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20 pages, 707 KiB

Open AccessArticle

Existence and Uniqueness Result for Fuzzy Fractional Order Goursat Partial Differential Equations

by Muhammad Sarwar, Noor Jamal, Kamaleldin Abodayeh, Chanon Promsakon and Thanin Sitthiwirattham

Fractal Fract. 2024, 8(5), 250; https://doi.org/10.3390/fractalfract8050250 - 25 Apr 2024

Cited by 2 | Viewed by 1633

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s

g H

-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s

g H

-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert [...] Read more.

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s

g H

-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s

g H

-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s

g H

-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform. Full article

(This article belongs to the Special Issue Advances in Fractional Order Derivatives and Their Applications, 2nd Edition)

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47 pages, 1399 KiB

Open AccessArticle

Novel Approaches for Solving Fuzzy Fractional Partial Differential Equations

by Mawia Osman, Yonghui Xia, Muhammad Marwan and Omer Abdalrhman Omer

Fractal Fract. 2022, 6(11), 656; https://doi.org/10.3390/fractalfract6110656 - 7 Nov 2022

Cited by 8 | Viewed by 2375

Abstract

In this paper, we present a comparison of several important methods to solve fuzzy partial differential equations (PDEs). These methods include the fuzzy reduced differential transform method (RDTM), fuzzy Adomian decomposition method (ADM), fuzzy Homotopy perturbation method (HPM), and fuzzy Homotopy analysis method (HAM). A distinguishing practical feature of these techniques is administered without the need to use discretion or restricted assumptions. Moreover, we investigate the fuzzy

(n + 1)

-dimensional fractional RDTM to obtain the solutions of fuzzy fractional PDEs. The much more distinctive element of this method is that it requires no predetermined assumptions, and reduces the computational effort. We apply the suggested techniques to a set of initial valued problems and get approximate numerical solutions for linear and nonlinear time-fractional PDEs. It is demonstrated that the fuzzy

(n + 1)

-dimensional fractional RDTM is both accurate and simple to use. The methods are based on gH-differentiability and fuzzy fractional derivatives. Some illustrative numerical examples are given to demonstrate the effectiveness of our proposed methods. The results show that the methods are powerful mathematical tools for solving fuzzy partial differential equations. Full article

(This article belongs to the Section General Mathematics, Analysis)

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24 pages, 766 KiB

Open AccessArticle

Initial Value Problems of Fuzzy Fractional Coupled Partial Differential Equations with Caputo gH-Type Derivatives

by Fan Zhang, Hai-Yang Xu and Heng-You Lan

Fractal Fract. 2022, 6(3), 132; https://doi.org/10.3390/fractalfract6030132 - 25 Feb 2022

Cited by 5 | Viewed by 2387

Abstract

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo

g H

-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and [...] Read more.

The purpose of this paper is to investigate a class of initial value problems of fuzzy fractional coupled partial differential equations with Caputo

g H

-type derivatives. Firstly, using Banach fixed point theorem and the mathematical inductive method, we prove the existence and uniqueness of two kinds of

g H

-weak solutions of the coupled system for fuzzy fractional partial differential equations under Lipschitz conditions. Then we give an example to illustrate the correctness of the existence and uniqueness results. Furthermore, because of the coupling in the initial value problems, we develop Gronwall inequality of the vector form, and creatively discuss continuous dependence of the solutions of the coupled system for fuzzy fractional partial differential equations on the initial values and

ε

-approximate solution of the coupled system. Finally, we propose some work for future research. Full article

(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)

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32 pages, 1730 KiB

Open AccessArticle

A Novel Treatment of Fuzzy Fractional Swift–Hohenberg Equation for a Hybrid Transform within the Fractional Derivative Operator

by Saima Rashid, Rehana Ashraf and Fatimah S. Bayones

Fractal Fract. 2021, 5(4), 209; https://doi.org/10.3390/fractalfract5040209 - 11 Nov 2021

Cited by 11 | Viewed by 2429

Abstract

This article investigates the semi-analytical method coupled with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). In addition, we apply this method to the time-fractional Swift–Hohenberg equation (SHe) with various initial conditions (IC) under gH-differentiability. Some aspects of the fuzzy Caputo fractional derivative (CFD) with the Elzaki transform are presented. Moreover, we established the general formulation and approximate findings by testing examples in series form of the models under investigation with success. With the aid of the projected method, we establish the approximate analytical results of SHe with graphical representations of initial value problems by inserting the uncertainty parameter

0 \leq ℘ \leq 1

with different fractional orders. It is expected that fuzzy EADM will be powerful and accurate in configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures. Full article

(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order II)

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31 pages, 1811 KiB

Open AccessArticle

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

by Saima Rashid, Rehana Ashraf, Ahmet Ocak Akdemir, Manar A. Alqudah, Thabet Abdeljawad and Mohamed S. Mohamed

Fractal Fract. 2021, 5(3), 113; https://doi.org/10.3390/fractalfract5030113 - 8 Sep 2021

Cited by 12 | Viewed by 2532

Abstract

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the [...] Read more.

g H

-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter

ζ \in [0, 1]

and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus)

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14 pages, 295 KiB

Open AccessArticle

Fuzzy Sawi Decomposition Method for Solving Nonlinear Partial Fuzzy Differential Equations

by Atanaska Georgieva and Albena Pavlova

Symmetry 2021, 13(9), 1580; https://doi.org/10.3390/sym13091580 - 27 Aug 2021

Cited by 12 | Viewed by 2292

Abstract

The main goal of this paper is to propose a new decomposition method for finding solutions to nonlinear partial fuzzy differential equations (NPFDE) through the fuzzy Sawi decomposition method (FSDM). This method is a combination of the fuzzy Sawi transformation and Adomian decomposition method. For this purpose, two new theorems for fuzzy Sawi transformation regarding fuzzy partial gH-derivatives are introduced. The use of convex symmetrical triangular fuzzy numbers creates symmetry between the lower and upper representations of the fuzzy solution. To demonstrate the effectiveness of the method, a numerical example is provided. Full article

(This article belongs to the Special Issue Research on Fuzzy Logic and Mathematics with Applications)

19 pages, 5552 KiB

Open AccessArticle

Fuzzy Solution to the Unconfined Aquifer Problem

by Christos Tzimopoulos, Kyriakos Papadopoulos, Christos Evangelides and Basil Papadopoulos

Water 2019, 11(1), 54; https://doi.org/10.3390/w11010054 - 29 Dec 2018

Cited by 10 | Viewed by 3803

Abstract

In this article, the solution to the fuzzy second order unsteady partial differential equation (Boussinesq equation) is examined, for the case of an aquifer recharging from a lake. In the examined problem, there is a sudden rise and subsequent stabilization of the lake’s water level, thus the aquifer is recharging from the lake. The aquifer boundary conditions are fuzzy and create ambiguities to the solution of the problem. Since the aforementioned problem concerns differential equations, the generalized Hukuhara (gH) derivative was used for total derivatives, as well as the extension of this theory concerning partial derivatives. The case studies proved to follow the generalized Hukuhara (gH) derivative conditions and they offer a unique solution. The development of the aquifer water profile was examined, as well as the calculation of the recharging fuzzy water movement profiles, velocity, and volume, and the results were depicted in diagrams. According to presented results, the hydraulic engineer, being specialist in irrigation projects or in water management, could estimate the appropriate water volume quantity with an uncertainty level, given by the α-cuts. Full article

(This article belongs to the Special Issue Insights on the Water–Energy–Food Nexus)

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