Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences

Jannelli, Alessandra

doi:10.3390/math8020215

Cited by 24 publications

(8 citation statements)

References 41 publications

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“…The most important reason is that fractional calculus allows for the appropriate modeling of real-world problems, which in turn depends on both the current and the preceding chronological period. The fractional differential equations (FDEs) have garnered attention from many scientific and engineering academics for their crucial role in comprehending various real-life processes in the natural sciences [ 4 , 5 ]. These processes include mechanical systems, wave propagation phenomena, earthquake modeling, image processing, and control theory.…”

Section: Introductionmentioning

confidence: 99%

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

Nadeem,

Alotaibi,

Alotaibi

et al. 2024

PLoS ONE

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This study suggests a strategy for calculating the fuzzy analytical solutions to a two-dimensional fuzzy fractional-order heat problem including a diffusion variable connected externally. We propose Sawi residual power series scheme (SRPSS) which is the amalgamation of Sawi transform and residual power series scheme under the Caputo fractional differential operator. We demonstrate three different examples to derive the fuzzy fractional series solution which is characterized by its rapid convergence and easy finding of the unknown coefficients using the concept of limit at infinity. The most significant aspect of this scheme is that it derives the results without time effort compared with the traditional residual power series approach. Our findings confirm that the SRPSS is a robust and valuable method for approximating the solution of fuzzy fractional problems. Furthermore, we provide 2D and 3D symbolic representations to present the physical behavior of fuzzy fractional problems under the lower and upper bounded solutions.

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Section: Introductionmentioning

confidence: 99%

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

Nadeem,

Alotaibi,

Alotaibi

et al. 2024

PLoS ONE

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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 Fractional Differential Equations Arising in Engineering Sciences

Cited by 24 publications

References 41 publications

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

Numerical investigation of two-dimensional fuzzy fractional heat problem with an external source variable

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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