, r) = \sum_{i = 0}^{n} {\underline{U}}_{i} (r) F_{i} (s) - \underline{f} (s, r) \\ - \sum_{i = 0}^{n} {\underline{U}}_{i} (r) \int_{0}^{s} F_{i} (t) dt, \\ \bar{u} (s, r) = \sum_{i = 0}^{n} {\bar{U}}_{i} (r) F_{i} (s) - \bar{f} (s, r) \\ - \sum_{i = 0}^{n} {\bar{U}}_{i} (r) \int_{0}^{s} F_{i} (t) dt . \end{matrix}$

Example 6.3 has been solved by assuming N = 3 and $s = \frac{1}{2}$ .

In this example, as in the previous ones, we have compared the error. In Fig. 7, the plot shows a comparison between the approximate solution and the exact solution. The absolute error of the approximate solution is obviously shown in Figs. 8 and 9. In Table 3, demonstrate the behavior of absolute errors of the presented method for $s = \frac{1}{2}$ and N = 3.

Fig. 7

The plot shows a comparison between the approximate solution and the exact solution of the Example 6.3 with N = 3.

Fig. 8

Absolute error $| \underline{u} (s, r) - {\underline{u}}_{3} (s, r) |$ .

Fig. 9

Absolute error $| \bar{u} (s, r) - {\bar{u}}_{3} (s, r) |$ .

Table 3

Numerical results of Example 6.3

	Exact solution	Approximate solution	Absolute error for presented method
r	$(\underline{u} (s, r), \bar{u} (s, r))$	$s = \frac{1}{2}$ and N = 3	$s = \frac{1}{2}$ and N = 3
0.0	(0.000, 3.297)	(0.000, 3.296)	(0.000e-0, 5.675e-4)
0.1	(0.164, 3.132)	(0.164, 3.132)	(2.837e-5, 5.391e-4)
0.2	(0.329, 2.967)	(0.329, 2.967)	(5.675e-5, 5.107e-4)
0.3	(0.494, 2.802)	(0.494, 2.802)	(8.511e-5, 4.824e-4)
0.4	(0.659, 2.637)	(0.659, 2.637)	(1.135e-4, 4.540e-4)
0.5	(0.824, 2.473)	(0.824, 2.472)	(1.418e-4, 4.256e-4)
0.6	(0.989, 2.308)	(0.989, 2.307)	(1.702e-4, 3.972e-4)
0.7	(1.154, 2.143)	(1.153, 2.142)	(1.986e-4, 3.689e-4)
0.8	(1.318, 1.978)	(1.318, 1.978)	(2.270e-4, 3.405e-4)
0.9	(1.483, 1.813)	(1.483, 1.813)	(2.553e-4, 3.121e-4)

7 Conclusion

As an alternative method to solve a second-order fuzzy Volterra integral equations based on the collocation method of Fibonacci polynomials, a numerical method was introduced. We showed that when the exact solution is a polynomial, the absolute error can be zero and the approximate solution is equal to the exact solution. Existence and uniqueness of the solution of the proposed method along with its convergence clearly has been illustrated. Using several numerical examples, the rational and tiny amount of error of the suggested method were expressed. The outcomes of the given examples showed that the convergence and the precision of the proposed method were in a good settlement with the analytical method. Regarding the analogy of numerical results, cited in figures and tables, our techniques improved the precision of the solutions, consequently can be widely used. Mathematica software was applied for calculations in this article. This method is generic with simple computations and yields very accurate outcomes. Using the concept of horizontal membership functions, a new definition of fuzzy integral called granular integral is defined. We intend to do new study based on this definition. Furthermore, we will adopt our approach in speed and exactness on nonlinear integral equations and other polynomials.

References

Park

J.H.

, Intuitionistic fuzzy metric spaces, Chaos Solitons & Fractals 22(5) (2004), 1039–1046.

Caldas

and Jafari

, θ-Compact fuzzy topological spaces, Chaos Solitons & Fractals 25(1) (2005), 229–232.

Feng

and Chen

, Adaptive control of discrete-time chaotic systems: a fuzzy control approach, Chaos Solitons & Fractals 23(2) (2005), 459–467.

Jiang

, Guo-Dong

and Bin

, H1 variable universe adaptive fuzzy control for chaotic system, Chaos Solitons & Fractals 24(4) (2005), 1075–1086.

Abbasbandy

and Allahviranloo

, Numerical solution of fuzzy differential equation by Runge-Kutta method, Journal of Science Teacher Training University 11(1) (2004), 117–129.

Abbasbandy

, Allahviranloo

, Lopez-Pouso

and Nieto

J.J.

, Numerical methods for fuzzy differential inclusions, Computers & Mathematics with Applications 48(10-11) (2004), 1633–1641.

Abbasbandy

, Nieto

J.J.

and Alavi

, Tuning of reachable set in one dimensional fuzzy differential inclusions, Chaos Solitons & Fractals 26(5) (2005), 1337–1341.

Dong

N.P.

, Long

H.V.

and Khastan

, Optimal control of a fractional order model for granular SEIR epidemic with uncertainty, Communications in Nonlinear Science and Numerical Simulation 88 (2020), 105312.

Son

N.T.K.

, Thao

H.T.P.

, Dong

N.P.

and Long

H.V.

, Fractional calculus of linear correlated fuzzy-valued functions related to Fréchet differentiability. Fuzzy Sets and Systems. In Press. (2020).

10.

Son

N.T.K.

, Long

H.V.

and Dong

N.P.

, Fuzzy delay differential equations under granular differentiability with applications. Computational and Applied Mathematics, 38(3) (2019), 107–136.

11.

Son

N.T.K.

, Dong

N.P.

, Abdel-Basset

, Manogaran

and Long

H.V.

, On the stabilizability for a class of linear time-invariant systems under uncertainty, Circuits Systems and Signal Processing 39(2) (2020), 919–960.

12.

Son

N.T.K.

, Dong

N.P.

and Long

H.V.

, Towards granular calculus of single-valued neutrosophic functions under granular computing, Multimedia Tools and Applications 79 (2019), 16845–16881.

13.

El Naschie

M.S.

, A review of E infinity theory and the mass spectrum of high energy particle physics, Chaos Solitons & Fractals 19(1) (2004), 209–236.

14.

El Naschie

M.S.

, On a fuzzy Kähler-like manifold which is consistent with the two slit experiment, International Journal of Nonlinear Sciences and Numerical Simulation 6(2) (2005), 95–98.

15.

El Naschie

M.S.

, The concepts of E infinity: An elementary introduction to the Cantorian-fractal theory of quantum physics, Chaos Solitons & Fractals 22(2) (2004), 495–511.

16.

Tanaka

, Mizuno

and Kado

, Chaotic dynamics in the Friedmann equation, Chaos Solitons & Fractals 24(2) (2005), 407–422.

17.

Dubois

and Prade

, Towards fuzzy differential calculus part 2: Integration on fuzzy intervals, Fuzzy Sets and Systems 8(2) (1982), 105–116.

18.

Goetschel

Jr and Voxman

, Elementary fuzzy calculus, Fuzzy Sets and Systems 18(1) (1986), 31–43.

19.

and Gong

, On Henstock integral of fuzzy-number-valued functions (I), Fuzzy Sets and Systems 120(3) (2001), 523–532.

20.

Bede

and Gal

S.G.

, Quadrature rules for integrals of fuzzy-number-valued functions, Fuzzy Sets and Systems 145(3) (2004), 359–380.

21.

Kaleva

, Fuzzy differential equations, Fuzzy Sets and Systems 24(3) (1987), 301–317.

22.

Seikkala

, On the fuzzy initial value problem, Fuzzy Sets and Systems 24(3) (1987), 319–330.

23.

Babolian

, Goghary

H.S.

and Abbasbandy

, Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Applied Mathematics and Computation 161(3) (2005), 733–744.

24.

Allahviranloo

, Khezerloo

, Ghanbari

and Khezerloo

, The homotopy perturbation method for fuzzy Volterra integral equations, International Journal of Computational Cognition 8(2) (2010), 31–37.

25.

Ghanbari

, Numerical solution of fuzzy linear Volterra integral equations of the second kind by homotopy analysis method, International Journal of Industrial Mathematics 2(2) (2010), 73–87.

26.

Allahviranloo

, Khalilzadeh

and Khezerloo

, Solving linear Fredholm fuzzy integral equations of the second kind by modified trapezoidal method. Journal of Applied Mathematics, Islamic Azad University of Lahijan 7(4) (2011), 25–37.

27.

Allahviranloo

and Behzadi

S.S.

, The use of airfoil and Chebyshev polynomials methods for solving fuzzy Fredholm integro-differential equations with Cauchy kernel, Soft Computing 18(10) (2014), 1885–1897.

28.

Hussain

E.A.

and Ali

A.W.

, Linear Volterra fuzzy integral equation solved by modified trapezoidal method, International Journal of Applied Mathematics & Statistical Sciences 2(2) (2013), 43–54.

29.

Mosleh

and Otadi

, Solution of fuzzy Volterra integral equations in a Bernstein polynomial basis, Journal of Advances in Information Technology 4(3) (2013), 148–155.

30.

Mirzaee

and Hoseini

S.F.

, Solving systems of linear Fredholm integro-differential equations with Fibonacci polynomials, Ain Shams Engineering Journal 5(1) (2014), 271–283.

31.

Behzadi

S.S.

, Allahviranloo

and Abbasbandy

, Fuzzy collocation methods for second-order fuzzy Abel-Volterra integro-differential equations, Iranian Journal of Fuzzy Systems 11(2) (2014), 71–88.

32.

Mirzaee

and Hoseini

S.F.

, A Fibonacci collocation method for solving a class of Fredholm-Volterra integral equations in two-dimensional spaces, Beni-Suef University Journal of Basic and Applied Sciences 3(2) (2014), 157–163.

33.

Narayanamoorthy

and Sathiyapriya

S.P.

, Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind, Springer Plus 5(1) (2016), 1–16.

34.

Hamaydi

and Qatanani

, Computational methods for solving linear fuzzy Volterra integral equation, Journal of Applied Mathematics (2017).

35.

Alijani

and Kangro

, Collocation method for fuzzy Volterra integral equations of the second kind, Mathematical Modelling and Analysis 25(1) (2020), 146–166.

36.

Allahviranloo

and Hajari

, Numerical methods for approximation of fuzzy data, Applied Mathematics and Computation 169(1) (2005), 16–33.

37.

Bede

and Gal

S.G.

, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems 151(3) (2005), 581–599.

38.

Hochstadt

, Integral equations, John Wiley & Sons (2011), 91.

39.

Congxin

and Ming

, On the integrals, series and integral equations of fuzzy set valued functions, Journal of Harbin Institute of Technology 21 (1990), 11–19.

40.

Brunner

and Lambert

J.D.

, Stability of numerical methods for Volterra integro-differential equations, Computing 12(1) (1974), 75–89.

41.

Crisci

M.R.

, Jackiewicz

, Russo

and Vecchio

, Global stability analysis of the Runge-Kutta methods for Volterra integral and integro-differential equations with degenerate kernels, Computing 45(4) (1990), 291–300.

42.

Falcon

and Plaza

, On k-Fibonacci sequences and polynomials and their derivatives, Chaos Solitons & Fractals 39(3) (2009), 1005–1019.

43.

Rainville

E.D.

, Special functions, Chelsea Publishing Company (1971).

44.

Sahu

P.K.

and Ray

S.S.

, A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein-Volterra delay integral equations, Fuzzy Sets and Systems 309 (2017), 131–144.

45.

Friedman

, Ming

and Kandel

, Fuzzy linear systems, Fuzzy Sets and Systems 96(2) (1998), 201–209.

46.

Ezzati

, Solving fuzzy linear systems, Soft Computing 15(1) (2011), 193–197.

47.

Abd-Elhameed

W.M.

, Youssri

Y.H.

, El-Sissi

and Sadek

, New hypergeometric connection formulae between Fibonacci and Chebyshev polynomials, The Ramanujan Journal 42(2) (2017), 347–361.

48.

Grimm

R.E.

, The autobiography of leonardo pisano, The Fibonacci Quarterly 11(1) (1973), 99–104.

49.

Allahviranloo

, Ghanbari

and Nuraei

, An application of a semi-analytical method on linear fuzzy Volterra integral equations, Journal of Fuzzy Set Valued Analysis 2014 (2014), 1–15.

50.

Wazwaz

A.M.

, Linear and nonlinear integral equations, Springer, Berlin, Heidelberg, (2011), 639.