# An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Overview of Fuzzy Calculus and Fuzzy Fractional Calculus

**Definition**

**1.**

**[28]**A fuzzy number$\rho $is a mapping such that$\rho :\mathbb{R}\to \left[0,1\right]$with the following properties:

- $\rho $is fuzzy convex, that is,$\rho \left(\lambda s+\left(1-\lambda \right)t\right)\ge min\left\{\rho \left(s\right),\rho \left(t\right)\right\}$for all$s,t\in \mathbb{R}$,$\lambda \in \left[0,1\right].$
- $\rho $is normal, that is,$\exists {s}_{*}\in \mathbb{R}$for which$\rho \left({s}_{*}\right)=1$.
- $\rho $is upper-semi continuous, that is,$\rho \left({s}_{*}\right)\ge li{m}_{s\to {s}_{*}^{+}}\rho \left(s\right)$for any${s}_{*}\in \mathbb{R}.$
- $supp\left(\rho \right)=\left\{s\in \mathbb{R}:\rho \left(s\right)>0\right\}$is the support of$\rho $, and$\overline{\left\{s\in \mathbb{R}:\rho \left(s\right)>0\right\}}$is compact, where$\overline{\{\ast \}}$denotes the closure of a subset.

**Definition**

**2.**

**[28]**A fuzzy number$\rho $in parametric form is a pair$\left({\rho}_{1},{\rho}_{2}\right)$of functions${\rho}_{1}\left(r\right),{\rho}_{2}\left(r\right)$, for$r\in \left[0,1\right]$, which satisfy the following requirements:

- ${\rho}_{1}\left(r\right)$is a bounded non-decreasing left continuous for each$r\in \left(0,1\right]$, and right continuous at$r=0$.
- ${\rho}_{2}\left(r\right)$is a bounded non-increasing left continuous for each$r\in \left(0,1\right],$and right continuous at$r=0$.
- ${\rho}_{1}\left(r\right)\le {\rho}_{2}\left(r\right)$, for each$r\in \left[0,1\right]$.

**Definition**

**3.**

- The H-differences$y\left({x}_{0}+\xi \right)\ominus y\left({x}_{0}\right),y\left({x}_{0}\right)\ominus y\left({x}_{0}-\xi \right)$exist, for each$\xi >0$sufficiently tends to 0 and$\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{y\left({x}_{0}+\xi \right)\ominus y\left({x}_{0}\right)}{\xi}={y}^{\prime}\left({x}_{0}\right)=\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{y\left({x}_{0}\right)\ominus y\left({x}_{0}-\xi \right)}{\xi},$
- The H-differences$y\left({x}_{0}\right)\ominus y\left({x}_{0}+\xi \right),y\left({x}_{0}-\xi \right)\ominus y\left({x}_{0}\right)$exist, for each$\xi >0$sufficiently tends to 0 and$\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{y\left({x}_{0}\right)\ominus y\left({x}_{0}+\xi \right)}{-\xi}={y}^{\prime}\left({x}_{0}\right)=\underset{\xi \to {0}^{+}}{\mathrm{lim}}\frac{y\left({x}_{0}-\xi \right)\ominus y\left({x}_{0}\right)}{-\xi}$.

**Definition**

**4.**

**Definition**

**5.**

**Theorem**

**1.**

- If$y$is (1)-differentiable on$\left[a,b\right]$, then${y}_{1r}$and${y}_{2r}$are two differentiable functions on$\left[a,b\right]$, and${[{D}_{1}^{1}y\left(x\right)]}^{r}=\left[{y}_{1r}^{\prime}\left(x\right),{y}_{2r}^{\prime}\left(x\right)\right],$
- If$y$is (2)-differentiable on$\left[a,b\right]$, then${y}_{1r}$and${y}_{2r}$are two differentiable functions on$\left[a,b\right]$, and${[{D}_{2}^{1}y\left(x\right)]}^{r}=\left[{y}_{2r}^{\prime}\left(x\right),{y}_{1r}^{\prime}\left(x\right)\right].$

**Theorem**

**2.**

**Definition**

**6.**

**Theorem**

**3.**

## 3. Formulation of Fuzzy Fractional Volterra IDEs

**Algorithm 1:**To obtain the $\left(n\right)$-solution of the FFVIDEs (1.1), there are two cases that will be discussed as follows:

**Case 1**: If $y\left(x\right)$ is Caputo [(1)-$\beta $]-differentiable, we convert the FFVIDEs (1) and (2) to the following OFVIDEs system:

- Step 1: Solve the system (4) and (5) for ${y}_{1r}\left(x\right)$ and ${y}_{2r}\left(x\right)$.
- Step 2: Ensure that $\left[{y}_{1r}\left(x\right),{y}_{2r}\left(x\right)\right]$ and $\left[{D}_{{a}^{+}}^{\beta}{y}_{1r}\left(x\right),{D}_{{a}^{+}}^{\beta}{y}_{2r}\left(x\right)\right]$ are valid level sets on $\left[a,b\right]$ or on a partial interval in$\text{}\left[a,b\right]$.
- Step 3: Construct a $\left(1\right)$-differentiable solution $y\left(x\right)$ whose $r$-cut representation is $\left[{y}_{1r}\left(x\right),{y}_{2r}\left(x\right)\right]$.

**Case 2**: If $y\left(x\right)$ is Caputo [(2)-$\beta $]-differentiable, we convert the FFVIDEs (1) and (2) to the following OFVIDEs system:

- Step 1: Solve the system (6) and (7) for ${y}_{1r}\left(x\right)$and ${y}_{2r}\left(x\right)$.
- Step 2: Ensure that $\left[{y}_{1r}\left(x\right),{y}_{2r}\left(x\right)\right]$ and $\left[{D}_{{a}^{+}}^{\beta}{y}_{2r}\left(x\right),{D}_{{a}^{+}}^{\beta}{y}_{1r}\left(x\right)\right]$ are valid level sets on $\left[a,b\right]$ or on a partial interval in$\text{}\left[a,b\right]$.
- Step 3: Construct a $\left(2\right)$-differentiable solution $y\left(x\right)$ whose $r$-cut representation is $\left[{y}_{1r}\left(x\right),{y}_{2r}\left(x\right)\right]$.

## 4. Description of the FRPS Technique

**Definition**

**7.**

**Theorem**

**4.**

## 5. Applications and Simulations

**Example**

**1.**

**Case1**: Under Caputo [(1)-$\beta $]-differentiability, the system of OFVIDEs corresponding to Caputo [(1)-$\beta $]-differentiable is

**Case2**: Under Caputo [(2)-$\beta $]-differentiability, the system of OFVIDEs corresponding to Caputo [(2)-$\beta $]-differentiable is

**Example**

**2.**

**Case 1:**Under Caputo [(1)-$\beta $]-differentiability, the system of OFVIDEs corresponding to Caputo [(1)-$\beta $]-differentiable is

**Case 2**: Under Caputo [(2)-$\beta $]-differentiability, the system of OFVIDEs corresponding to Caputo [(2)-$\beta $]-differentiable is

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**(

**a**) The core and the support of fuzzy (1)-approximated solutions under Caputo [(1)-$\beta $]-differentiable; (

**b**) The core and the support of derivative of fuzzy (1)-approximated solution under Caputo [(1)-$\beta $]-differentiable, at $\beta =1$ of Example 1: gray support and red core.

**Figure 2.**(

**a**) The core and the support of fuzzy (2)-approximated solutions under Caputo [(2)-$\beta $]-differentiable; (

**b**) The core and the support of fuzzy (2)-approximated Caputo derivatives solution under Caputo [(2)-$\beta $]-differentiable, at $\beta =1$ of Example 1: gray support and red core.

**Figure 3.**(

**a**) Surface plot exact solution, case 1; (

**b**) Surface plot of 10th-FRPS approximated solutions, case 1;

**(c)**Surface plot exact solution, case 2;

**(d)**Surface plot of 10th-FRPS approximated solutions, case 2, of Example 2 at $\beta =1$, for all $t\in \left[0,1\right]$ and $r\in \left[0,1\right]$: (yellow and blue are the upper and lower solution, respectively).

**Figure 4.**(

**a**) Plots of $r$-cut representations of exact and ${\phi}_{10,1r}\left(t\right)$, of Example 2, case 1; (

**b**) Plots of $r$-cut representations of exact and ${\phi}_{10,2r}\left(t\right)$, of Example 2, case 2, for $\beta =1,$ $t\in \left[0,1\right]$ with different values of $r$. in parametric form: red $r=0$, dashed blue $r=0.25$, dashed green $r=0.5$, darker red-dashed $r=0.75$, blue $r=1$.

**Table 1.**The (1)-approximated solution of Example 1, case 1 for different values of $\beta $ with $r=0.5$.

${\mathit{\beta}}_{\mathit{i}}$ | 7th-FRPS Approximated Solutions |
---|---|

$\frac{1}{4}$ | ${y}_{7}\left(x\right)=\frac{3\Gamma \left(1/4\right)\sqrt{x}}{4\sqrt{\pi}}+\frac{3\Gamma \left(1/4\right)x}{8}+\frac{\Gamma \left(1/4\right){x}^{3/2}}{2\sqrt{\pi}}+\frac{3{x}^{1/4}}{2\Gamma \left(5/4\right)}+\frac{3{x}^{3/4}}{2\Gamma \left(7/4\right)}+\frac{3{x}^{5/4}}{2\Gamma \left(9/4\right)}+\frac{3{x}^{7/4}}{2\Gamma \left(11/4\right)}$ |

$\frac{1}{2}$ | ${y}_{7}\left(x\right)=\frac{3\sqrt{x}}{\sqrt{\pi}}+\frac{3\sqrt{\pi}x}{4}+\frac{2{x}^{3/2}}{\sqrt{\pi}}+\frac{3\sqrt{\pi}{x}^{2}}{8}+\frac{4{x}^{5/2}}{5\sqrt{\pi}}+\frac{\sqrt{\pi}{x}^{3}}{8}+\frac{8{x}^{7/2}}{35\sqrt{\pi}}$ |

$\frac{3}{4}$ | ${y}_{7}\left(x\right)=\frac{3\Gamma \left(3/4\right){x}^{3/2}}{2\sqrt{\pi}}+\frac{3\Gamma \left(3/4\right){x}^{3}}{16}+\frac{4\Gamma \left(3/4\right){x}^{9/2}}{105\sqrt{\pi}}+\frac{3{x}^{3/4}}{2\Gamma \left(7/4\right)}+\frac{3{x}^{9/4}}{2\Gamma \left(13/4\right)}+\frac{3{x}^{15/4}}{2\Gamma \left(19/4\right)}+\frac{3{x}^{21/4}}{2\Gamma \left(25/4\right)}$ |

$1$ | ${y}_{7}\left(x\right)=\frac{3}{2}x+\frac{3}{4}{x}^{2}+\frac{1}{4}{x}^{3}+\frac{1}{16}{x}^{4}+\frac{1}{80}{x}^{5}+\frac{1}{480}{x}^{6}+\frac{1}{3360}{x}^{7}$ |

**Table 2.**The (2)-approximated solution of Example 5.1, case 2 for different values of $\beta $ with $r=0.5$.

${\mathit{\beta}}_{\mathit{i}}$ | 7th-FRPS Approximated Solutions |
---|---|

$\frac{1}{4}$ | ${y}_{7}\left(x\right)=\frac{3\Gamma \left(1/4\right)\sqrt{x}}{4\sqrt{\pi}}+\frac{3\Gamma \left(1/4\right)x}{8}+\frac{\Gamma \left(1/4\right){x}^{3/2}}{2\sqrt{\pi}}+\frac{3{x}^{1/4}}{2\Gamma \left(5/4\right)}+\frac{3{x}^{3/4}}{2\Gamma \left(7/4\right)}+\frac{3{x}^{5/4}}{2\Gamma \left(9/4\right)}+\frac{3{x}^{7/4}}{2\Gamma \left(11/4\right)}$ |

$\frac{1}{2}$ | ${y}_{7}\left(x\right)=\frac{3\sqrt{x}}{\sqrt{\pi}}+\frac{3\sqrt{\pi}x}{4}+\frac{2{x}^{3/2}}{\sqrt{\pi}}+\frac{3\sqrt{\pi}{x}^{2}}{8}+\frac{4{x}^{5/2}}{5\sqrt{\pi}}+\frac{\sqrt{\pi}{x}^{3}}{8}+\frac{8{x}^{7/2}}{35\sqrt{\pi}}$ |

$\frac{3}{4}$ | ${y}_{7}\left(x\right)=\frac{3\Gamma \left(3/4\right){x}^{3/2}}{2\sqrt{\pi}}+\frac{3\Gamma \left(3/4\right){x}^{3}}{16}+\frac{4\Gamma \left(3/4\right){x}^{9/2}}{105\sqrt{\pi}}+\frac{3{x}^{3/4}}{2\Gamma \left(7/4\right)}+\frac{3{x}^{9/4}}{2\Gamma \left(13/4\right)}+\frac{3{x}^{15/4}}{2\Gamma \left(19/4\right)}+\frac{3{x}^{21/4}}{2\Gamma \left(25/4\right)}$ |

$1$ | ${y}_{7}\left(x\right)=\frac{3}{2}x+\frac{3}{4}{x}^{2}+\frac{1}{4}{x}^{3}+\frac{1}{16}{x}^{4}+\frac{1}{80}{x}^{5}+\frac{1}{480}{x}^{6}+\frac{1}{3360}{x}^{7}$ |

${\mathit{y}}_{1\mathit{r}}\left(\mathit{x}\right)$ | |||

${\mathit{x}}_{\mathit{i}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ |

$0.2$ | $5.273559366\times {10}^{-16}$ | $7.216449660\times {10}^{-16}$ | $1.054711873\times {10}^{-15}$ |

$0.4$ | $1.086908341\times {10}^{-12}$ | $1.630362511\times {10}^{-12}$ | $2.173816682\times {10}^{-12}$ |

$0.6$ | $9.565170878\times {10}^{-11}$ | $1.434776741\times {10}^{-10}$ | $1.913034175\times {10}^{-10}$ |

$0.8$ | $2.304785251\times {10}^{-9}$ | $3.457177877\times {10}^{-9}$ | $4.609570503\times {10}^{-9}$ |

${\mathit{y}}_{2\mathit{r}}\left(\mathit{x}\right)$ | |||

${\mathit{x}}_{\mathit{i}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ |

$0.2$ | $1.054711873\times {10}^{-15}$ | $7.216449660\times {10}^{-16}$ | $5.273559366\times {10}^{-16}$ |

$0.4$ | $2.173816682\times {10}^{-12}$ | $1.630362511\times {10}^{-12}$ | $1.086908341\times {10}^{-12}$ |

$0.6$ | $1.913034175\times {10}^{-10}$ | $1.434776741\times {10}^{-10}$ | $9.565170877\times {10}^{-11}$ |

$0.8$ | $4.609570503\times {10}^{-9}$ | $3.457177877\times {10}^{-9}$ | $2.304785251\times {10}^{-9}$ |

${\mathit{y}}_{1\mathit{r}}\left(\mathit{x}\right)$ | |||

${\mathit{x}}_{\mathit{i}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ |

$0.2$ | $1.137978600\times {10}^{-15}$ | $8.326672685\times {10}^{-16}$ | $4.996003611\times {10}^{-16}$ |

$0.4$ | $2.172595439\times {10}^{-12}$ | $1.630362512\times {10}^{-12}$ | $1.088018564\times {10}^{-12}$ |

$0.6$ | $1.910848146\times {10}^{-10}$ | $1.434776742\times {10}^{-10}$ | $9.587064476\times {10}^{-11}$ |

$0.8$ | $4.600237524\times {10}^{-9}$ | $3.457178099\times {10}^{-9}$ | $2.314118674\times {10}^{-9}$ |

${\mathit{y}}_{2\mathit{r}}\left(\mathit{x}\right)$ | |||

${\mathit{x}}_{\mathit{i}}$ | $\mathit{r}=0$ | $\mathit{r}=0.5$ | $\mathit{r}=1$ |

$0.2$ | $5.551115123\times {10}^{-16}$ | $8.326672685\times {10}^{-16}$ | $1.193489751\times {10}^{-15}$ |

$0.4$ | $1.088018564\times {10}^{-12}$ | $1.630362512\times {10}^{-12}$ | $2.172595437\times {10}^{-12}$ |

$0.6$ | $9.587064476\times {10}^{-11}$ | $1.434776742\times {10}^{-10}$ | $1.910848146\times {10}^{-10}$ |

$0.8$ | $2.314118674\times {10}^{-9}$ | $3.457178099\times {10}^{-9}$ | $4.600237524\times {10}^{-9}$ |

${\mathit{y}}_{1\mathit{r}}\left(\mathit{x}\right)$ | |||||

${\mathit{r}}_{\mathit{i}}$ | ${\mathit{x}}_{\mathit{i}}$ | $\mathit{\beta}=1$ | $\mathit{\beta}=0.9$ | $\mathit{\beta}=0.8$ | $\mathit{\beta}=0.7$ |

$0.5$ | $0.2$ | $-0.100668001$ | $-0.123692685$ | $-0.151695235$ | $-0.186264438$ |

$0.4$ | $-0.205376163$ | $-0.238161466$ | $-0.277064940$ | $-0.324760477$ | |

$0.6$ | $-0.318326791$ | $-0.359432909$ | $-0.408753265$ | $-0.470283639$ | |

$0.8$ | $-0.444052990$ | $-0.494515208$ | $-0.556205023$ | $-0.634532156$ | |

$0.75$ | $0.2$ | $-0.050334001$ | $-0.061846342$ | $-0.075847617$ | $-0.093132219$ |

$0.4$ | $-0.102688081$ | $-0.119080733$ | $-0.138532470$ | $-0.162380238$ | |

$0.6$ | $-0.159163396$ | $-0.179716454$ | $-0.204376633$ | $-0.235141819$ | |

$0.8$ | $-0.222026495$ | $-0.247257604$ | $-0.278102512$ | $-0.317266078$ | |

${\mathit{y}}_{2\mathit{r}}\left(\mathit{x}\right)$ | |||||

${\mathit{r}}_{\mathit{i}}$ | ${\mathit{x}}_{\mathit{i}}$ | $\mathit{\beta}=1$ | $\mathit{\beta}=0.9$ | $\mathit{\beta}=0.8$ | $\mathit{\beta}=0.7$ |

$0.5$ | $0.2$ | $0.100668001$ | $0.123692685$ | $0.151695235$ | $0.186264438$ |

$0.4$ | $0.205376163$ | $0.238161466$ | $0.277064940$ | $0.324760477$ | |

$0.6$ | $0.318326791$ | $0.359432909$ | $0.408753265$ | $0.470283639$ | |

$0.8$ | $0.444052990$ | $0.494515208$ | $0.556205023$ | $0.634532156$ | |

$0.75$ | $0.2$ | $0.050334001$ | $0.061846342$ | $0.075847617$ | $0.093132219$ |

$0.4$ | $0.102688081$ | $0.119080733$ | $0.138532470$ | $0.162380238$ | |

$0.6$ | $0.159163396$ | $0.179716454$ | $0.204376633$ | $0.235141819$ | |

$0.8$ | $0.222026495$ | $0.247257604$ | $0.278102512$ | $0.317266078$ |

${\mathit{y}}_{1\mathit{r}}\left(\mathit{x}\right)$ | |||||

${\mathit{r}}_{\mathit{i}}$ | ${\mathit{x}}_{\mathit{i}}$ | $\mathit{\beta}=1$ | $\mathit{\beta}=0.9$ | $\mathit{\beta}=0.8$ | $\mathit{\beta}=0.7$ |

$0.5$ | $0.2$ | $0.099334665$ | $0.120583966$ | $0.144646654$ | $0.170765057$ |

$0.4$ | $0.194709171$ | $0.217958708$ | $0.239847027$ | $0.258218270$ | |

$0.6$ | $0.282321238$ | $0.299033752$ | $0.310147377$ | $0.313819870$ | |

$0.8$ | $0.358678047$ | $0.363060372$ | $0.359026685$ | $0.346425845$ | |

$0.75$ | $0.2$ | $0.049667333$ | $0.060291983$ | $0.072323327$ | $0.085382529$ |

$0.4$ | $0.097354586$ | $0.108979354$ | $0.119923514$ | $0.129109135$ | |

$0.6$ | $0.141160618$ | $0.149516876$ | $0.155073688$ | $0.156909935$ | |

$0.8$ | $0.179339023$ | $0.181530186$ | $0.179513342$ | $0.173212923$ | |

${\mathit{y}}_{2\mathit{r}}\left(\mathit{x}\right)$ | |||||

${\mathit{r}}_{\mathit{i}}$ | ${\mathit{x}}_{\mathit{i}}$ | $\mathit{\beta}=1$ | $\mathit{\beta}=0.9$ | $\mathit{\beta}=0.8$ | $\mathit{\beta}=0.7$ |

$0.5$ | $0.2$ | $-0.099334665$ | $-0.120583966$ | $-0.144646654$ | $-0.170765057$ |

$0.4$ | $-0.194709171$ | $-0.217958708$ | $-0.239847027$ | $-0.258218270$ | |

$0.6$ | $-0.282321238$ | $-0.299033752$ | $-0.310147377$ | $-0.313819870$ | |

$0.8$ | $-0.358678047$ | $-0.363060372$ | $-0.359026685$ | $-0.346425845$ | |

$0.75$ | $0.2$ | $-0.049667333$ | $-0.060291983$ | $-0.072323327$ | $-0.085382529$ |

$0.4$ | $-0.097354586$ | $-0.108979354$ | $-0.119923514$ | $-0.129109135$ | |

$0.6$ | $-0.141160618$ | $-0.149516876$ | $-0.155073688$ | $-0.156909935$ | |

$0.8$ | $-0.179339023$ | $-0.181530186$ | $-0.179513342$ | $-0.173212923$ |

${\mathit{y}}_{1\mathit{r}}\left(\mathit{x}\right)$ | ||

${\mathit{x}}_{\mathit{i}}$ | FRPS Method | HW Method |

$0.1$ | $2.76167\times {10}^{-15}$ | $4.205\times {10}^{-6}$ |

$0.2$ | $1.41145\times {10}^{-12}$ | $5.305\times {10}^{-6}$ |

$0.3$ | $5.42854\times {10}^{-11}$ | $2.07\times {10}^{-6}$ |

${\mathit{y}}_{2\mathit{r}}\left(\mathit{x}\right)$ | ||

${\mathit{x}}_{\mathit{i}}$ | FRPS Method | HW Method |

$0.1$ | $2.76167\times {10}^{-15}$ | $2.876\times {10}^{-6}$ |

$0.2$ | $1.41145\times {10}^{-12}$ | $1.836\times {10}^{-6}$ |

$0.3$ | $5.42854\times {10}^{-11}$ | $1.295\times {10}^{-6}$ |

^{1}Results of HW method referred in [35].

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alaroud, M.; Al-Smadi, M.; Rozita Ahmad, R.; Salma Din, U.K.
An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations. *Symmetry* **2019**, *11*, 205.
https://doi.org/10.3390/sym11020205

**AMA Style**

Alaroud M, Al-Smadi M, Rozita Ahmad R, Salma Din UK.
An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations. *Symmetry*. 2019; 11(2):205.
https://doi.org/10.3390/sym11020205

**Chicago/Turabian Style**

Alaroud, Mohammad, Mohammed Al-Smadi, Rokiah Rozita Ahmad, and Ummul Khair Salma Din.
2019. "An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations" *Symmetry* 11, no. 2: 205.
https://doi.org/10.3390/sym11020205