Linear Fuzzy Differential Equations Research Articles

Analysis of Second-Order Linear Fuzzy Differential Equation Under an Innovative Fuzzy Derivative Approach and Its Application

Fuzzy derivatives are a concept based on fuzzy calculus, which extends classical calculus to handle uncertainty and vagueness, habitually represented using fuzzy sets or fuzzy numbers. In fuzzy calculus, the derivative incorporates fuzziness, meaning both the input and output can have imprecise or uncertain values. In this paper, the step size is also considered as a fuzzy number. The impact of fuzziness in step size is studied here for first- and second-order fuzzy derivatives. Under this proposed derivative, differentiation of fuzzy exponential function is estimated. The concept is applied for solving second-order linear fuzzy differential equation. The proposed approaches are applied to spring mass system dynamics in fuzzy environment. Lastly, numerical illustrations on second-order fuzzy differential equations are shown using analytical technique and graphically the results are displayed.

The Solution of the Intuitionistic Fuzzy Linear Fractional Partial Differential Equations Using the Homotopy Analysis Method

The Solution of the Intuitionistic Fuzzy Linear Fractional Partial Differential Equations Using the Homotopy Analysis Method

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

Solutions of periodic boundary value problems for first-order linear fuzzy differential equations under new conditions

The BVP of a class of second order linear fuzzy differential equations is solved by Green function method under the concept of granular differentiability

The BVP of a class of second order linear fuzzy differential equations is solved by Green function method under the concept of granular differentiability

Representation of solutions to fuzzy linear fractional differential equations with a piecewise constant argument

This study marks the first exploration of fuzzy linear fractional differential equations with a piecewise constant argument (FLFDEs-PCA), incorporating the concept of Caputo’s type gH-differentiability with the order α ∈ (0, 1]. Such problems are noteworthy as they represent hybrid systems, blending the characteristics of continuous and discrete dynamical systems and integrating aspects from both differential and difference equations. The primary objective of this research is to establish a standardized framework for deriving explicit solution formulas for FLFDEs-PCA under various scenarios. Additionally, illustrative examples are provided to demonstrate the practical implications of our theoretical findings.

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.

Incommensurate non-homogeneous system of fuzzy linear fractional differential equations using the fuzzy bunch of real functions

This article aims to introduce and investigate the analytical fuzzy solution of the incommensurate non-homogeneous system of fuzzy linear fractional differential equations (INS-FLFDEs) using trivariate Mittag-Leffler functions. Entries of the coefficient matrix of the given system are treated as real numbers, initial-values are triangular fuzzy numbers (TFNs), and the forcing function is a fuzzy set (or a bunch) of real function. We extract the potential solution in the form of a fuzzy bunch of real functions (FBoRFs) rather than the solution of fuzzy-valued functions. We formulate the fuzzy initial value problem as a set of classical initial value problems by taking the forcing function from the class of FBoRFs and the initial value from the collection of TFNs (as a special case). The solution of this system is in the form of a trivariate Mittag-Leffler function. We interpret this solution as an element of the fuzzy solution set and assign the minimum value of membership that takes from the forcing function and the initial value in the fuzzy set. The originality of the proposed technique is that the uncertainty is smaller compared to the uncertainty extracted from other techniques. In addition, generalized derivatives increase the order and dimension of the system. Therefore, the proposed technique is better in terms of complexity because it reduces the order and dimension of the system. Finally, to grasp the proposed technique, we solve the electrical network and multiple mass-spring systems as applications and analyze their graphs to visualize and support theoretical results.

System of linear fuzzy differential equations in the space of vector-valued linearly correlated fuzzy numbers

The present paper is concerned with the solution of system of linear fuzzy differential equations where the unknown function is restricted to a vector-valued linearly correlated fuzzy function. First, the concept of the space of vector-valued linearly correlated fuzzy numbers is proposed as an extension to the space of linearly correlated fuzzy numbers. Meantime, the LC-difference and its related properties are also introduced and proved for vector-valued linearly correlated fuzzy numbers, respectively. Second, some analytical properties of vector-valued linearly correlated fuzzy functions such as limit, continuity, and LC-differentiability are examined. Finally, the existence conditions and solutions of the linear fuzzy system are further studied in a special space of vector-valued linearly correlated fuzzy numbers. Some examples are provided to demonstrate the proposed method.

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 Unsteady-State Drainage Solution for Land Reclamation

Very well-drained lands could have a positive impact in various soil health indicators such as soil erosion and soil texture. A drainage system is responsible for properly aerated soil. Until today, in order to design a drainage system, a big challenge remained to find the subsurface drain spacing because many of the soil and hydraulic parameters present significant uncertainties. This fact also creates uncertainties to the overall physical problem solution, which, if not included in the preliminary design studies and calculations, could have bad consequences for the cultivated lands and soils. Finding the drain spacing requires the knowledge of the unsteady groundwater movement, which is described by the linear Boussinesq equation (Glover-Dumm equation). In this paper, the Adomian solution to the second order unsteady linear fuzzy partial differential one-dimensional Boussinesq equation is presented. The physical problem concerns unsteady drain spacing in a semi-infinite unconfined aquifer. The boundary conditions, with an initially horizontal water table, are considered fuzzy and the overall problem is translated to a system of crisp boundary value problems. Consequently, the crisp problem is solved using an Adomian decomposition method (ADM) and useful practical results are presented. In addition, by application of the possibility theory, the fuzzy results are translated into a crisp space, enabling the decision maker to make correct decisions about both the drain spacing and the future soil health management practices, with a reliable degree of confidence.

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

<abstract> <p>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.</p></abstract>

Necessary and sufficient condition for the existence of solutions to two-point boundary value problem of fuzzy linear multi-term fractional differential equations

Necessary and sufficient condition for the existence of solutions to two-point boundary value problem of fuzzy linear multi-term fractional differential equations

Using Fuzzy Aboodh Transform to Solve A System of First Order Fuzzy Linear Differential Equations

In this paper, fuzzy Aboodh transform used to solve system of fuzzy linear differential equations of first order in dimension two. Moreover, support work with an applied example in one of the institution.

Solving the BVP to a class of second-order linear fuzzy differential equations under granular differentiability concept1

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.

USER-ORIENTED INTELLIGENCE MINING UNDER THE EXISTENCE OF SOLUTIONS TO INTEGRAL BOUNDARY VALUE PROBLEMS FOR FUZZY PARTIAL FRACTIONAL DIFFERENTIAL EQUATIONS

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

Representation of solutions to fuzzy linear conformable differential equations

In this paper, we study fuzzy linear conformable differential equations using the generalized fuzzy conformable fractional differentiability concept. We give an explicit representation of q(1)- differentiable and q(2)-differentiable solutions for appropriate differential equations. Finally, we give some examples to illustrate our theoretical results.

Numerical Solution of Boundary Value Problems for Second Order Fuzzy Linear Differential Equations

In this paper, the numerical solution of the boundary value problem for second order fuzzy linear differential equations is discussed. We consider the fuzzy difference equation to replace the fuzzy differential equation. The numerical solution of the boundary value problem is obtained by calculating the fuzzy difference equation. Finally, an example is given, to verify the effectiveness of this method.

Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

In this paper, we provide the generalization of two predefined concepts under the name fuzzy conformable differential equations. We solve the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order $\Psi$, we use two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order $\Psi$ by a variation of the constant formula. Moreover, we generalize our results to solve fuzzy conformable ordinary differential equations of a higher order. Further, we provide some examples in each section for the sake of demonstration of our results.

Solution of Linear Fuzzy Fractional Differential Equations Using Fuzzy Natural Transform

The purpose of this paper is to apply the fuzzy natural transform (FNT) for solving linear fuzzy fractional ordinary differential equations (FFODEs) involving fuzzy Caputo’s H-difference with Mittag-Leffler laws. It is followed by proposing new results on the property of FNT for fuzzy Caputo’s H-difference. An algorithm was then applied to find the solutions of linear FFODEs as fuzzy real functions. More specifically, we first obtained four forms of solutions when the FFODEs is of order α∈(0,1], then eight systems of solutions when the FFODEs is of order α∈(1,2] and finally, all of these solutions are plotted using MATLAB. In fact, the proposed approach is an effective and practical to solve a wide range of fractional models.

Fuzzy linear singular differential equations under granular differentiability concept

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.