Fuzzy Fractional Differential Equations Research Articles

Best Proximity Point Results via Simulation Function with Application to Fuzzy Fractional Differential Equations

In this study, we prove the existence and uniqueness of a best proximity point in the setting of non-Archimedean modular metric spaces via the concept of simulation functions. A non-Archimedean metric modular is shaped as a parameterized family of classical metrics; therefore, for each value of the parameter, the positivity, the symmetry, the triangle inequality, or the continuity is ensured. Also, we demonstrate how analogous theorems in modular metric spaces may be used to generate the best proximity point results in triangular fuzzy metric spaces. The utility of our findings is further demonstrated by certain examples, illustrated consequences, and an application to fuzzy fractional differential equations.

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

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev’s scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev’s is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes.

Solutions of fuzzy fractional differential equations

Solutions of fuzzy fractional differential equations

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

Objectives: In applied sciences and engineering, fuzzy fractional differential equations (FFDEs) and fuzzy fractional integral equations (FFIEs) are a crucial topic. The main objective of this work is to discover an analytical approximate solution for the fuzzy fractional Volterra-Fredholm integro differential equations (FFVFIDE). In the Caputo concept, fractional derivatives are regarded. Methods: The Shehu transform is challenging to exist for nonlinear problems. So, the Shehu transform is combined with the Adomian decomposition method is called the Shehu Adomian decomposition method (SHADM) and has been proposed to solve both linear and nonlinear FFVFIDEs. Findings: Both linear and nonlinear FFVIFIDEs can be solved using this technique. For nonlinear terms, Adomian polynomials have been used. The main benefit of this approach is that it converges quickly to the exact solution. Figures and numerical examples demonstrate the expertise of the suggested approach. Novelty: The comparison between the exact solution and numerical solution is shown in figures for various values of fractional order . The numerical evolution demonstrates the efficiency and reliability of the proposed SHADM. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes. Keywords: Fractional calculus, fuzzy number, Mittag ­ Leffler function, Shehu Adomian decomposition method, fuzzy fractional Volterra­-Fredholm integro differential equation

Numerical solution of fuzzy fractional volterra integro differential equations with boundary conditions

In this study, the boundary value problem of fuzzy fractional nonlinear Volterra integro differential equations of order 1 < ϱ ≤ 2 is addressed. Fuzzy fractional derivatives are defined in the Caputo sense. To show the existence result, the Krasnoselkii theorem from the theory of fixed points is used, where as the well-known contraction mapping concept is utilized in order to show the solution is unique to the proposed problem. Moreover, a novel Adomian decomposition method is utilized to get numerical solution; the approach behind deriving the solution is from Adomian polynomials, and it is organized according to the recursive relation that is obtained. The proposed method significantly decreases the numerical computations by obtaining solutions without the need of discretization or constrictive assumptions. According to the results, there is substantial agreement between the series solutions produced by the fuzzy Adomian decomposition method. Finally, using MATLAB, the symmetry between the lower and upper-cut representations of the fuzzy solutions is demonstrated in the numerical result.

Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator

Abstract This article presents a new approach for solving the fuzzy fractional Degasperis–Procesi (FFDP) and Camassa–Holm equations using the iterative transform method (ITM). The fractional Degasperis–Procesi (DP) and Camassa–Holm equations are extended from the classical DP and Camassa–Holm equations by incorporating fuzzy sets and fractional derivatives. The ITM is a powerful technique widely used for solving nonlinear differential equations. This approach transforms the fuzzy fractional differential equations into a series of ordinary differential equations, which are then solved iteratively using a recursive algorithm. Numerical simulations demonstrate the proposed approach’s accuracy and effectiveness. The results show that the ITM provides an efficient and accurate method for solving the FFDP and Camassa–Holm equations. The proposed method can be extended to solve other fuzzy fractional differential equations.

A new approach to the fractional Abel k−integral equations and linear fractional differential equations in a fuzzy environment

This paper introduces some recent approaches to the fuzzy fractional Abel integral equations with respect to another function, which is called the fuzzy fractional Abel k−integral equations. The problem proposed here allows for the interpolation of different types of fuzzy classical fractional Abel integral equations, and the solvability of each type of integral equation is also discussed. Additionally, a new approach to the linear fuzzy fractional differential equations in the space of generalized fuzzy functions is introduced. Unlike previous approaches with gH-differentiability, this approach avoids making any prior assumptions about the solution's monotonic behavior during the solving process. It also overcomes the problem of multiple solutions for fuzzy fractional differential equations. To demonstrate the benefits of this approach, two initial value problems of linear fuzzy differential equations are compared using two different concepts of fractional derivatives: the Caputo fractional gH-differentiability and the Caputo fractional derivative in the generalized fuzzy space. Furthermore, some applications are provided in a viscoelastic material model and a population growth with harvesting model.

A Fuzzy Fractional Power Series Approximation and Taylor Expansion for Solving Fuzzy Fractional Differential Equation

Fuzzy fractional differential has the strength to capture the senses of memory and uncertainty simultaneously involved in dynamical systems. However, a solution for fuzzy fractional differential equations is not always found regularly. This paper discusses a numerical solution approach for the fuzzy fractional differential equation using power series approximation with a fuzzy fractional counterpart of Taylor’s theorem. Caputo’s definition of the fractional derivative and generalized Hukuhara difference are used to describe the fuzzy differential equation in this paper. Utilization of the generalized Hukuhara difference for the fuzzy valued function ensures the uniqueness and boundedness of the fuzzy solution in parametric form.

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

In this study, we examine the numerical solutions of nonlinear fuzzy fractional partial differential equations under the Caputo derivative utilizing the technique of fuzzy Adomian decomposition. This technique is used as an alternative method for obtaining approximate fuzzy solutions to various types of fractional differential equations and also investigated some new existence and uniqueness results of fuzzy solutions. Some examples are given to support the effectiveness of the proposed technique. We present the numerical results in graphical form for different values of fractional order and uncertainty γ∈0,1.

Local Fuzzy Fractional Partial Differential Equations in the Realm of Fractal Calculus with Local Fractional Derivatives

In this study, local fuzzy fractional partial differential equations (LFFPDEs) are considered using a hybrid local fuzzy fractional approach. Fractal model behavior can be represented using fuzzy partial differential equations (PDEs) with local fractional derivatives. The current methods are hybrids of the local fuzzy fractional integral transform and the local fuzzy fractional homotopy perturbation method (LFFHPM), the local fuzzy fractional Sumudu decomposition method (LFFSDM) in the sense of local fuzzy fractional derivatives, and the local fuzzy fractional Sumudu variational iteration method (LFFSVIM); these are applied when solving LFFPDEs. The working procedure shows how effective solutions for specific LFFPDEs can be obtained using the applied approaches. Moreover, we present a comparison of the local fuzzy fractional Laplace variational iteration method (LFFLIM), the local fuzzy fractional series expansion method (LFFSEM), the local fuzzy fractional variation iteration method (LFFVIM), and the local fuzzy fractional Adomian decomposition method (LFFADM), which are applied to obtain fuzzy fractional diffusion and wave equations on Cantor sets. To demonstrate the effectiveness of the used techniques, some examples are given. The results demonstrate the major advantages of the approaches, which are equally efficient and simple to use in order to solve fuzzy differential equations with local fractional derivatives.

Hybrid fuzzy Laplace-like transforms for solving fractional-order fuzzy differential equations

The main purpose of this paper is to introduce a novel technique called the hybrid fuzzy Laplace-like transform method (HFLTM) for solving nonlinear forms of fuzzy fractional differential equations (FFDEs) and fuzzy partial differential equations (FPDEs). The HFLTM combines the Adomian decomposition method with the Laplace-like transform, utilizing the framework of gH-differentiability. General algorithms based on the HFLTM are presented for solving nonlinear FFDEs and FPDEs. The effectiveness and accuracy of this new method are demonstrated through examples and simulations.

Fuzzy solutions of some variants of the fractional order Korteweg-de-Vries equations via an analytical method

Under uncertainty, the analytical behaviour of fractional partial differential equations is frequently puzzling and challenging to predict. Therefore, in order to address these problems, it is essential to create a proper, extensive, and highly effective theory. The theory of fuzzy fractional partial differential equations is a relatively new idea with applications in applied mathematics and engineering. In this article, we examined the different types of fuzzy fractional-order Korteweg-de-Vries (KdV) equations along with their analytical fuzzy solutions. To construct a series-type solution under a fuzzy notion for the relevant research, we utilise the fractional reduced differential transform method (FRDTM) in the Caputo sense. The solutions of various kinds of fuzzy fractional KdV equations that have been developed are more broadly applicable. The fuzzy concept is used to overrule the uncertainty in physical models based on the parametric form of the fuzzy number. The numerical and graphical presentation demonstrates the symmetry between the lower and upper cut representations of the fuzzy solutions and may be helpful in improving understanding of automatic control models, measure theory, physics, biology, computer science, quantum optics, economics, artificial intelligence, and other domains, as well as non-financial analysis.

Solutions of non-homogeneous system of fuzzy fractional differential equations: a novel approach

Solutions of non-homogeneous system of fuzzy fractional differential equations: a novel approach

Fractional derivatives of fuzzy complex-variable functions and application to fuzzy fractional complex differential equations

In this paper, through theory of fuzzy complex analysis, we introduce some representations of r-cuts of trivial fuzzy zero, nontrivial fuzzy zero and fuzzy complex numbers in polar form on complex plane and establish some of their basic properties. Based on this, the addition and scalar multiplication operations of fuzzy complex numbers are established and the calculus of fuzzy complex-variable functions is studied. Moreover, we introduce a notion of fuzzy Ortigueira fractional derivative of fuzzy complex-variable functions and develop a theory of fuzzy fractional calculus on complex plane including some geometric features. As applications, the existence and uniqueness theorem of solutions for nonlinear fuzzy fractional complex-variable differential equations is established and some examples are provided.

Singular fuzzy fractional quadratic regulator problem

This study aims to create a fuzzy optimal control system for a singular fuzzy fractional quadratic regulator problem, which is governed by singular fuzzy fractional differential equations. Additionally, this paper introduces new terms such as fuzzy stability, asymptotic stability, stabilizability, and detectability in the context of singular fractional control systems. Theoretical concepts regarding the stabilizability and detectability of these systems are also outlined, as well as a theorem and related algorithm for determining the fuzzy optimal control and trajectory of the singular fuzzy fractional quadratic regulator problem. The proposed method is tested through simulation on an electrical circuit modeled as a singular fuzzy fractional system, with results demonstrating the efficacy of the approach.

Fuzzy fractional differential equations with interactive derivative

In this manuscript we present the concept of fuzzy fractional differential equation considering the fuzzy relationship of interactivity. We develop the theory of integral equation and fractional differentiability from the point of view of interactivity, that is, the fuzzy Caputo fractional derivative is given by fuzzy interactive derivative. We propose the fractional initial value problem via the interactive fuzzy Caputo fractional derivative and we comment on the usefulness of the interactive fuzzy derivative in the fractional context, since there are two fuzzy processes involved. Finally, we apply the results in two models of population dynamics, namely the Malthusian model and the Verhulst model, where analytical and numerical solutions are presented.

Solving fuzzy fractional differential equations with applications

In this article, we proposed several methods to solve the nonlinear fuzzy fractional differential equation. The methods include the fuzzy Adomian decomposition method (fuzzy ADM), fuzzy homotopy perturbation method (fuzzy HPM), fuzzy homotopy analysis method (fuzzy HAM), and fuzzy Laplace decomposition method (fuzzy LDM). Moreover, the comparisons between these methods are presented. The fuzzy LDM is the combined form of the fuzzy Laplace transform method and the fuzzy ADM. The proposed scheme finds the solutions without any discretization or restrictive assumptions and therefore, reduces the numerical computations to a great extent. The results show that the solutions obtained by the fuzzy LDM have a close agreement with the series solutions obtained with the help of the fuzzy ADM. Finally, we apply the fuzzy ADM and HAM to obtain the solutions of fuzzy multi-term linear and nonlinear fractional diffusion-wave equations. The techniques are investigated based on fuzzy Caputo’s fractional derivative. Applying the obtained methods to the fuzzy fractional diffusion-wave equations, we obtained several new results. Some illustrative numerical examples are given to demonstrate the effectiveness of the proposed methods. The results reveal that the methods are very effective, convenient, and quite accurate mathematical tools for solving fuzzy fractional differential equations.

Approximate Solutions of Fuzzy Fractional Differential Equations via Homotopy Analysis Method

Approximate Solutions of Fuzzy Fractional Differential Equations via Homotopy Analysis Method

A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations

&lt;abstract&gt; &lt;p&gt;The purpose of this study is to extend and determine the analytical solution of a two-dimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized $ H $-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the Mittag-Leffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.&lt;/p&gt;&lt;/abstract&gt;

Some results on the study of -Hilfer type fuzzy fractional differential equations with time delay

This paper is concerned with the finite-time stability of -Hilfer type fuzzy fractional differential equations (FFDEs) with time delay. By applying standard theorems and a hypothetical condition, we explore the existence of solution and stabilty results.