Hukuhara Differentiability Research Articles

Generalized Ulam-Hyer stability results for fuzzy fractional stochastic system under Caputo fractional generalized Hukuhara differentiability concept

Generalized Ulam-Hyer stability results for fuzzy fractional stochastic system under Caputo fractional generalized Hukuhara differentiability concept

Ulam–Hyers Stability of Fuzzy Fractional Non-instantaneous Impulsive Switched Differential Equations Under Generalized Hukuhara Differentiability

Ulam–Hyers Stability of Fuzzy Fractional Non-instantaneous Impulsive Switched Differential Equations Under Generalized Hukuhara Differentiability

Fuzzy analysis of 2-D wave equation through Hukuhara differentiability coupled with AOS technique

The research article in hand provides a new mechanism that deals with the investigation of the triangular analytical fuzzy solutions (TAFS) of the two-dimensional fuzzy fractional order wave equation (2-D FFWE) through the Hukuhara conformable fractional derivative (HCFD) along with the concept of [gH] and [gH−p] differentiability. The mechanism consists of a fuzzy traveling wave method coupled with additive operating splitting (AOS). The procedure for the aforesaid mechanism starts with the extension of the HCFD to the fuzzy conformable fractional derivative (FCFD). Furthermore, some properties of FCFD that play a vital role in this study like, ([i−gH],[ii−gH],[i−p],[ii−p])-differentiability, switching point, fuzzy chain rule, and traveling wave method are discussed in detail. Further, fuzzy traveling wave method coupled with the procedure of the additive operating splitting (AOS) method is adopted to investigate the triangular analytical fuzzy solution of the Two-dimensional fuzzy wave equation (2-D FWE). Finally, some examples are provided that support our arguments.

Solving first order fuzzy partial differential equations using fuzzy Aboodh transform

In this article, we extend the concept of the fuzzy Aboodh transform to the solution of fuzzy partial differential equations using Hukuhara differentiability. In order to create the solution to first-order fuzzy partial differential equations, we first provide the fuzzy Aboodh transform and then present the basic definitions and theorems for the fuzzy Aboodh transform of fuzzy partial derivatives. Finally, the method is illustrated with an example to show the ability of the proposed method.

Existence of generalized Hukuhara differentiable solutions to a class of first-order fuzzy differential equations in dual form

This article deals with the problem of the existence of solution for a class of first-order fuzzy differential equations in dual form in the sense that the derivative of the unknown function appears on both sides of the equation. The problem is studied from the point of view of the generalized Hukuhara differentiability and sufficient conditions for the existence of a unique solution are provided, in details. Also, iterative sequences converging to the solutions are presented and applied to solve some numerical examples. Finally, a practical application of the results is brought.

Solution of generalized fractional Jaulent–Miodek model with uncertain initial conditions

This paper analyses a coupled system of generalized coupled system of fractional Jaulent–Miodek equations, including uncertain initial conditions with fuzzy extension. In this regard, an extension of the homotopy with a generalized integral algorithm is adopted for a class of time-fractional fuzzy Jaulent–Miodek models by mixing the fuzzy q-homotopy analysis algorithm with a generalized integral transform and Caputo fractional derivative. The triangular fuzzy numbers (TFNs)are expressed in double parametric form using κ-cut and r-cut and utilized to explain the uncertainties arising in the initial conditions of highly nonlinear differential equations with generalized Hukuhara differentiability (gH-differentiability). The TFNs are controlled by the κ-cut and r-cut, and the variability of uncertainty is examined using a “triangular membership function” (TMF). The results are analyzed by finding the solutions for different spatial coordinate values of time with κ-cut and r-cut for both lower and upper bounds and validated through numerical and graphical representations in crisp cases. Finally, it can be seen that the uncertain probability density function rapidly decreases at the left and right edges when the fractional order is increased, and it is observed that the obtained solutions are more accurate than the existing results through the Hermite wavelet method in the literature.

Fuzzy Caputo [formula omitted]-fractional linear equations on the time scale [formula omitted

In this paper, we explore various concepts in q-fractional calculus and extend them to the fuzzy domain. Specifically, we define fuzzy q-exponential and fuzzy q-Mittag-Leffler functions using r-cuts. Additionally, we evaluate the fuzzy Caputo q-derivatives of the fuzzy q-exponential functions and introduce the concept of fuzzy q-Laplace transform. Furthermore, we provide a formula for the fuzzy q-Laplace transform of the fuzzy Caputo q-fractional derivative.We also tackle the problem of solving fuzzy Caputo q-fractional linear differential equations under the generalized Hukuhara differentiability conditions, with a focus on specific forcing functions. The key technique employed for solving these equations is the fuzzy q-Laplace transform. To illustrate the practicality of our theoretical results, we present several numerical examples.This paper contributes to the development of q-fractional calculus in the context of fuzzy mathematics and provides valuable insights into solving fuzzy differential equations using the proposed techniques.

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

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.

Explicit analytical solutions of an incommensurate system of fractional differential equations in a fuzzy environment

Fuzzy fractional models have attracted considerable attention because of their comprehensive and broader understanding of real-world problems. Analytical studies of these models are often complex and difficult. Therefore, it is beneficial to develop a suitable and comprehensive technique to solve these models analytically. In this paper, an explicit analytical technique for solving two-dimensional incommensurate linear fuzzy systems of fractional Caputo differential equations (FLSoCFDEs) considering generalized Hukuhara differentiability (gH-differentiability) is presented and demonstrated. This extracted explicit solution is presented for different classes of such systems, including homogeneous and non-homogeneous cases with commensurate and incommensurate fractional orders. Moreover, the potential solution of FLSoCFDEs in terms of the Mittag-Leffler function involving double series is presented. The originality of the proposed technique is that the fuzzy Cauchy problem is transformed into a system of fuzzy linear Volterra integral equations of second kind and then the solution is extracted using the iterative Picard scheme based on the Banach fixed point theorem. Moreover, several interesting results are derived from FLSoCFDEs in terms of the Mittage-Leffler function for both homogeneous and inhomogeneous cases. To understand the proposed technique, we solve a diffusion process problem (a biological model) and several mass-spring systems as applications. Their graphs are analyzed to illustrate and support the theoretical results.

Gradient-based descent linesearch to solve interval-valued optimization problems under gH-differentiability with application to finance

In this article, gradient based descent line search scheme is proposed to solve interval optimization problems under generalized Hukuhara differentiability. The innovation and importance of these concepts are presented from practical and computational point of view. The necessary condition for existence of critical point is presented in inclusion form of interval-valued gradient, without transforming it into real valued function. Suitable efficient descent direction is chosen based on the monotonic property of the interval-valued function and specific interval ordering. Mathematical convergence of the scheme is proved under the assumption of Inexact line search for interval optimization problem. The theoretical developments are implemented with a set of interval test problems in different dimensions. A possible application in finance is provided and solved by our proposed scheme.

A Neutrosophic differential equation approach for modelling glucose distribution in the bloodstream using neutrosophic sets

Neutrosophic sets are a generalization of fuzzy sets and intuitionistic fuzzy sets. They play an important role in addressing uncertainty, vagueness, and indeterminacy in problem-solving. Differential equations in a neutrosophic environment under Hukuhara differentiability are used to model the amount of glucose distribution and absorption rates in blood. Trapezoidal neutrosophic numbers are used to describe the initial conditions and equation parameters. Equilibrium points, along with their stability analysis, have been performed. Exact solutions are found, and corresponding numerical simulations for different (α,β,γ)-cut values of trapezoidal neutrosophic numbers are performed using MATLAB.

Tracking Control of a Class of Fuzzy Dynamical Systems under the Concept of Granular Differentiability

In this work, the tracking control of a class of uncertain linear dynamical systems is investigated. The uncertainty is considered to be represented as fuzzy numbers, and hence, these uncertain dynamical systems are referred to as fuzzy linear dynamical systems, which are presented in the form of fuzzy differential equations (FDEs). The solution of an FDE is found using an approach called relative-distance-measure fuzzy interval arithmetic and under the granular differentiability concept. The control objective is to provide a control law such that the output of the system tracks a desired reference input in the presence of uncertainties. To this end, a theorem is proposed, which suggests that the control law should take the form of a feedback of fuzzy states with fuzzy gains and a fuzzy pre-compensator. However, since the fuzzy states of the system may not always be measurable, a fuzzy observer is designed for the estimation of such fuzzy states. It is also clearly shown that the generalized Hukuhara differentiability concept is unable for solving the problem examined in this study. Finally, the efficiency of the approach is examined for a plane landing control problem.

New Solutions of Fuzzy-Fractional Fisher Models via Optimal He–Laplace Algorithm

Fuzzy differential equations have gained significant attention in recent years due to their ability to model complex systems in the presence of uncertainty or imprecise information. These equations find applications in various fields, such as biomathematics, horological processes, production inventory models, epidemic models, fluid models, and economic investments. The Fisher model is one such example, which studies the dynamics of a population with uncertain growth rates. This study proposes a dual parametric extension of the He–Laplace algorithm for solving time-fractional fuzzy Fisher models. The proposed methodology uses triangular fuzzy numbers to introduce uncertainty in highly nonlinear fractional differential equations under the generalized Hukuhara differentiability concept. The obtained solutions are validated against existing results in crisp form and are found to be more accurate. The results are analyzed by finding solutions with different values of spatial coordinate ξ , time η , and ς -cut for both upper and lower bounds, and they are presented graphically. The analysis reveals that the uncertain probability density function gradually decreases at left and right boundaries when the fractional order is increased. The study provides valuable insights into the behavior of population growth under uncertain conditions and demonstrates the effectiveness of the proposed methodology in solving fuzzy-fractional models.

A Study of Some Properties of Fuzzy Laplace Transform with Their Applications in Solving the Second-Order Fuzzy Linear Partial Differential Equations

In this paper, several results and theorems about the high-order strongly generalized Hukuhara differentiability of function defined via the fuzzy Riemann improper integral (in the sense of Wu) have been established. Then, some properties dealing with the partial derivatives of fuzzy Laplace transform for a fuzzy function of two real variables have been proved. Afterwards, an algorithm of fuzzy Laplace transform for solving second-order fuzzy partial differential equations has been proposed. Finally, two numerical examples, including the heat equation under fuzzy initial conditions, have been studied to justify the efficiency of the algorithm.

Fuzzy SHmath.Mbio-transform generalization and application to skin cancer imaging (distributed diseases)

Fuzzy integral transformations are an important methodology for solving the relevant fuzzy boundary (initial) value issues that arise from solving fuzzy differential and integral equations. In this study, the way we classify the-for the corresponding-order fuzzy derivative, fuzzy Shmath.mbio.M-transform. Studies of this process are conducted on the assumption of substantial generalized Hukuhara differentiability. When comparing the suggested transform to another transform, the fuzzy Laplace transform, the effectiveness of the proposed transform in extracting melanoma edges was shown to be much higher. The accuracy and efficiency of the proposed transformation may be enhanced even more. statistical parameters were used to corroborate the findings.

Generalization of Fuzzy SHA-Transform with Medical Application

Fuzzy differential and integral equations and their associated fuzzy boundary (initial) value problems have been solved using a variety of methods., fuzzy integral transforms are significant method for this purpose. In this research, we extend and use the formula of third- order fuzzy derivative of fuzzy SHA-transform (FSHAT) to generalize the nth-order fuzzy derivative of this integral transform. In fact, this process studies under strongly generalized Hukuhara differentiability. In fact, a drug distribution means the amount of drug in the body which is measured by the proportional of this drug to its amount in the plasma or blood, so corresponding to this issue, a drug concentration in an organ equation solves using this proposed fuzzy SHA-transform.

Manifestation of interval uncertainties for fractional differential equations under conformable derivative

We propose a generalization of conformable calculus for Type-2 interval-valued functions. We investigated the differentiability and integrability properties of such functions. The conformable generalized Hukuhara (gH) differentiability of fractional order is introduced in this study. We prove a number of essential theorems on the conformable differentiability of the sum, gH difference, and product in a Type 2 interval setting. Furthermore, we define conformable Laplace transformation of Type-2 interval-valued functions. We interpret uncertain linear differential equations by using proposed theories. Several examples are given in detail to illustrate and clarify these rules and theorems. Applications to solving Type-2 interval differential equations with conformable derivatives are shown. Type-2 interval generalizes the interval uncertainty. On the other hand, conformable calculus extends the notion of integer calculus. This paper contributes a generalized theory that includes several existing results of classical integral and differential calculus and their conformable extensions in crisp and interval environments.

Formula omitted]-Laplace transforms and [formula omitted]-differential equations of the arbitrary-order, theory and applications

This paper provides a framework to study a class of arbitrary-order uncertain differential equations known as arbitrary-order Z+-differential equations. For this purpose, we first present the parametric form of Z+-numbers. Then, we introduce the basic algebraic operations on Z+-numbers, including addition, scalar multiplication, and Hukuhara difference. Definitely, these operations lead us to define the Z+-valued function. Afterward, the limit and continuity concepts of a Z+-valued function are provided under the definition of a metric on the space of Z+-numbers. Furthermore, the concepts of Z+-differentiability, Z+-integral, and Z+-Laplace transform with the convergence theorem for the Z+-valued function and its nth-order derivatives are introduced in detail. Considering all these concepts, a Z+-differential equation (Z+DE) can be expressed in the form of a bimodal differential equation combining a fuzzy initial value problem (FIVP) and a random differential equation (RDE). To this end, we use a combination of “FIVP under strongly generalized Hukuhara differentiability (SGH-differentiability)” and “random differential equation under mean-square differentiability (ms-differentiability)” to define the nth-order differential equations with Z+-number initial values. Further, the existence and uniqueness of the Z+-differential equations are examined by presenting several theorems. Finally, the effectiveness of the approaches is illustrated by solving two examples.

The relationship of three difference operations for fuzzy numbers to three kinds of derivative

The difference operation for fuzzy number is an essential concept for the fuzzy set theory. There are several differences proposed: generalized difference, generalized Hukuhara difference and granule difference. Based on these differences, generalized differentiability, generalized Hukuhara differentiability and granule differentiability are also proposed, respectively. In this paper, the relations among these three kinds of differences and that of related three kinds of differentiability are clarified.

Necessary and sufficient conditions for interval‐valued differentiability

This paper presents necessary and sufficient conditions for generalized Hukuhara differentiability of interval‐valued functions and counterexamples of some equivalences previously presented in the literature, for which important results are based on. Moreover, applications of interval generalized Hukuhara differentiability are presented.