Dynamical control of computations using the finite differences method to solve fuzzy boundary value problem

Abstract

In this paper, by using the fuzzy CESTAC method and the CADNA library a procedure is proposed to control the step size for solving the fuzzy differential equation with fuzzy boundary conditions based on the finite differences method under generalized H-differentiability (gH-differentiability). An algorithm is presented to implement the discrete stochastic arithmetic for solving the given fuzzy boundary value problem on the C++ code via the CADNA library. Also, a theorem is proved to show the accuracy of results based on the concept of the common proximity of two fuzzy numbers. Finally, some examples are solved by using the proposed algorithm to illustrate the effectiveness of applying the stochastic arithmetic (SA) in place of the floating-point arithmetic (FPA) to validate the results and find the optimal solution.

Keywords

Finite differences method Fuzzy boundary value problem (FBVP)Stochastic arithmetic CESTAC method CADNA Library

1 Introduction

The study of fuzzy differential equations (FDEs) forms a suitable setting for the mathematical modeling of real world problems in which uncertainty or vagueness pervades. Hence solving FDEs in both initial and boundary conditions are important topics in fuzzy mathematics and its applications [15 , 26]. In recent years, different numerical and analytical approaches were proposed to find the fuzzy solution of a given FDE such as [6 , 31]. In these works, the exact solution of the FDE must be determined in order to compare with the computed results when a numerical scheme has been considered. Since the calculations are based on the FPA, a fixed number of iterations or a constant step size is assumed to implement the algorithms by a package like Matlab, Maple or Mathematica. In this case, the accuracy of the results cannot be verified and optimized. Finite precision computations may affect the stability of algorithms and the accuracy of computed solutions. In fact, the numerical result provided by an algorithm is affected by a global error, which consists of both a truncation error and a round-off error. Computation of the solution of many problems in scientific computing involves a step size h. As h decreases, the truncation error also decreases, but the round-off error in the method may increase. Here, the problem is to find the optimal step size h_opt which minimizes the global error. In general, it is difficult to estimate the optimal step size in an algorithm. So, it needs to replace a new arithmetic to validate the results of numerical algorithms. Vignes and La Porte [32 –34] proposed the SA to substitute for the common computer arithmetic. Vignes planned the CESTAC method [35, 36] to implement the discrete stochastic arithmetic (DSA). In the sequel, Chesneaux designated a library called CADNA [17, 25], to implement the CESTAC method automatically on a code written by ADA, Fortran or C++. This library has been applied to validate the results of different algorithms for solving various problems such as [1–5 , 28]. In [21], the authors proposed a procedure to obtain the optimal step size for solving fuzzy differential equation by using fuzzy Runge-Kutta method.

In this paper, a procedure is presented to control the step size for solving fuzzy differential equations with fuzzy boundary conditions and under gH-differentiability based on the finite differences method [9]. Also, a characterization theorem presented by Bede [11] and Bede and Gal [13] is used, which describes that an FDE under the H-differentiability and gH-differentiability is equivalent to a system of ODEs under certain conditions that are appropriate for solving FDEs numerically. It is shown, by using the CESTAC method and the CADNA library, the optimal iteration of the fuzzy finite differences method can be dynamically computed and its accuracy can be estimated. By using this library, we determine the optimal step size, and we find the approximate solution of the FDE at the nodal points. Then, the CESTAC method and the SA are used to validate the results and implement the numerical examples.

Let the following fuzzy linear two-point boundary value problem (FBVP) $\begin{matrix} {\tilde{x}}_{gH}^{''} = g_{1} (t) ⊙ {\tilde{x}}_{gH}^{'} \oplus g_{2} (t) ⊙ \tilde{x} \oplus f (t), \\ \tilde{x} (a) = \tilde{A}, \tilde{x} (b) = \tilde{B}, \end{matrix}$ (1) where g₁ (t), g₂ (t) and f (t) are real continuous functions and the values of g₂ (t) are positive on [a, b] and $\tilde{A}$ and $\tilde{B}$ are fuzzy numbers [9]. This paper is organized as follows, in Section 2, the basic definitions of fuzzy numbers and fuzzy sets are presented briefly. In Section 3, the main idea is given, including a brief description of fuzzy finite differences method. Also, a theorem is proved to illustrate the accuracy of the proposed method. The fuzzy CESTAC method in companion with an algorithm which is implemented by the CADNA library to solve Eq. (1) are given in Section 4. In Section 5, two numerical examples are solved to illustrate the importance of using the stochastic arithmetic to validate the results based on the given algorithm.

2 Preliminary

The basic definitions of fuzzy mathematics are given in [8 , 23]. In this section, we review some of them.

Definition 1. [23] Given the definition domain U, the mapping X: U ⟶ [0, 1], is called a fuzzy set. X (u) is denoted as the membership function, whose values locate in a closed interval [0, 1]. If only the point values 0 and 1 are suitable, the fuzzy set X degenerates into a classical one, which means that the classical set is a special form of the fuzzy sets.

Definition 2. [23] Given a fuzzy set X, for any r ∈ [0, 1], the classical set [X] ^r = {u: u ∈ U, X (u) ≥ r} is defined as the r-cut set, where r is the cut level. Usually, the cut sets are considered as intervals of confidence, since in the case of convex fuzzy sets, they are closed intervals associated with a gradation of confidence between [0, 1]. Also, support of X is supp (X) = {u ∈ U: X (u) >0}. The addition X ⊕ Y and the scalar multiplication $α ⊙ X, α \in ℝ$ are defined as having the level cuts $\begin{matrix} [X \oplus Y]^{r} = [X]^{r} + [Y]^{r} = {x + y : x \in [X]^{r}, y \in [Y]^{r}}, \\ [α ⊙ X]^{r} = α [X]^{r}, r \in [0, 1] . \end{matrix}$ So, the product and division of two fuzzy numbers $X, Z \in F (ℝ)$ is $\begin{matrix} {[X \otimes Z]}^{r} & = & [min_{s, t \in {+, -}} x^{s} (r) . z^{t} (r), \\ max_{s, t \in {+, -}} x^{s} (r) . z^{t} (r)], \\ {[X ø Z]}^{r} & = & [min_{s, t \in {+, -}} \frac{x^{s} (r)}{z^{t} (r)}, max_{s, t \in {+, -}} \frac{x^{s} (r)}{z^{t} (r)}], \\ 0 \notin supp (Z) . \end{matrix}$ For a fuzzy interval X, its r-cuts are closed intervals in $ℝ$ and we denote them by [X] ^r = [x⁻ (r), x⁺ (r)]. Two widely used of fuzzy intervals are triangular and trapezoidal fuzzy intervals.

Let us denote by $F (ℝ)$ the set of all fuzzy numbers, however, fuzzy numbers are generalized closed intervals.

Definition 3. The generalized Hukuhara difference of two intervals, $X, Y \in F (ℝ)$ , (gH -difference) is defined as follows [23]: $\begin{matrix} X \underset{gH}{⊖} Y = Z \Leftrightarrow X = Y + Z & or & Y = X + (- Z) . \end{matrix}$

Consider [Z] ^r = [z⁻ (r), z⁺ (r)] then z⁻ (r) = min {x⁻ (r) - y⁻ (r), x⁺ (r) - y⁺ (r)} and z⁺ (r) = max {x⁻ (r) - y⁻ (r), x⁺ (r) - y⁺ (r)}.

Definition 4. [23] Define $D : F (ℝ) \times F (ℝ) ⟶ ℝ^{+}$ by $\begin{matrix} D (u, v) : = \\ sup_{r \in [0, 1]} max {abs u^{-} (r) - v^{-} (r), abs u^{+} (r) - v^{+} (r)} \\ = sup_{r \in [0, 1]} D_{H} ([u]^{r}, [v]^{r}), \end{matrix}$ (2) where is Hausdorff distance between two fuzzy intervals $u, v \in F (ℝ)$ with the properties $\begin{matrix} D (u \oplus w, v \oplus w) = D (u, v), \forall u, v, w \in F (ℝ), \\ D (k ⊙ u, k ⊙ v) = abs k D (u, v), \forall k \in ℝ, u, v \in F (ℝ), \\ D (k_{1} ⊙ u, k_{2} ⊙ u) = abs k_{1} - k_{2} D (u, \tilde{0}), \\ \forall k_{1}, k_{2} \geq 0, u \in F (ℝ), \\ D (u \oplus v, w \oplus z) \leq D (u, w) + D (v, z), \\ \forall u, v, w, z \in F (ℝ) . \end{matrix}$

Definition 5. [9, 23] A function $Y : ℝ ⟶ F (ℝ)$ is called a fuzzy function. If for arbitrary fixed $t_{0} \in ℝ$ and ɛ > 0, a δ > 0 such that $| t - t_{0} | < δ \Rightarrow D (Y (t), Y (t_{0})) < ɛ,$ (3) exists, Y is said to be continuous.

Generalized derivative: [23, 29] Let $Y^{(j)} : ℝ ⟶ F (ℝ)$ (jth-order differential of Y) for all j = 0, 1, …, n − 1 and $t_{0} \in ℝ$ . Y^(j) is called gH-differentiable (gH-differentiable for short) at t₀, if there exists $Y^{(j)} (t_{0}) \in F (ℝ)$ for all j = 0, 1, …, n − 1, such that $lim_{h \to 0} \frac{1}{h} ⊙ (Y^{(j)} (t_{0} + h) \underset{gH}{⊖} Y^{(j)} (t_{0})) = Y_{gH}^{(j + 1)} (t_{0}),$ also we say that Y^(j), j = 0, 1, …, n − 1 is gH_i-differentiable at t₀ if $[Y_{{gH}_{i}}^{(j)} (t)]^{r} = [y^{-^{(j)}} (t; r), y^{+^{(j)}} (t; r)],$ and we say that Y^(j), j = 0, 1, …, n − 1 is gH_ii-differentiable at t₀ if $[Y_{{gH}_{ii}}^{(j)} (t)]^{r} = [y^{+^{(j)}} (t; r), y^{-^{(j)}} (t; r)] .$

Fuzzy Taylor formula: [8] Let $Y^{(j)} : T \subseteq ℝ ⟶ F (ℝ)$ are gH-differentiable for all j = 0, 1, …, n − 1 and fuzzy continuous on T. Then for s ≥ a; s, a ∈ T $\begin{matrix} Y (s) = Y (a) \oplus Y^{'} (a) ⊙ \frac{(s - a)}{1!} \oplus . . . \oplus \\ Y^{(n - 1)} (a) ⊙ \frac{(s - a)^{n - 1}}{(n - 1)!} \oplus R_{n} (a, s) \\ = \sum_{i = 0}^{* n - 1} Y^{(i)} (a) ⊙ \frac{(s - a)^{i}}{(i)!} \oplus R_{n} (a, s), \end{matrix}$ where $R_{n} (a, s) : = \int_{a}^{s} (\int_{a}^{s_{1}} . . . (\int_{a}^{s_{n - 1}} f^{(n)} (s_{n}) {ds}_{n}) {ds}_{n - 1}) . . . {ds}_{1} .$ Here R_n (a, s) is fuzzy continuous on T and ∑^* denotes the fuzzy summation.

Theorem 1. [23, 27] Let us consider y^′′ = F (t, x (t), x′ (t)) where $F : [t_{0}; t_{0} + a] \times F (ℝ) \times F (ℝ) ⟶ F (ℝ)$ is such that

[F (t, x (t), x′ (t))] ^r = [f⁻ (t, x, x′) (r), f⁺ (t, x, x′) (r)] for each r ∈ [0, 1].

The functions f⁻ (t, x, x′) (r),and f⁺ (t, x, x′) (r) are equi-continuous, i.e. for any ɛ > 0 there exists δ > 0 such that for any $(t, y, z), (t_{1}, y_{1}, z_{1}) \in [t_{0}, t_{0} + a, t_{0} + b] \times ℝ^{2},$ we have absf⁻ (t, y, z) (r) - f⁻ (t₁, y₁, z₁) (r) < ɛ and absf⁺ (t, y, z) (r) - f⁺ (t₁, y₁, z₁) (α) < ɛ whenever norm (t, y, z) - (t₁, y₁, z₁) < δ.

[F (t, x (t), x′ (t))] ^r is named uniformly bounded on any bounded set, if exist l > 0 such that $\begin{matrix} abs f^{-} (t_{1}, y_{1}, z_{1}) (r) - f^{-} (t_{2}, y_{2}, z_{2}) (r) < \\ l max {abs t_{1} - t_{2}, abs y_{1} - y_{2}, abs z_{1} - z_{2}}, \end{matrix}$ and also $\begin{matrix} abs f^{+} (t_{1}, y_{1}, z_{1}) (r) - f^{+} (t_{2}, y_{2}, z_{2}) (r) < \\ l max {abs t_{1} - t_{2}, abs y_{1} - y_{2}, abs z_{1} - z_{2}} . \end{matrix}$ Then, for the FBVP (1) and the corresponding BVPs systems are equivalent.

3 Main idea

In order to solve Eq. (1) by the finite differences method, the interval [a, b] is divided into N equal subintervals by the grid points t_i = t₀ + ih, i = 0, 1, …, N where h = (b - a)/N. At the interior mesh points, t_i, for i = 1, …, N, the solution of a differential equation is ${\tilde{x}}_{gH}^{''} (t_{i}) = g_{1} (t_{i}) ⊙ {\tilde{x}}_{gH}^{'} (t_{i}) \oplus g_{2} (t_{i}) ⊙ \tilde{x} (t_{i}) \oplus f (t_{i}) .$ Expanding $\tilde{x}$ in a third fuzzy Taylor formula [9] and applying definition of standard difference and definition 2, we get $\begin{array}{l} {\tilde{x}}^{″}_{g H} (t_{i}) = \frac{1}{h^{2}} ⊙ ((\tilde{x} (t_{i + 1}) θ_{g H} 2 ⊙ \tilde{x} (t_{i}) \oplus \tilde{x} (t_{i - 1})) \\ θ_{g H} \frac{h^{2}}{12} ⊙ ({\tilde{x}}^{(4)} (ξ_{i})), ξ_{i} \in (t_{i - 1}, t_{i + 1}), \end{array}$ so $\begin{array}{l} \tilde{x} (t_{i}) = \frac{1}{h^{2}} ⊙ ((\tilde{x} (t_{i + 1}) θ_{g H} \tilde{x} (t_{i - 1})) \\ θ_{g H} \frac{h^{2}}{6} ⊙ ({\tilde{x}}^{″}^{'} (η_{i})), η_{i} \in (t_{i - 1}, t_{i + 1}), \end{array}$ thus $\begin{array}{l} \frac{1}{h^{2}} ⊙ ((\tilde{x} (t_{i + 1}) θ_{g H} 2 ⊙ \tilde{x} (t_{i}) \tilde{x} (t_{i - 1})) = \\ g_{1} (t_{i}) ⊙ \frac{1}{2 h} ⊙ (\tilde{x} (t_{i + 1}) θ_{g H} \tilde{x} (t_{i - 1})) \oplus g_{2} (t_{i}) ⊙ \tilde{x} (t_{i}) \\ \oplus f (t_{i}) θ_{g}_{H} \frac{h^{2}}{12} ⊙ (2 ⊙ g_{1} (t_{i}) ⊙ {\tilde{x}}^{″}^{'} (η_{i}) θ_{g H} {\tilde{x}}^{(4)} (ξ_{i})) . \end{array}$ for each i = 1, 2, …, N. A finite differences method with truncation error of order $O (h^{2})$ results by using above equation together with the boundary conditions $\tilde{x} (a) = \tilde{A}$ and $\tilde{x} (b) = \tilde{B}$ to define the system of linear equations

g₁ (t_i) >0 for i = 1, …, N $\begin{array}{l} \tilde{w} i - 1 ⊖_{g H} h^{2} ⊙ g_{2} (t_{i}) ⊙ {\tilde{w}}_{i} \oplus {\tilde{w}}_{i + 1} \\ ⊖_{g H} \frac{h}{2} ⊙ g_{1} (t_{i}) ⊙ {\tilde{w}}_{i + 1} ⊖_{g H} 2 ⊙ {\tilde{w}}_{i} = h^{2} ⊙ f (t_{i}) . \end{array}$ (4)

g₁ (t_i) <0 for i = 1, …, N $\begin{array}{l} (1 + \frac{h}{2} ⊙ g_{1} (t_{i})) ⊙ {\tilde{w}}_{i - 1} ⊖_{g H} h^{2} ⊙ g_{2} (t_{i}) ⊙ \tilde{w_{i}} \oplus {\tilde{w}}_{i + 1} \\ \oplus \frac{h}{2} ⊙ g_{1} (t_{i}) {\tilde{w}}_{i + 1} ⊖_{g H} 2 ⊙ \tilde{w_{i}} = h^{2} ⊙ f (t_{i}), \end{array}$ (5) where ${\tilde{w}}_{i}, s$ are approximation of $\tilde{x} (t_{i})$ . Also, the ${\tilde{w}}_{i} (h), s$ are approximation of $\tilde{x} (t_{i})$ for i = 1, 2, …, N with the step size h, similarly ${\tilde{w}}_{i} (\frac{h}{k})$ for step size $\frac{h}{k}$ . The local truncation error with step size h can be written as $w ˜ i (h) ⊖ g H \tilde{x} (t_{i}) = \tilde{M} ⊙ h^{2} + O (h^{4}),$ (6) where $\tilde{M}$ is independent of h. With step size $\frac{h}{k}, 1 \neq k \in ℕ,$ ${\tilde{w}}_{i} (h k) ⊖ g H \tilde{x} (t_{i}) = \tilde{M} ⊙ \frac{h^{2}}{k^{2}} + O (h^{4}),$ (7) however, we can rewrite ${\tilde{w}}_{i} (h) ⊖_{g H} {\tilde{w}}_{i} (\frac{h}{k}) = \tilde{M} ⊙ (1 - \frac{1}{k^{2}}) h^{2} + O (h^{4}) .$ (8)

Bede [11] defined the Characterization Theorem that provides certain conditions under which an FDE is equivalent to a system of ODEs with respect to H-differentiability. Thereafter, Bede and Gal [13] proposed another version of this theorem for solving FDEs under gH-differentiability. Both of these characterization theorems are employed to transfer the FDEs under fuzzy differentiability to ODEs systems. So, A. Khastan, J.J. Nieto [27] by using Characterization Theorem under gH-differentiability, present a method for translating a FBVP to a system of BVPs. Based on Definition 2, gH_(i)-differentiable, gH_(ii)-differentiable and the Characterization Theorem 1 [27], each of Eqs. (2) and (3), under certain conditions, we may replace by two equivalent BVPs systems.

In any case, we have system of linear equations with 2N equations as $\begin{matrix} \tilde{H} \tilde{W} = \tilde{Q}, \tilde{W} = [\tilde{w_{1}}, \tilde{w_{2}}, . . ., \tilde{w_{N}}]^{T}, \end{matrix}$ (9) where elements of the matrix $\tilde{H}$ and vector $\tilde{Q}$ , are obtained by Eqs. (2) and (3). The solution of this system of linear equations gives the finite differences solution of Eq. (1) satisfying its fuzzy boundary conditions.

Definition 6. For two distinct fuzzy numbers $\tilde{x}$ and $\tilde{z}$ in $F (ℝ)$ the notation $C_{\tilde{x}, \tilde{z}}$ means the common proximity between $\tilde{x}$ and $\tilde{z}$ , which is defined as, $C_{\tilde{x}, \tilde{z}} = \log D (\tilde{x} \emptyset (\tilde{x} θ_{g}_{_{H}} \tilde{z}) - \frac{1}{2}, \tilde{0}) .$ (10)

Remark 1. The defuzzification, the 1-cut of Eq. (10) is as follows $\begin{matrix} C_{x, z} = log abs \frac{x}{x - z} - \frac{1}{2}, \end{matrix}$ where the notation C_x,z is estimation of the number of common significant digits between two distinct real numbers x and z [1,15, 1,15]. For instance, if C_x,z = 3, the relative difference between x and z is of the order of 10⁻³, which means that x and z have three significant decimal digits in common.

Example 1. Assume $\tilde{x} = (3.754749 + 0.000025 r, 3.754899 - 0.000125 r)$ and $\tilde{z} = (3.754421 + 0.000013 r, 3.754655 - 0.000221 r)$ are two fuzzy numbers, then based on Definition 6, we easily see that common proximity between $\tilde{x}$ and $\tilde{z}$ become actually different from the fourth position namely $C_{\tilde{x}, \tilde{z}} = 4$ .

We prove the following theorem for computing of the common proximity of each corresponding components of the computed solution and the exact solution for a FBVP using finite differences method.

Theorem 2. Suppose ${\tilde{w}}_{i} (h)$ , ${\tilde{w}}_{i} (h / k),$ $1 \neq k \in ℕ,$ are the approximate values of $\tilde{x} (t_{i})$ in two successive iterations respectively obtained from the fuzzy finite differences method to solve FBVP (1). Then $C_{{\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / k)} = C_{{\tilde{w}}_{i} (h), \tilde{x} (t_{i})} + log (\frac{k^{2}}{k^{2} - 1}) + O (h^{2}) .$ (11)

Proof. By using Eq. (6), we deduce $\begin{array}{l} {\tilde{w}}_{i} (h) ⊘ ({\tilde{w}}_{i} (h) ⊖_{g H} \tilde{x} (t_{i})) \\ = {\tilde{w}}_{i} (h) ⊘ (\tilde{M} ⊙ h^{2} + O (h^{4})) {\tilde{w}}_{i} (h) ⊘ (\tilde{M} ⊙ h^{2} ⊙ (1 + O (h^{2}))) \\ = (1 + O (h^{2})) ⊙ {\tilde{w}}_{i} (h) ⊘ (\tilde{M} ⊙ h^{2}) \\ = {\tilde{w}}_{i} (h) ⊘ (\tilde{M} ⊙ h^{2}) + O (1), \end{array}$ then $\begin{array}{l} {\tilde{w}}_{i} (h) \emptyset ({\tilde{w}}_{i} (h) θ_{g H} \tilde{x} (t_{i})) - \frac{1}{2} = \\ {\tilde{w}}_{i} (h) \emptyset (\tilde{M} ⊙ h^{2}) + O (1) . \end{array}$ (12) Similarly, from Eq. (8), we get $\begin{array}{l} {\tilde{w}}_{i} (h) \emptyset ({\tilde{w}}_{i} (h) θ_{g H} {\tilde{w}}_{i} (h / k)) - \frac{1}{2} = \\ {\tilde{w}}_{i} (h) \emptyset (\tilde{M} ⊙ h^{2}) ⊙ (\frac{k^{2}}{k^{2} - 1}) + O (1) . \end{array}$ (13) From Definition 6 and Eq. (12) we have $\begin{matrix} C_{{\tilde{w}}_{i} (h), \tilde{x} (t_{i})} \\ = log D ((1 + O (h^{2})) ⊙ {\tilde{w}}_{i} (h) ø (\tilde{M} ⊙ h^{2}), \tilde{0}) \\ = log ((1 + O (h^{2})) D ({\tilde{w}}_{i} (h) ø (\tilde{M} ⊙ h^{2}), \tilde{0})) \\ = log D ({\tilde{w}}_{i} (h) ø (\tilde{M} ⊙ h^{2}), \tilde{0}) + log (1 + O (h^{2})) . \end{matrix}$ Therefore $\begin{matrix} C_{{\tilde{w}}_{i} (h), \tilde{x} (t_{i})} = log D ({\tilde{w}}_{i} (h) ø (\tilde{M} ⊙ h^{2}), \tilde{0}) + O (h^{2}) . \end{matrix}$ (14) Similarly, by using Definition 6 and Eq. (13) the following formula can be obtained as $\begin{matrix} C_{{\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / k)} = \\ log D ({\tilde{w}}_{i} (h) ø (\tilde{M} ⊙ h^{2}), \tilde{0}) + log (\frac{k^{2}}{k^{2} - 1}) + O (h^{2}) . \end{matrix}$ (15) Finally $C_{{\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / k)} = C_{{\tilde{w}}_{i} (h), \tilde{x} (t_{i})} + log (\frac{k^{2}}{k^{2} - 1}) + O (h^{2}) . □$

This theorem shows, if h is small enough, then the number of common proximity between ${\tilde{w}}_{i} (h)$ and ${\tilde{w}}_{i} (h / k)$ is almost equal to the number of the common proximity between ${\tilde{w}}_{i} (h)$ and $\tilde{x} (t_{i})$ . It is necessary to mention that $0 < log (\frac{k^{2}}{k^{2} - 1}) < 1 .$ According to Eq. (11),

For k = 2: $C_{{\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / 2)} = C_{{\tilde{w}}_{i} (h), \tilde{x} (t_{i})} + log (\frac{4}{3}) + O (h^{2}),$

For k = 3: $C_{{\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / 3)} = C_{{\tilde{w}}_{i} (h), \tilde{x} (t_{i})} + log (\frac{9}{8}) + O (h^{2}) .$

4 CESTAC method

In many numerical iterative and approximate recursive schemes, the absolute error is applied to show the accuracy of the method. In order to apply the absolute error, existence of the exact solution is necessary. In the iterative algorithms which are constructed based on the common FPA, the termination criterion depends on a tolerance parameter like epsilon (eps) as follows: $abs Z_{m} - ψ < eps,$ (16) where ψ is the exact solution and Z_m is the m-th computed result obtained from the iterative algorithm. This criterion may not be acceptable. If eps is chosen very large, the iterations are stopped before getting access to a suitable approximation. If eps is chosen very small then, unnecessary iterations are done without improving the accuracy of the results. So, it can not been determined the optimal iteration m by applying a tolerance in the stopping criterion. Also, if one applies the criterion absZ_m - Z_m+1 < eps, in place of the condition (1), it is not any guarantee to confirm Z_m is an approximation of ψ without knowing the exact solution. This is because of the nature of the FPA. Therefore, it needs to be suggested other way which is independent of this tolerance. For this purpose, we must replace the FPA by a new arithmetic so that it is able to omit the tolerance epsilon in the termination criterion. This new environment is the stochastic arithmetic. The details of this arithmetic can be found in [1–5 , 32–36]. In order to use this arithmetic in discrete case, the CESTAC method should be applied [20 , 36]. This method is able to validate the results and detect any instability step by step during the run of a code. The algorithm of this method is based on the concept of common significant digits. In this case, the following termination criterion is substituted which is independent of the value epsilon. $abs Z_{m} - Z_{m + 1} = @ . 0,$ (17) where the symbol @.0 is denoted as the informatical zero which means the corresponding value (difference between two sequential results) has not any significant digits and hence in the point of computational view we can say Z_m ≅ Z_m+1. Eq. (17) means when the difference between two sequential results is an informatical zero, the computations must be stopped and any calculations after m-th iteration is redundant. In order to implement the CESTAC method, the CADNA library was designated by Chesnueax [17, 25]. This library is able to perform the discrete stochastic arithmetic and the CESTAC method automatically on any C++ or Fortran code. So, in order to apply it, the main code must be written in one of this programming languages.

The CESTAC method was developed by La Porte and Vignes [34]. This method is able to detect numerical instabilities which occur during the run and estimate accuracy of the computed results. During the run, as soon as the number of the significant digits of any result becomes zero, an informatical zero is detected and the result is printed by the notation @.0.

The basic idea of the CESTAC method is to replace the usual FPA with a random arithmetic. Consequently, each result appears as a random variable. This approach leads toward two concepts: stochastic numbers and stochastic arithmetic.

The applying of the CESTAC method in a scientific program has the following advantages:

The accuracy of any numerical result is estimated, during the running of a program.

The numerical instabilities are detected and the branching are checked.

Unnecessary iterations are eliminated, which the FPA is not able to distinguish them. In some cases, the termination criterion of iterative methods is not suitable so that, the implementation of the algorithm is continued without improvement in the accuracy of the result. In the SA, instead of the termination criterion, a criterion that directly reflects the mathematical condition, is replaced, that must be satisfied by the solution.

It is able to find the optimal step of the iterative methods, which after this step, the accuracy of the result does not increases or maybe decrease, because of the rounding error accumulation.

It is an effective and powerful tool that helps to achieve the validation of scientific programs and gives them a reliable.

Let F be a set of real values which are reproduced by computer and the arbitrary value ψ is demonstrated as Ψ ∈ F. In the personal computer (PC), Ψ with the binary FPA, ρ mantissa bits and the rounding error term is shown by $Ψ = ψ - ɛ 2^{E - θ} ρ,$ (18) where 2^−θρ is the missing segment of mantissa which is obtained from round-off error, ɛ is the sign of ψ, E is the binary power of the outcome. If ρ is a random parameter on [-1, 1] which is distributed uniformly and is applied to perturb on the last mantissa bit of Ψ then, the random result of Ψ can be calculated where mean (μ) and standard deviation (σ) are applied to guarantee the precision of results [32 –36].

In PC, for θ = 24, 53 the results can be obtained by single or double precisions respectively. By L times performing the process for Ψ_i, i = 1, …, L the distribution of them is in the quasi Gaussian form. Therefore, the mean of them is equal with the exact value of ψ and the values of μ and σ can be estimated by these L samples. The following algorithm of CESTAC method is presented where τ_δ is the value of T distribution with L − 1 degree of freedom and confidence interval 1 - δ.

Algorithm 1.

1- Find L samples for Ψ as Φ ={Ψ₁, Ψ₂, …, Ψ_L} by means of the perturbation of the last bit of mantissa.

2- Compute $Ψ_{ave} = \frac{\sum_{i = 1}^{L} Ψ_{i}}{L}$ .

3- Calculate $σ^{2} = \frac{\sum_{i = 1}^{L} (Ψ_{i} - Ψ_{ave})^{2}}{L - 1}$ .

4- Compute $C_{Ψ_{ave}, Ψ} = {log}_{10} \frac{\sqrt{L} | Ψ_{ave} |}{τ_{δ} σ},$ as the common significant digits between Ψ and Ψ_ave.

5- If C_{Ψ_ave,Ψ} ≤ 0 or Ψ_ave = 0, then write Ψ =@.0.

In the sequel, Algorithm 1 is developed in fuzzy case.

4.1 Fuzzy CESTAC method

Let $\tilde{y}$ be a fuzzy number in $F (ℝ)$ . Then, $\tilde{y}$ is represented as $\tilde{v}$ in the computer. It can be shown that: $\begin{matrix} \tilde{v} = \tilde{y} \underset{gH}{⊖} ɛ 2^{E - P} ⊙ \tilde{z}, \end{matrix}$ (19) where, ɛ is the sign of y^± (r), and 2^−Pz^± (r) is the lost part of the mantissa due to round-off error for r ∈ [0, 1] and E is the binary exponent of the results. In the fuzzy CESTAC method, z^± (r), 0 ≤ r ≤ 1 is considered as fuzzy random variable uniformly distributed on [-1, 1]. In order to find L samples for the obtained random variables, we perturb the last mantissa bit of the values v^± (r), 0 ≤ r ≤ 1. The algorithm of the fuzzy CESTAC method is as follows where τ_β is the value of T distribution with L − 1 degree of freedom and confidence interval 1 - β. If L = 3 and β = 0.05 then τ_β = 4.303.

Algorithm 2.

Find L samples for $\tilde{v}$ as ${\tilde{v}}_{1}, {\tilde{v}}_{2}, . . ., {\tilde{v}}_{L} .$

Compute ${\tilde{v}}_{ave} = \frac{1}{L} ⊙ \sum_{i = 1}^{* L} {\tilde{v}}_{i} .$

Compute ${\tilde{S}}^{* 2} = \frac{1}{L - 1} ⊙ \sum_{i = 1}^{* L} ({\tilde{v}}_{i} \underset{g H}{⊖} {\tilde{v}}_{a v e}) \otimes ({\tilde{v}}_{i} ⊖_{g H} {\tilde{v}}_{a v e}) .$ ${\tilde{S}}^{* 2} = \tilde{S} \otimes \tilde{S}$ denotes the fuzzy product.

Compute $C_{{\tilde{v}}_{ave}, \tilde{v}} = {log}_{10} D ((\sqrt{L} ⊙ {\tilde{v}}_{ave}) ø (τ_{β} ⊙ \tilde{S}), \tilde{0}),$ $C_{{\tilde{v}}_{ave}, \tilde{v}}$ as the common estimated proximity between the exact value $\tilde{v}$ and the approximate value ${\tilde{v}}_{ave}$ .

If $C_{{\tilde{v}}_{ave}, \tilde{v}} \leq 0 or v_{ave}^{\pm} (r) = 0,$ then write v^± (r) = @.0, r ∈ [0, 1].

4.2 CADNA library

The CADNA library is a tool to implement the SA automatically. The first goal of this library is the estimation of the accuracy of each computed result. The CADNA detects numerical instabilities (informatical zero) during the run of the program. The CADNA works in Fortran or C++ codes on the Linux operating system [25]. When a result is a stochastic zero (i.e. is insignificant), the symbol @.0 is printed. The CADNA detects numerical instabilities during the run of the program. For more details about this library, we refer the reader to "http://www-pequan.lip6.fr/cadna".

We divide the interval [0, T] into $N = k^{n}, 1 \neq k \in ℕ$ sub-intervals, and then obtain the optimal h by increasing n when n = 1, 2, 3, … by using following algorithm. According to the previous results, by using the Fuzzy CESTAC method and the CADNA library, we consider $D ({\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / k)) = @ . 0$ as the stopping criterion. In order to solve FBVP (1) in the stochastic arithmetic, we use the following algorithm by applying fuzzy finite differences method. Suppose ${\tilde{w}}_{i} (h)$ , ${\tilde{w}}_{i} (h / k)$ are the approximate values of $\tilde{x} (t_{i})$ in two successive iterations obtained from the fuzzy finite differences method. For the termination criterion, we consider the Hausdorff distance to be an informatical zero (@.0).

Algorithm 3.

1- Input h = T/N and let $1 \neq k \in ℕ .$

2- Find ${\tilde{w}}_{i} (h)$ and ${\tilde{w}}_{i} (h / k)$ by using system (9),

3- If $D ({\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / k)) = @ . 0$ then stop.

4- Else h: = h/k and goto step 2.

5- EndIf.

5 Numerical Examples

According to Algorithm 3 and Theorem 2, by using the CESTAC method and the CADNA library, we apply $D ({\tilde{w}}_{i} (h), {\tilde{w}}_{i} (h / k)) = @ . 0$ as a stopping criterion which allows us to estimate the optimal step size as soon as the difference between ${\tilde{w}}_{i} (h)$ and ${\tilde{w}}_{i} (h / k)$ is equal to the computational zero. In this case, by the theoretical results presented in Section 3, the common proximity between ${\tilde{w}}_{i} (h)$ and ${\tilde{w}}_{i} (h / k)$ are also the common proximity between ${\tilde{w}}_{i} (h)$ and $\tilde{x} (t_{i})$ , up to one digit. By the proposed method, the best approximation of $\tilde{x} (t_{i})$ provided by the computer is chosen. It is necessary to mention that in all, the presented examples ${\tilde{w}}_{i} (h)$ have been computed by the system (40). Let us now present the examples and the results which we obtain by the C++ code of the fuzzy finite differences method combined with the CADNA library on Linux in double precision. The data_st function of the CADNA library has been used to take into account the assignment error of some data of the examples which are not integer numbers and cannot be exactly coded for computer.

Example 2. Consider the fuzzy two-point boundary value problem [18] $\begin{matrix} {\tilde{x}}_{{gH}_{ii}}^{''} (t) & = & \frac{- 2}{t} ⊙ {\tilde{x}}^{'} (t) \oplus \frac{2}{t^{2}} ⊙ \tilde{x} (t) \oplus \frac{2 t^{2} - 1}{t^{2}}, \\ t \in [1, 3], \end{matrix}$ (20) with the boundary conditions $\begin{matrix} \tilde{x} (1) & = & (1.5 + 0.5 r, 3 - r), \\ \tilde{x} (3) & = & (3 + r, 4.5 - 0.5 r), r \in [0, 1], \end{matrix}$ where in case 1-cut, the exact crisp solution is $x_{cr} (t) = \frac{1}{2} (t^{2} + 1) + \frac{1}{26} (\frac{36}{t^{2}} - 10 t) .$

The analytical solution according to presented method in [18] is $\begin{matrix} x^{-} (t; r) = \frac{1}{2} (t^{2} + 1) + \frac{1}{26} (\frac{36}{t^{2}} - 10 t) + \\ \frac{1}{26} ((\frac{27}{x^{2}} - t) (- 0.5 + 0.5 r) + (9 t - \frac{9}{t^{2}}) (- 1 + r)), \\ x^{+} (t; r) = \frac{1}{2} (t^{2} + 1) + \frac{1}{26} (\frac{36}{t^{2}} - 10 t) + \\ \frac{1}{26} ((\frac{27}{x^{2}} - t) (1 - r) + (9 t - \frac{9}{t^{2}}) (0.5 - 0.5 r)) . \end{matrix}$ To solve Eq. (20), we use the fuzzy finite differences method. For k = 2, we divide the interval [1, 2] into N = 2ⁿ sub-intervals, and then obtain the optimal h by increasing n when n = 1, 2, 3, … based on algorithm 2. Fuzzy solution generates according to r-cuts by using fuzzy CESTAC method and CADNA library a band in tx-plane (Fig. 1).

Fig. 1

The fuzzy solution for Example 2 and its r-cuts, h = 2^-13.

We compare the solution obtained by our method with the solution obtained by an analytical method that uses in the [18] For comparison, the results of these two methods at the point t = 2 are shown in Fig. 2. The numerical solution for different r,s are observed in columns w⁻ (2; r) and w⁺ (2; r) which is optimized at n = 14, and optimal step size h_opt = 2⁻¹³. In optimal step case, the Hausdorff distance between the solutions of two successive iterations equals to informatical zero (@.0). In Fig. 3 the error of iterative methods with Hausdorff distance is displayed. The results of the numerical method at the point t = 2 are shown in Table 1.

Fig. 2

Comparison of Example 2, t = 2, h = 2^-13.

Fig. 3

Hausdorff distance of Example 2, t = 2, h = 2^-n.

Table 1

Numerical results of Example 2.

n	$h = \frac{2}{2^{n}}$	r	w⁻ (2; r)	w⁺ (2; r)	$D (\tilde{w} (h), \tilde{w} (h / 2))$
2	0.5	−	−	−	−
3	0.25	0.0	0.132173767713205E+001	0.250601760745425E+001
		0.1	0.13914756759707E+001	0.245732761326073E+001
		0.2	0.146121367480946E+001	0.240863761906722E+001	0.100854521270E-001
		⋮	⋮	⋮
		1	0.201911766551909E+001	0.20191176655190E+001
4	0.125	0.0	0.13200362074344E+001	0.25031381971966E+001
		0.1	0.138975457393044E+001	0.245454636471650E+001
		0.2	0.14594729404264E+001	0.24059545322363E+001	0.28794102575E-002
		⋮	⋮	⋮
		1	0.20172198723947E+001	0.20172198723947E+001
5	0.62500E-001	0.0	0.13195936332165E+001	0.25023892254433E+001
		0.1	0.13893068930870E+001	0.24538229260911E+001
		0.2	0.14590201529574E+001	0.24052566267389E+001	0.7489717532E-003
		⋮	⋮	⋮
		1	0.20167262319210E+001	0.20167262319210E+001
⋮	⋮	⋮	⋮	⋮	⋮
12	0.48828125E-003	0.0	0.1319444453E+001	0.2502136767E+001
		0.1	0.13891559921E+001	0.24535790747E+001
		0.2	0.14588675307E+001	0.24050213819E+001	0.463E-007
		⋮	⋮	⋮
		1	0.20165598392E+001	0.20165598392E+001
13	0.244140625E-003	0.0	0.1319444446E+001	0.2502136755E+001
		0.1	0.1389155985E+001	0.2453579063E+001
		0.2	0.14588675236E+001	0.24050213711E+001	0.11E-007
		⋮	⋮	⋮
		1	0.20165598316E+001	0.20165598316E+001
14	0.1220703125E-003	0.0	0.1319444444E+001	0.2502136753E+001
		0.1	0.1389155983E+001	0.2453579060E+001
		0.2	0.1458867521E+001	0.2405021368E+001	@.0
		⋮	⋮	⋮
		1	0.201655982E+001	0.201655982E+001

Example 3. Consider the fuzzy two-point boundary value problem $\begin{matrix} {\tilde{x}}_{{gH}_{i}}^{''} (t) = {\tilde{x}}^{'} (t) \oplus 2 ⊙ \tilde{x} (t) \oplus cos (t), t \in [0, \frac{π}{2}], \end{matrix}$ (21) with the boundary conditions $\tilde{x} (0) = (- 1, - 0.3, 0.5), \tilde{x} (\frac{π}{2}) = (- 0.5, - 0.1, 0.2),$ where in case 1-cut, the exact crisp solution is $x_{cr} (t) = \frac{- 3}{10} cos (t) - \frac{1}{10} sin (t) .$

The analytical solution according to presented method in [18] is $\begin{matrix} x^{-} (t; r) = \frac{- 3}{10} cos (t) - \frac{1}{10} sin (t) + \frac{1}{e^{π} - e^{- π / 2}} \\ ((e^{π - t} - e^{2 t - π / 2}) (0.7 r - 0.7) + (e^{2 t} - e^{- t}) (0.4 r - 0.4)), \\ x^{+} (t; r) = \frac{- 3}{10} cos (t) - \frac{1}{10} sin (t) + \frac{1}{e^{π} - e^{- π / 2}} \\ ((e^{π - t} - e^{2 t - π / 2}) (0.8 - 0.8 r) + (e^{2 t} - e^{- t}) (0.3 - 0.3 r)) . \end{matrix}$

Fuzzy solution generates according to r-cuts by using fuzzy CESTAC method and CADNA library a band in tx-plane (Fig. 4).

Fig. 4

The fuzzy solution for Example 3 and its r-cuts, $h = \frac{π}{2^{14}} .$

The results of finite differences method in the case k = 2 at

t = \frac{π}{4}

is shown in Table 2. According to Table 3, it has been found the optimal step size h_opt = 2⁻¹⁵π, when the Hausdoff distance equals to @.0.

Table 2

Numerical results of Example 3.

n	$h = \frac{π}{2^{n + 1}}$	r	w⁻ (π/4; r)	w⁺ (π/4; r)	$D (\tilde{w} (h), \tilde{w} (h / 2))$
1	0.785398163397448	−	−	−	−
2	0.392699081698724	0.0	−0.65291261E+000	0.1101605E+000
		0.1	−0.6159119E+000	0.7085395E-001
		0.2	−0.5789112E+000	0.315473E-001	0.1829058E+000
		⋮	⋮	⋮
		1	−0.2829058E+000	−0.2829058E+000
3	0.196349540849362	0.0	−0.6509902E+000	0.1080245E+000
		0.1	−0.61417742E+000	0.6893585E-001
		0.2	−0.57736461E+000	0.298471E-001	0.21360E-002
		⋮	⋮	⋮
		1	−0.2828620E+000	−0.2828620E+000
4	0.981747704246810E-001	0.0	−0.65049004E+000	0.1074777E+000
		0.1	−0.61372581E+000	0.6844523E-001
		0.2	−0.57696159E+000	0.294126E-001	0.54672E-003
		⋮	⋮	⋮
		1	−0.2828477E+000	−0.2828477E+000
⋮	⋮	⋮	⋮	⋮	⋮
12	0.383495196971410E-003	0.0	−0.65032152E+000	0.1072943E+000
		0.1	−0.61357364E+000	0.6828067E-001
		0.2	−0.57682576E+000	0.292669E-001	0.26E-007
		⋮	⋮	⋮
		1	−0.2828427E+000	−0.2828427E+000
13	0.191747598485705E-003	0.0	−0.65032151E+000	0.1072943E+000
		0.1	−0.61357363E+000	0.6828066E-001
		0.2	−0.57682575E+000	0.292669E-001	0.4E-008
		⋮	⋮	⋮
		1	−0.2828427E+000	−0.2828427E+000
14	0.958737992428525E-004	0.0	−0.65032151E+000	0.1072943E+000
		0.1	−0.61357363E+000	0.6828066E-001
		0.2	−0.57682575E+000	0.292669E-001	@.0
		⋮	⋮	⋮
		1	−0.2828427E+000	−0.2828427E+000

The numerical results between presented method and analytical method [18] for $t = \frac{π}{4}$ are compared in Fig. 5. In Fig. 6 the error of iterative methods with Hausdorff distance is displayed.

Fig. 5

Comparison of Example 3, $t = \frac{π}{4}, h = \frac{π}{2^{14}} .$

Fig. 6

Hausdorff distance of Example 3, $t = \frac{π}{4}, h = \frac{π}{2^{n + 1}}$ .

6 Conclusion

In this work, a novel arithmetic called discrete stochastic arithmetic was suggested to implement the numerical algorithm to control the step size in the fuzzy finite differences method for solving the linear fuzzy two-point boundary value problem under gH-differentiability. For this purpose, the CESTAC method and the CADNA library were applied to control the computations dynamically and find the optimal step size. Numerical examples showed that the proposed method is effective and is able to rely the computed results of the mentioned algorithm. Many real world problems within fields such as information knowledge, engineering technologies, environmental sciences, social sciences andmedical sciences, contain uncertainties in various forms. It is known that these types of uncertainties cannot be modeled by the customary analytical techniques. Soft set theory and rough set theory [30 , 37–40] are mathematical tools for dealing with uncertainties and are closely related. The main purpose of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. Rough set theory has attracted worldwide attention of many researchers and practitioners, who have contributed essentially to its development and applications. Rough set theory overlaps with many other theories. Despite this, rough set theory may be considered as an independent discipline in its own right. The rough set approach seems to be of fundamental importance in artificial intelligence and cognitive sciences, especially in research areas such as machine learning, intelligent systems, inductive reasoning, pattern recognition, image processing, signal analysis, knowledge discovery, decision analysis, and expert systems. As well, soft set theory can have a bearing on making decisions in many fields. Particularly important is parameter reduction of soft sets, an essential topic both for information sciences and artificial intelligence. Therefore in the future, we can investigate by using the CESTAC method and the CADNA library some algorithms of parameter reduction based on some types of soft sets.

Footnotes

1

Control et Estimation Stochastique des Arrondis de Calculs.

2

Control of Accuracy and Debugging for Numerical Application.

Acknowledgment

The authors would like to thank the anonymous reviewers for their constructive comments to improve the quality of this work.

References

Abbasbandy

and Araghi Fariborzi

M.A.

, A Stochastic scheme for solving definite integrals, Appl Num Math 55 (2005), 125–136.

Abbasbandy

and Araghi Fariborzi

M.A.

, The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree, Appl Numer Math 50 (2004), 279–290.

Abbasbandy

and Araghi Fariborzi

M.A.

, A reliable method to determine the ill-condition functions using stochastic arithmetic, Southwest J Pure Appl Math 1 (2002), 33–38.

Abbasbandy

and Fariborzi Araghi

M.A.

, Numerical solution of improper integrals with valid implementation, Math Comput Appl 7 (2002), 83–91.

Abbasbandy

and Araghi Fariborzi

M.A.

, The valid implementation of numerical integration methods, Far East J Appl Math 8 (2002), 89–101.

Abbasbandy

, Allahviranloo

, Lopez-Pouso

and Nieto

J.J.

, Numerical methods for fuzzy differential inclusions, Journal of Computer and Mathematics with Applications 48 (2004), 1633–1641.

Ahmadian

, Salahshour

, Chan

C.S.

and Baleanu

, Numerical solutions of fuzzy differential equations by an efficient Runge-Kutta method with generalized differentiability, Fuzzy Sets and Systems 331 (2018), 47–67.

Anastassiou

G.A.

, Fuzzy Mathematics: Approximation Theory. Springer, Berlin (2010).

Allahviranloo

and Khalilpour

, A numerical method for two-point fuzzy boundary value problems, World Appl Sci J 13 10 (2011), 2137–2147.

10.

Balooch

M.R.

Shahriyar, Ismail

, Aghabeigi

, Ahmadian

and Salahshour

, An eigenvalue-eigenvector method for solving a systemof fractional differential equations with uncertainty, Mathematical Problems in Engineering (2013). ID 579761.

11.

Bede

, Note on Numerical solution of fuzzy differential equations by predictor-corrector method, Inform Sci 178 (2008), 1917–1922.

12.

Bede

and Gal

S.G.

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

13.

Bede

and Gal

S.G.

, Solutions of fuzzy differential equations based on generalized differentiability, Commun Math Anal 9 (2010), 22–41.

14.

Sh. Behzadi

, Vahdani

, Allahviranloo

and Abbasbandy

, Application of fuzzy Picard method for solving fuzzy quadratic Riccati and fuzzy Painleve I equations, AppMath Mod 40 (2016), 8125–8137.

15.

Buckley

J.J.

and Feuring

, Fuzzy differential equations, Fuzzy Sets Syst 110 (2000), 43–54.

16.

Chesneaux

J.M.

and Jezéquel

, Dynamical control of computations using the trapezodial and simpsons rules, J Universal Comput Sci 4 (1998), 2–10.

17.

Chesneaux

J.M.

, CADNA: An ADA tool for round-off errors analysis and for numerical debugging, In ADA in Aerospace, Barcelone, Spain, 1990.

18.

Gasilova

, Emrah

Amrahovb and Fatullayeva

A.G.

, Solution oflinear differential equations with fuzzy boundary values, Fuzzy Sets and Systems 257 (2014), 169–183.

19.

Chalco-Cano

and Romtan-Flores

, On new solutions of fuzzy differential equations, Chaos, Solitons and Fractals 38 (2008), 112–119.

20.

Fariborzi Araghi

M.A.

, The methods of valid implementation of the numerical algorithms, Phd dissertation thesis, Science andresaerch branch, Islamic Azad University, 2002.

21.

Fariborzi

M.A.

Araghi and Barzegar Kelishami

, Dynamical control of accuracy in the fuzzy Runge-Kutta methods to estimate the solution of a fuzzy differential equation, Journal of Fuzzy Set Valued Analysis 2016 SI 1 (2016), 71–84.

22.

Fariborzi

M.A.

Araghi and Fattahi

, Solving fuzzy Linear systems in the stochastic arithmetic by applying CADNA library, In Proceedings of the International Conference on Evolutionary Computation Theory and Applications, 2011, pp. 446–450.

23.

Gomes

L.T.

, Barros

L.C.

and Bede

, Fuzzy Differential Equations in Various Approaches, Springer, ISBN: 978-3319-22574-6, 2015.

24.

Jézéquel

, Contrôle dynamique de méthodes d’ approximation, Habilitation á diriger des recherches, Université Pierre et Marie Curie, Paris, 2005.

25.

Jezequel

and Chesneaux

J.M.

, CADNA: A library for estimating round-off error propagation, Computer Physics Communications 178 (2008), 933–955.

26.

Kaleva

, A note on fuzzy differential equations, Nonlinear Analysis 64 (2006), 895–900.

27.

Khastan

and Nieto

J.J.

, A boundary value problem for second order fuzzy differential equations, Nonlinear Anal 72 (2010), 3583–3593.

28.

Khojasteh Salkuyeh

, Toutounian

and Shariat Yazdi

, A procedure with stepsize control for solving n onedimensional IVPs, Math Comput in Simulation 79 (2008), 167–176.

29.

Liu

H.-K.

, Comparison results of two-point fuzzy boundary value problems, Int J Comput Math Sci 5 (2011), 1–7.

30.

, Zhan

, Ali

M.I.

and Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49(4) (2018), 511–529.

31.

Seikkala

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

32.

Vignes

, A stochastic arithmetic for reliable scientific computation, Math Comp Simul 35 (1993), 233–261.

33.

Vignes

, New methods for evaluating the validity of the results of mathematical computations, Math Comp Simul 20 (1978), 227–249.

34.

Vignes

and La Porte

, Error analysis in computing, in: Information Processing 1974, North-Holland, 1974, pp. 610–614.

35.

Vignes

, A stochastic approach to the analysis of roundoff error propagation. A survey of the CESTAC method, in: Proc 2nd Real Numbers and Computers Conference, Marseille, France, 1996, pp. 233–251.

36.

Vignes

, Discrete stochastic arithmetic for validating results of numerical software, Num Alg 37 (2004), 377–390.

37.

Zhang

and Zhan

, Fuzzy soft β-covering based fuzzy rough sets and corresponding decision-making applications, Int J Mach Learn Cybern (2018). DOI: 10.1007/s13042-018-0828-3

38.

Zhan

and Wang

, Certain types of soft coverings based rough sets with applications, Int J Mach Learn Cybern (2018). 10.1007/s13042-018-0785-x .

39.

Zhan

and J. Alcantud

C.R.

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018). 10.1007/s10462-018-9617-3

40.

Zhan

and J. Alcantud

C.R.

, A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review (2018). DOI 10.1007/s10462-017-9592-0