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The stability analysis is one of the basic problems in the field of system, prey–predator, production inventory control, signal processing etc. The goal of stability analysis for prey–predator and production inventory control model is to determine the region of the different variables and parameters at which the system is stable or unstable. This paper explores an uncertain prey–predator model where both prey and predator populations are fuzzy variables. Also we have considered all the prey–predator parameters of the system are imprecise. This fuzzy prey–predator model is converted to an equivalent crisp model using granular differentiability approach. The condition of local stability is obtained mathematically by analysis the eigenvalues of characteristic equation. And also we introduced a production inventory model as a case study of fuzzy dynamical system to illustrate the efficiency of the result and practical aspects. [ABSTRACT FROM AUTHOR]

This paper considers fuzzy boundary value problems associated with second-order fuzzy differential equations under granular differentiability. Using a horizontal membership function, we present the notion of second-order granular differentiability for fuzzy functions. Using the granular differentiability concept, we interpret two kinds of two-point boundary value problems for second-order fuzzy differential equations. Sufficient conditions are established for the existence and uniqueness of solutions to these fuzzy boundary value problems. An algorithm is presented for solving non-linear fuzzy boundary value problems under granular differentiability. We provide one example and two engineering applications to demonstrate the algorithm’s effectiveness and results.

In this paper, the second-order fuzzy homogeneous differential equation is transformed into a more special simplest form under the condition that the solution of the boundary value problem of the equation exists and is unique. Then the eigenvalues of the boundary value problem of the second-order simplest fuzzy homogeneous differential equation are studied and the theorems that make the eigenvalues exist are proposed and then illustrated with examples. Finally, it is proved that when the second-order fuzzy coefficient p ˜ (t) in the second-order fuzzy homogeneous differential equation is a fuzzy number, the solution set of its corresponding second-order granular homogeneous differential equation becomes larger, that is, the solution set of fuzzy differential equations with real numbers is a subset of the solution set with fuzzy coefficients as fuzzy numbers. [ABSTRACT FROM AUTHOR]

In this paper, the second-order fuzzy homogeneous differential equation is transformed into a more special simplest form under the condition that the solution of the boundary value problem of the equation exists and is unique. Then the eigenvalues of the boundary value problem of the second-order simplest fuzzy homogeneous differential equation are studied and the theorems that make the eigenvalues exist are proposed and then illustrated with examples. Finally, it is proved that when the second-order fuzzy coefficient p ˜ (t) in the second-order fuzzy homogeneous differential equation is a fuzzy number, the solution set of its corresponding second-order granular homogeneous differential equation becomes larger, that is, the solution set of fuzzy differential equations with real numbers is a subset of the solution set with fuzzy coefficients as fuzzy numbers. [ABSTRACT FROM AUTHOR]

In this paper, under the condition of granular differentiation, we consider the fuzzy optimization problems with the general fuzzy function as the objective function. Firstly, we introduce the concept of granular convexity, and propose the properties of the granular convex fuzzy functions. Secondly, we present the Karush-Kuhn-Tucker type optimality conditions of the fuzzy relative optimal solution of more general fuzzy programming problems and some test examples. Finally, the relationships between a class of variational inequalities and the fuzzy optimization problems are established. [ABSTRACT FROM AUTHOR]

In this research, we investigate a novel class of granular type optimality guidelines for the fuzzy multi-objective optimizations based on guidelines of vector granular convexity and granular differentiability. Firstly, the concepts of vector granular convexity is introduced to the vector fuzzy-valued function. Secondly, several properties of vector granular convex fuzzy-valued functions are provided. Thirdly, the granular type Karush-Kuhn-Tucker(KKT) optimality guidelines are derived for the fuzzy multi-objective optimizations. [ABSTRACT FROM AUTHOR]

Based on the granular derivative and the horizontal membership function of fuzzy-number-valued-function, the existence and uniqueness of solutions of two-point boundary value problem (BVP) for a class of second-order linear fuzzy ordinary differential equations are given, including the homogeneous BVP, the semi-homogeneous BVP and the non-homogeneous BVP. However, this is somewhat different from ordinary differential equations. In fact, we can think of it simply as the transformation from a number to a set, or we can think of it macroscopically as the transformation from a point to a plane. At the same time, appropriate examples are given to illustrate this conclusion. [ABSTRACT FROM AUTHOR]

Based on the granular derivative and the horizontal membership function of fuzzy-number-valued-function, the existence and uniqueness of solutions of two-point boundary value problem (BVP) for a class of second-order linear fuzzy ordinary differential equations are given, including the homogeneous BVP, the semi-homogeneous BVP and the non-homogeneous BVP. However, this is somewhat different from ordinary differential equations. In fact, we can think of it simply as the transformation from a number to a set, or we can think of it macroscopically as the transformation from a point to a plane. At the same time, appropriate examples are given to illustrate this conclusion. [ABSTRACT FROM AUTHOR]

In this paper, fuzzy linear singular differential equations under so-called granular differentiability are investigated in which the coefficients and initial conditions are as fuzzy numbers. Some new notions such as fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix are introduced. Moreover, we compare the proposed method with other ones based on other derivatives, using some examples. [ABSTRACT FROM AUTHOR]

Since the term “Fuzzy differential equations” (FDEs) emerged in the literature in 1978, prevailing research effort has been dedicated not only to the development of the concepts concerning the topic, but also to its potential applications. This paper presents a chronological survey on fuzzy differential equations of integer and fractional orders. Attention is concentrated on the FDEs in which a definition of fuzzy derivative of a fuzzy number-valued function has been taken into account. The chronological rationale behind considering FDEs under each concept of fuzzy derivative is highlighted. The pros and cons of each approach dealing with FDEs are also discussed. Moreover, some of the proposed FDEs applications and methods for solving them are investigated. Finally, some of the future perspectives and challenges of fuzzy differential equations are discussed based on our personal view point.

In this paper, a new type of equation namely fuzzy fractional stochastic Pantograph delay differential system (FSPDDS) is proposed. In our previous work, a first extension of fuzzy stochastic differential system into fuzzy fractional stochastic differential system by using Granular differentiability has been established. Here we study the existence and uniqueness results for the fuzzy FSPDDS which are obtained by using generalized Granular differentiability and contraction principle with weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the fuzzy variational problem and fuzzy optimal control problem are considered. The granular Euler–Lagrange condition for the fuzzy variational problem and necessary conditions of Pontryagin type for fixed and free final state fuzzy optimal control problem are derived based on the concepts of horizontal membership function (HMF) and granular differentiability with the calculus of variations. Further, based on the proposed solution method, the solutions of fuzzy optimal control problem, i.e., optimal fuzzy control, and corresponding optimal fuzzy state are always fuzzy functions. Finally, the proposed algorithm used to summarize the main steps of solving the fuzzy variational problem and fuzzy optimal control problem numerically using He's variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

In this research, we investigate a novel class of granular type optimality guidelines for the fuzzy multi-objective optimizations based on guidelines of vector granular convexity and granular differentiability. Firstly, the concepts of vector granular convexity is introduced to the vector fuzzy-valued function. Secondly, several properties of vector granular convex fuzzy-valued functions are provided. Thirdly, the granular type Karush-Kuhn-Tucker(KKT) optimality guidelines are derived for the fuzzy multi-objective optimizations.

The Black–Scholes option pricing model is one of the most significant achievements in modern investment science. However, many factors are constantly fluctuating in the actual financial market option pricing, such as risk-free interest rate, stock price, option underlying price, and security price volatility may be inaccurate in the real world. Therefore, it is of great practical significance to study the fractional fuzzy option pricing model. In this paper, we proposed a reliable approximation method, the Elzaki transform homotopy perturbation method (ETHPM) based on granular differentiability, to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Firstly, the fuzzy function is converted to a real number function based on the horizontal membership function (HMF). Secondly, the specific steps of the ETHPM are given to solve the fuzzy time-fractional Black–Scholes European option pricing equations. Finally, some examples demonstrate that the new approach is simple, efficient, and accurate. In addition, the fuzzy approximation solutions have been visualized at the end of this paper.

In this paper, we have created a single-period fuzzy production inventory control model on the finite time horizon. A new nonlinear demand function has been introduced, which depends on the stock, selling price and product quality. The realistic reasons come from logistical function and entry into the initial demand for the product and the reliance on uncertain advertising rates, uncertain stock rates, uncertain selling prices and uncertain product quality. In order to control and test the stability, the model needs to be defuzzified. The concept of granular differentiability has been applied to defuzzification. Granular differentiability is the new definition of fuzzy derivatives based on the function of horizontal membership. For the first time in this paper, we have used the granular differentiation method in production inventory systems. We analyzed vaguely optimized controls in terms of granular differentiation analytically and numerically. [ABSTRACT FROM AUTHOR]

Stochastic differential equations represent equipment in modeling of a dynamic systems operating with fuzzy settings driven by stochastic noise. In this manuscript, a new kind of equation, namely fuzzy fractional stochastic differential system (FSDS) is proposed. It is defined by stochastic integral of a fuzzy process with respect to the m-dimensional Brownian motion. In this paper, the existence and uniqueness results are obtained for impulsive fuzzy FSDS with granular derivative using contraction principle. Finally, we provide a numerical example for the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

The mathematical models are usually considered for the assessment of a dynamical system performance. Using such models often is based on an assumption which takes all the system parameter values as crisp values into account. However, the parameter values may not be always determined exactly. Since fuzzy logic has been well known as an effective tool in modeling uncertainty, in this paper, fuzzy numbers are used in modeling of dynamical systems. In this paper, concentrate on second-order dynamical systems with fuzzy parameters as they play a major role in many aspects of control theory. New performance definitions such as fuzzy rise time, fuzzy settling time, and fuzzy maximum overshoot are presented. The new definitions for such systems provide a more comprehensive view of the system output which helps for designing a better control. Also, the solutions of an RLC circuit and intelligent transformer with fuzzy parameter based on these definitions are considered. [ABSTRACT FROM AUTHOR]

In this paper, a class of uncertain linear dynamical systems called fuzzy fractional linear dynamical systems is investigated. The aim is to find control inputs to keep the states of the fuzzy fractional dynamical systems near the zero in an optimal manner. The optimality criterion is in a form of a granular fuzzy integral whose integrand is a quadratic function of the state variables and control inputs. The fuzzy fractional dynamical system is described using fuzzy fractional differential equations (FFDEs). In order to achieve the aim, an effective approach for solving FFDEs should be at disposal. Due to some restrictions imposed by the previous approaches dealing with FFDEs, a new approach is proposed. The proposed approach is based on the granular derivative and the so-called relative-distance-measure fuzzy interval arithmetic. New definitions of fuzzy fractional derivatives and integral called left and right granular Riemann–Liouville fuzzy fractional derivatives, left and right granular Caputo fuzzy fractional derivatives, and the left and right granular fuzzy fractional integral are also presented. In addition, the concepts of granular fuzzy partial derivative and granular fuzzy chain rule are introduced. By the approximations of the granular fuzzy fractional integral and the granular Caputo fuzzy fractional derivative, the approximation solution to the FFDEs is obtained. Consequently, based on the new concepts and theorems, the solution to the fuzzy fractional quadratic regulator problem is given by a theorem. This paper closes with an example of regulating the motion of Boeing 747 in longitudinal direction with the presence of uncertainty in the initial conditions and the coefficients. [ABSTRACT FROM AUTHOR]

This paper deals with sub-optimal control of a fuzzy linear dynamical system. The aim is to keep the state variables of the fuzzy linear dynamical system close to zero in an optimal manner. In the fuzzy dynamical system, the fuzzy derivative is considered as the granular derivative; and all the coefficients and initial conditions can be uncertain. The criterion for assessing the optimality is regarded as a granular integral whose integrand is a quadratic function of the state variables and control inputs. Using the relative-distance-measure (RDM) fuzzy interval arithmetic and calculus of variations, the optimal control law is presented as the fuzzy state variables feedback. Since the optimal feedback gains are obtained as fuzzy functions, they need to be defuzzified. This will result in the sub-optimal control law. This paper also sheds light on the restrictions imposed by the approaches which are based on fuzzy standard interval arithmetic (FSIA), and use strongly generalized Hukuhara and generalized Hukuhara differentiability concepts for obtaining the optimal control law. The granular eigenvalues notion is also defined. Using an RLC circuit mathematical model, it is shown that, due to their unnatural behavior in the modeling phenomenon, the FSIA-based approaches may obtain some eigenvalues sets that might be different from the inherent eigenvalues set of the fuzzy dynamical system. This is, however, not the case with the approach proposed in this study. The notions of granular controllability and granular stabilizability of the fuzzy linear dynamical system are also presented in this paper. Moreover, a sub-optimal control for regulating a Boeing 747 in longitudinal direction with uncertain initial conditions and parameters is gained. In addition, an uncertain suspension system of one of the four wheels of a bus is regulated using the sub-optimal control introduced in this paper. [ABSTRACT FROM AUTHOR]

In this study, we present a general formulation for the optimal control problem to a class of fuzzy fractional differential systems relating to SIR and SEIR epidemic models. In particular, we investigate these epidemic models in the uncertain environment of fuzzy numbers with the rate of change expressed by granular Caputo fuzzy fractional derivatives of order β  ∈ (0, 1]. Firstly, the existence and uniqueness of solution to the abstract fractional differential systems with fuzzy parameters and initial data are proved. Next, the optimal control problem for this fractional system is proposed and a necessary condition for the optimality is obtained. Finally, some examples of the fractional SIR and SEIR models are presented and tested with real data extracted from COVID-19 pandemic in Italy and South Korea., (© 2020 Elsevier B.V. All rights reserved.)

In this paper, we consider a new setting of fuzzy delay differential equations by employing the concepts of granular differentiability. Firstly, we establish some basic properties of complete granular metric space. Then we study the local and global existence and uniqueness of integral fuzzy solution. For the first task, we use successive approximation technique combined with Gronwall's lemma. The second task will be solved by Banach contraction principle in weighted metric space defined by granular distance. Additionally, we demonstrate the theoretical results by solving some real-world models in both analytical method and numerical method. The first model is often referred to the exponential growth, namely time-delay growth Malthusian model. The second model is one type of dynamical systems, which is used for studies involving large amounts of tissues, called Ehrlich ascites tumor model with delay. Furthermore, we introduce the Mackey–Glass model which is considered as a useful tool for times series prediction in the neutral network and fuzzy logic fields. Finally, we represent the surface of fuzzy solution for visible identification mark-on. [ABSTRACT FROM AUTHOR]