Numerical solutions of fuzzy time fractional advection‐diffusion equations in double parametric form of fuzzy number

Zureigat, Hamzeh; Ismail, Ahmad Izani; Sathasivam, Saratha

doi:10.1002/mma.5573

Cited by 11 publications

(3 citation statements)

References 37 publications

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“…Using the same procedure, we can show that the Saulyev's scheme in Equation ( 29) is unconditionally stable [34]. □…”

Section: The Stability Analysismentioning

confidence: 90%

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

Al-Khateeb

2024

Axioms

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In this study, we explore fractional partial differential equations as a more generalized version of classical partial differential equations. These fractional equations have shown promise in providing improved descriptions of certain phenomena under specific circumstances. The main focus of this paper comprises the development, analysis, and application of two explicit finite difference schemes to solve an initial boundary value problem involving a fuzzy time fractional convection–diffusion equation with a fractional order in the range of 0≤ ξ ≤ 1. The uniqueness of this problem lies in its consideration of fuzziness within both the initial and boundary conditions. To handle the uncertainty, we propose a computational mechanism based on the double parametric form of fuzzy numbers, effectively converting the problem from an uncertain format to a crisp one. To assess the stability of our proposed schemes, we employ the von Neumann method and find that they demonstrate unconditional stability. To illustrate the feasibility and practicality of our approach, we apply the developed scheme to a specific example.

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“…Using the same procedure, we can show that the Saulyev's scheme in Equation ( 29) is unconditionally stable [34]. □…”

Section: The Stability Analysismentioning

confidence: 90%

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

Al-Khateeb

2024

Axioms

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“…A fuzzy partial differential equation has been used to describe the behavior of many time-dependent phenomena, including fuzzy heat conduction and fuzzy particle diffusion, in which uncertainty or vagueness exists. The fuzzy heat equation is considered one of the most significant fuzzy parabolic partial differential equations used to describe how a fuzzy quantity, such as heat, diffuses through a given region [7][8][9][10][11][12][13][14][15][16][17][18][19]. In general, the exact analytical solution for the fuzzy heat equations is difficult to obtain.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fuzzy Heat Equation with Complex Dirichlet Conditions

Zureigat

Alkouri

Al-Khateeb

et al. 2023

IJFIS

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Recently, complex fuzzy sets have become powerful tools for generalizing the range of fuzzy sets to wider ranges that lie on a unit disk in the complex plane. In this study, complex fuzzy numbers are discussed and applied for the first time to solve a complex fuzzy partial differential equation involving a complex fuzzy heat equation under Hukuhara differentiability. Subsequently, an explicit finite difference scheme, referred to as the forward time-center space (FTCS), was implemented to solve the complex fuzzy heat equations. The imprecision of the issue is evident in the initial and boundary conditions, as well as in the amplitude and phase terms' coefficients, where the convex normalized triangular fuzzy numbers are extended to the unit disk in the complex plane. The proposed numerical methods utilized the properties and benefits of the complex fuzzy set theory. Furthermore, a new proof of consistency, stability, and convergence was established under this theory. A numerical example was provided to illustrate the reliability and feasibility of the proposed approach. The results obtained using the proposed approach are in adequate agreement with the exact solution and related theoretical aspects.

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“…The fuzzy partial differential equation is commonly utilized to explain the behavior of dynamic phenomena in which imprecision or indeterminacy is present. This includes fuzzy heat conduction and fuzzy particle diffusion, with the fuzzy heat equation being one of the most important fuzzy parabolic partial differential equations for describing how a fuzzy quantity such as heat diffuses through a given area [9][10][11][12][13][14][15][16][17][18][19][20][21]. While exact analytical solutions for fuzzy heat equations may be challenging to obtain, numerical techniques are needed to achieve the solution.…”

Section: Introductionmentioning

confidence: 99%

A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method

Zureigat,

Tashtoush,

Jassar

et al. 2023

Advances in Mathematical Physics

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Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem’s fuzziness arises from the coefficients in both amplitude and phase terms, as well as in the initial and boundary conditions, with the Convex normalized triangular fuzzy numbers extended to the unit disk in the complex plane. The researchers take advantage of the properties and benefits of CFS theory in the proposed numerical methods and subsequently provide a new proof of the stability under CFS theory. A numerical example is presented to demonstrate the proposed approach’s reliability and feasibility, with the results showing good agreement with the exact solution and relevant theoretical aspects.

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Numerical solutions of fuzzy time fractional advection‐diffusion equations in double parametric form of fuzzy number

Cited by 11 publications

References 37 publications

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

Efficient Numerical Solutions for Fuzzy Time Fractional Convection Diffusion Equations Using Two Explicit Finite Difference Methods

Numerical Solution of Fuzzy Heat Equation with Complex Dirichlet Conditions

A Solution of the Complex Fuzzy Heat Equation in Terms of Complex Dirichlet Conditions Using a Modified Crank–Nicolson Method

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