# Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form

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## Abstract

**:**

## 1. Introduction

## 2. The FEW in General From

- (1)
- It exists an element ${\mathrm{f}}^{\prime}({\mathrm{t}}_{0})\u03f5{\mathrm{F}}^{\mathrm{n}}$ such that, for all $\mathrm{h}>0$ sufficiently near to 0, there are $\mathrm{F}\left({\mathrm{t}}_{0}+\mathrm{h}\right)-\mathrm{F}\left({\mathrm{t}}_{0}\right),\mathrm{F}\left({\mathrm{t}}_{0}\right)-\mathrm{F}\left({\mathrm{t}}_{0}-\mathrm{h}\right)$ and the limit$$\underset{\mathrm{h}\to 0+}{\mathrm{lim}}\frac{\mathrm{F}\left({\mathrm{t}}_{0}+\mathrm{h}\right)-\mathrm{F}\left({\mathrm{t}}_{0}\right)}{\mathrm{h}}=\underset{\mathrm{h}\to 0+}{\mathrm{lim}}\frac{\mathrm{F}\left({\mathrm{t}}_{0}\right)-\mathrm{F}\left({\mathrm{t}}_{0}-\mathrm{h}\right)}{\mathrm{h}}={\mathrm{F}}^{\prime}({\mathrm{t}}_{0})$$
- (2)
- It exists an element ${\mathrm{f}}^{\prime}({\mathrm{t}}_{0})\u03f5{\mathrm{F}}^{\mathrm{n}}$ such that, for all $\mathrm{h}<0$ sufficiently near to 0, there are $\mathrm{F}\left({\mathrm{t}}_{0}+\mathrm{h}\right)-\mathrm{F}\left({\mathrm{t}}_{0}\right),\mathrm{F}\left({\mathrm{t}}_{0}\right)-\mathrm{F}\left({\mathrm{t}}_{0}-\mathrm{h}\right)$ and the limit$$\underset{\mathrm{h}\to 0-}{\mathrm{lim}}\frac{\mathrm{F}\left({\mathrm{t}}_{0}+\mathrm{h}\right)-\mathrm{F}\left({\mathrm{t}}_{0}\right)}{\mathrm{h}}=\underset{\mathrm{h}\to 0-}{\mathrm{lim}}\frac{\mathrm{F}\left({\mathrm{t}}_{0}\right)-\mathrm{F}\left({\mathrm{t}}_{0}-\mathrm{h}\right)}{\mathrm{h}}={\mathrm{F}}^{\prime}({\mathrm{t}}_{0})$$

## 3. CTCS Scheme for Solving the FWE

## 4. General Implicit Scheme for Solving FEW

## 5. Fuzzy Stability Analysis

#### 5.1. The Stability of CTCS for Fuzzy Wave Equation

**Theorem**

**1.**

**Proof.**

#### 5.2. The Consistency and Convergence of CTCS for Fuzzy Wave Equation

**Theorem**

**2:**

**Proof.**

## 6. Numerical Examples and Solution Analysis

## 7. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Amoddeo, A. Moving mesh partial differential equations modelling to describe oxygen induced effects on avascular tumour growth. Cogent Phys.
**2015**, 2, 1050080. [Google Scholar] [CrossRef] - Holmes, E.E.; Lewis, M.A.; Banks, J.E.; Veit, R.R. Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics. Ecology
**1994**, 75, 17–29. [Google Scholar] [CrossRef] - Stockar, S.; Canova, M.; Guezennec, Y.; Rizzoni, G. A Lumped-Parameter Modeling Methodology for One-Dimensional Hyperbolic Partial Differential Equations Describing Nonlinear Wave Propagation in Fluids. J. Dyn. Syst. Meas. Control.
**2014**, 137, 011002. [Google Scholar] [CrossRef] - Macías-Díaz, J.; Tomasiello, S. A differential quadrature-based approach à la Picard for systems of partial differential equations associated with fuzzy differential equations. J. Comput. Appl. Math.
**2016**, 299, 15–23. [Google Scholar] [CrossRef] - Sarmad, A.A.; Ali, F.J.; Azizan, S. A Single Convergent Control Parameter Optimal Homotopy Asymptotic Method Approximate-Analytical Solution of Fuzzy Heat Equation. ASM Sci. J.
**2019**, 12, 42–47. [Google Scholar] - Nemati, K.; Matinfar, M. An implicit method for fuzzy parabolic partial differential equations. J. Nonlinear Sci. Appl.
**2008**, 1, 61–71. [Google Scholar] [CrossRef] [Green Version] - Allahviranloo, T.; Kermani, A.M. Numerical Methods for Fuzzy Partial Differential Equations under New Defini-tion For Derivative. Iran. J. Fuzzy Syst.
**2010**, 7, 33–50. [Google Scholar] - Zureigat, H.H.; Ismail, A.I.M. Numerical solution of fuzzy heat equation with two different fuzzifications. In Proceedings of the 2016 SAI Computing Conference (SAI), London, UK, 13–15 July 2016; pp. 85–90. [Google Scholar]
- Abdi, M.; Allahviranloo, T. Fuzzy finite difference method for solving fuzzy Poisson’s equation. J. Intell. Fuzzy Syst.
**2019**, 37, 5281–5296. [Google Scholar] [CrossRef] - Aminzadeh, F. Applications of AI and soft computing for challenging problems in the oil industry. J. Pet. Sci. Eng.
**2005**, 47, 5–14. [Google Scholar] [CrossRef] - He, J.-H. Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B
**2006**, 20, 1141–1199. [Google Scholar] [CrossRef] [Green Version] - Wang, F.Y.; Liu, D. Networked Control Systems. In Theory and Applications; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Zheng, Y.-J. Water wave optimization: A new nature-inspired metaheuristic. Comput. Oper. Res.
**2015**, 55, 1–11. [Google Scholar] [CrossRef] [Green Version] - Long, H.V.; Nieto, J.J.; Son, N.T.K. New approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces. Fuzzy Sets Syst.
**2018**, 331, 26–46. [Google Scholar] [CrossRef] - Allahviranloo, T.; Abbasbandy, S.; Rouhparvar, H. The exact solutions of fuzzy wave-like equations with variable coefficients by a variational iteration method. Appl. Soft Comput.
**2011**, 11, 2186–2192. [Google Scholar] [CrossRef] - Chadli, L.S.; Harir, A.; Melliani, S. Solutions of fuzzy wave-like equations by variational iteration method. Int. Ann. Fuzzy Math. Inform.
**2014**, 8, 527–547. [Google Scholar] - Hashemi, M.; Malekinagad, J. Series solution of fuzzy wave-like equations with variable coefficients. J. Intell. Fuzzy Syst.
**2013**, 25, 415–428. [Google Scholar] [CrossRef] - Bayrak, M.A. Approximate Solution of Wave Equation using Fuzzy Number. Int. J. Comput. Appl.
**2013**, 68, 975. [Google Scholar] - Zureigat, H.; Ismail, A.I.; Sathasivam, S. A compact Crank–Nicholson scheme for the numerical solution of fuzzy time fractional diffusion equations. Neural Comput. Appl.
**2019**, 32, 6405–6412. [Google Scholar] [CrossRef] - Cheng, C. Fuzzy Solutions to Partial Differential Equations: Adaptive Approach. IEEE Trans. Fuzzy Syst.
**2009**, 17, 116–127. [Google Scholar] [CrossRef] - Bodjanova, S. Median alpha-levels of a fuzzy number. Fuzzy Sets Syst.
**2006**, 157, 879–891. [Google Scholar] [CrossRef] - George, J.; Bo, Y. Fuzzy Sets and Fuzzy Logic, Theory and Applications; Prentice Hall Publishing: Upper Saddle River, NJ, USA, 1995. [Google Scholar]
- Zadeh, L.A. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst.
**1978**, 1, 3–28. [Google Scholar] [CrossRef] - Zadeh, L.A. Toward a generalized theory of uncertainty (GTU)—An outline. Inf. Sci.
**2005**, 172, 1–40. [Google Scholar] [CrossRef] - Kermani, M.A. Numerical method for solving fuzzy wave equation. AIP Conf. Proc.
**2013**, 1558, 2444–2447. [Google Scholar] [CrossRef] - Allahviranloo, T.; Gouyandeh, Z.; Armand, A.; Hasanoglu, A. On fuzzy solutions for heat equation based on gen-eralized Hukuhara differentiability. Fuzzy Sets Syst.
**2015**, 265, 1–23. [Google Scholar] [CrossRef] - Chakraverty, S.; Tapaswini, S.; Behera, D. Fuzzy Differential Equations and Applications for Engineers and Scientists; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Oishi, C.M.; Yuan, J.Y.; Cuminato, J.A.; Stewart, D.E. Stability analysis of Crank–Nicolson and Euler schemes for time-dependent diffusion equations. BIT Numer. Math.
**2015**, 55, 487–513. [Google Scholar] [CrossRef] - Allahviranloo, T. Difference Methods for Fuzzy Partial Differential Equations. Comput. Methods Appl. Math.
**2002**, 2, 233–242. [Google Scholar] [CrossRef] - Alhayani, W. Exact solutions for heat-like and wave-like equations with variable coefficients by daftardar-jafari method. Far East J. Appl. Math.
**2014**, 87, 191. [Google Scholar] - Smarandache, F. Neutrosophic logic-a generalization of the intuitionistic fuzzy logic. Multispace Multistruct.
**2010**, 4, 396. [Google Scholar] [CrossRef] [Green Version] - Aslam, M. Neutrosophic analysis of variance: Application to university students. Complex Intell. Syst.
**2019**, 5, 403–407. [Google Scholar] [CrossRef] [Green Version]

**Figure 3.**The exact and numerical solution of Equation (31) by CTCS and implicit schemes at $t=0.05$ and $x=0.9$ for all $r,\beta \in \left[0,1\right]$.

**Figure 4.**The exact and numerical solution of Equation (31) by implicit at $t=0.5$ and $x=0.9$ for all $r,\beta \in \left[0,1\right]$.

**Figure 5.**The exact solution (lower solution) of Equation (33) at $t=0.01,x0.1$, $r=0\mathrm{and}\beta =0$.

**Figure 6.**The exact solution (upper solution) of Equation (33) at $t=0.01,x0.1$, $r=0\mathrm{and}\beta =0.$

**Figure 7.**The exact and numerical solution of Equation (33) implicit at $t=0.05$ and $x=0.9$ for all $r,\beta \in \left[0,1\right]$.

**Table 1.**Numerical solution of Equation (31) by CTCS and implicit at $t=0.05$ and $x=0.9$ for $r,\beta \in \left[0,1\right]$.

CTCS | Implicit | ||||
---|---|---|---|---|---|

$\beta $ | $r$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ |

Lower $\beta =0$ | $0$ | $-0.30897918419461035$ | $1.14671\times {10}^{-4}$ | $-0.3088657654966003$ | $1.25235\times {10}^{-6}$ |

$0.2$ | $-0.247183347355688$ | $9.17368\times {10}^{-5}$ | $-0.24709261239728061$ | $1.00188\times {10}^{-6}$ | |

$0.4$ | $-0.1853875105167659$ | $6.88026\times {10}^{-5}$ | $-0.1853194592979603$ | $7.51413\times {10}^{-7}$ | |

$0.6$ | $-0.123591673677844$ | $458684\times {10}^{-5}$ | $-0.12354630619864031$ | $5.00942\times {10}^{-7}$ | |

$0.8$ | $-0.06179583683892198$ | $2.29342\times {10}^{-5}$ | $-0.06177315309932013$ | $2.50471\times {10}^{-7}$ | |

$1$ | 0 | 0 | 0 | 0 | |

Upper $\beta =1$ | 0 | $0.30897918419461035$ | $1.14671\times {10}^{-4}$ | $0.3088657654966003$ | $1.25235\times {10}^{-6}$ |

0.2 | $0.247183347355688$ | $9.17368\times {10}^{-5}$ | $0.24709261239728061$ | $1.00188\times {10}^{-6}$ | |

0.4 | $0.1853875105167659$ | $6.88026\times {10}^{-5}$ | $0.1853194592979603$ | $7.51413\times {10}^{-7}$ | |

0.6 | $0.123591673677844$ | $458684\times {10}^{-5}$ | $0.12354630619864031$ | $5.00942\times {10}^{-7}$ | |

0.8 | $0.06179583683892198$ | $2.29342\times {10}^{-5}$ | $0.06177315309932013$ | $2.50471\times {10}^{-7}$ | |

1 | 0 | 0 | 0 | 0 |

**Table 2.**Numerical solution of Equation (31) by CTCS and implicit at $t=0.05$ and $x=0.9$ for $r,\beta \in \left[0,1\right].$

CTCS | Implicit | ||||
---|---|---|---|---|---|

$\beta $ | $r$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ |

$\beta =0$.4 | $0$ | $-0.06179583683892198$ | $2.29342\times {10}^{-5}$ | $-0.06177315309932013$ | $2.50471\times {10}^{-7}$ |

$0.2$ | $-0.04943666947113756$ | $1.83474\times {10}^{-5}$ | $-0.0494185224794561$ | $2.00377\times {10}^{-7}$ | |

$0.4$ | $-0.0370775021033532$ | $1.37605\times {10}^{-5}$ | $-0.03706389185959205$ | $1.50283\times {10}^{-7}$ | |

$0.6$ | $-0.02471833473556878$ | $9.17368\times {10}^{-6}$ | $-0.02470926123972805$ | $1.00188\times {10}^{-7}$ | |

$0.8$ | $-0.012359167367784408$ | $4.58684\times {10}^{-6}$ | $-0.012354630619864042$ | $5.00942\times {10}^{-8}$ | |

$1$ | 0 | 0 | 0 | 0 | |

$\beta =0.6$ | 0 | $0.06179583683892198$ | $2.29342\times {10}^{-5}$ | $0.06177315309932013$ | $2.50471\times {10}^{-7}$ |

0.2 | $0.04943666947113756$ | $1.83474\times {10}^{-5}$ | $0.0494185224794561$ | $2.00377\times {10}^{-7}$ | |

0.4 | $0.0370775021033532$ | $1.37605\times {10}^{-5}$ | $0.03706389185959205$ | $1.50283\times {10}^{-7}$ | |

0.6 | $0.02471833473556878$ | $9.17368\times {10}^{-6}$ | $0.02470926123972805$ | $1.00188\times {10}^{-7}$ | |

0.8 | $0.012359167367784408$ | $4.58684\times {10}^{-6}$ | $0.012354630619864042$ | $5.00942\times {10}^{-8}$ | |

1 | 0 | 0 | 0 | 0 |

**Table 3.**Numerical solution of Equation (33) by CTCS and implicit at $t=0.05$ and $x=0.9$ for $r,\beta \in \left[0,1\right].$

CTCS | Implicit | ||||
---|---|---|---|---|---|

$\beta $ | $r$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ |

Lower $\beta =0$ | $0$ | $-0.30553524551329963$ | $1.635507\times {10}^{-2}$ | $-0.2940107401726$ | $1.4547205\times {10}^{-4}$ |

$0.2$ | $-0.23535441946397$ | $9.0580554\times {10}^{-3}$ | $-0.2352085921381$ | $1.157764\times {10}^{-4}$ | |

$0.4$ | $-0.1833535114597978$ | $6.85104\times {10}^{-3}$ | $-0.1764064884410$ | $8.853232\times {10}^{-5}$ | |

$0.6$ | $-0.12513322097319855$ | $4.544027\times {10}^{-3}$ | $-0.117604296069$ | $5.858821\times {10}^{-5}$ | |

$0.8$ | $-0.0651351048659927$ | $2.247013\times {10}^{-3}$ | $-0.058802148034$ | $2.954411\times {10}^{-5}$ | |

$1$ | 0 | 0 | 0 | 0 | |

Upper $\beta =1$ | 0 | $0.30513524329963$ | $1.143507\times {10}^{-2}$ | $0.2940107401726$ | $1.457205\times {10}^{-4}$ |

0.2 | $0.2441946397$ | $9.408054\times {10}^{-3}$ | $0.2352085921381$ | $1.175764\times {10}^{-4}$ | |

0.4 | $0.183145597978$ | $6.8163404\times {10}^{-3}$ | $0.1764064884410$ | $8.83232\times {10}^{-5}$ | |

0.6 | $0.1220957319855$ | $4.54027\times {10}^{-3}$ | $0.117604296069$ | $5.88821\times {10}^{-5}$ | |

0.8 | $0.0610548659927$ | $2.2705413\times {10}^{-3}$ | $0.058802148034$ | $2.94411\times {10}^{-5}$ | |

1 | 0 | 0 | $0$ | 0 |

**Table 4.**Numerical solution of Equation (33) by CTCS and implicit at $t=0.05$ and $x=0.9$ for $r,\beta \in \left[0,1\right]$.

CTCS | Implicit | ||||
---|---|---|---|---|---|

$\beta $ | $r$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ | $\tilde{u}\left(0.9,0.5;r,\beta \right)$ | $\tilde{E}\left(0.9,0.5;r,\beta \right)$ |

$\beta =0$.4 | $0$ | $-0.06145361048659927$ | $2.270113\times {10}^{-3}$ | $-0.05880796628017$ | $2.94411\times {10}^{-5}$ |

$0.2$ | $-0.04156651458838927942$ | $1.8154611\times {10}^{-3}$ | $-0.04704637302416$ | $2.35528\times {10}^{-5}$ | |

$0.4$ | $-0.0513536629195956$ | $1.34656208\times {10}^{-3}$ | $-0.0354545628477976810$ | $1.76646\times {10}^{-5}$ | |

$0.6$ | $-0.0251564419463971$ | $9.0458054\times {10}^{-4}$ | $-0.02354654552318651207$ | $1.17764\times {10}^{-5}$ | |

$0.8$ | $-0.01531532209731985$ | $4.54027\times {10}^{-4}$ | $-0.01176159325603$ | $5.88821\times {10}^{-6}$ | |

$1$ | 0 | 0 | 0 | 0 | |

$\beta =0.6$ | 0 | $0.061051548659927$ | $2.27013\times {10}^{-3}$ | $0.05880796628017$ | $2.94411\times {10}^{-5}$ |

0.2 | $0.048838927942$ | $1.81611\times {10}^{-3}$ | $0.04704637302416$ | $2.35528\times {10}^{-5}$ | |

0.4 | $0.0365445629195956$ | $1.3654208\times {10}^{-3}$ | $0.0355428477976810$ | $1.764646\times {10}^{-5}$ | |

0.6 | $0.02441549415463971$ | $9.0854054\times {10}^{-4}$ | $0.02345652318651207$ | $1.14567764\times {10}^{-5}$ | |

0.8 | $0.012204559731985$ | $4.544027\times {10}^{-4}$ | $0.01154576159325603$ | $5.8468821\times {10}^{-6}$ | |

1 | 0 | 0 | 0 | 0 |

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**MDPI and ACS Style**

Almutairi, M.; Zureigat, H.; Izani Ismail, A.; Fareed Jameel, A.
Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form. *Mathematics* **2021**, *9*, 667.
https://doi.org/10.3390/math9060667

**AMA Style**

Almutairi M, Zureigat H, Izani Ismail A, Fareed Jameel A.
Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form. *Mathematics*. 2021; 9(6):667.
https://doi.org/10.3390/math9060667

**Chicago/Turabian Style**

Almutairi, Maryam, Hamzeh Zureigat, Ahmad Izani Ismail, and Ali Fareed Jameel.
2021. "Fuzzy Numerical Solution via Finite Difference Scheme of Wave Equation in Double Parametrical Fuzzy Number Form" *Mathematics* 9, no. 6: 667.
https://doi.org/10.3390/math9060667