Linear Fuzzy Differential Equations Research Articles

Analytical and numerical solutions of fuzzy differential equations

In this paper, we study analytical and numerical solutions of fuzzy differential equations based on the extension principle. For linear fuzzy differential equations, we state some results on the behaviour of the solutions and study their relationship with the generalised Hukuhara derivative. In order to approximate the solutions of linear and non-linear fuzzy differential equations, we propose a new fuzzification of the classical Euler method and then incorporate an unconstrained optimisation technique. This combination offers a powerful tool to tackle uncertainty in any numerical method. An efficient computational algorithm is also provided to guarantee the convexity of fuzzy solutions on the time domain. Several illustrative examples are given.

Numerical Solution of n-th Order Fuzzy Linear Differential Equations by Homotopy Perturbation Method

This paper targets to investigate the numerical solution of n th order fuzzy differential equations with fuzzy environment using Homotopy Perturbation Method (HPM). Triangular fuzzy convex normalized sets are used for the fuzzy parameter and variables. Obtained results are compared with the existing solution depicted in term of plots to show the efficiency of the applied method.

Approximate Solution ofnth-Order Fuzzy Linear Differential Equations

The approximate solution ofnth-order fuzzy linear differential equations in which coefficient functions maintain the sign is investigated by the undetermined fuzzy coefficients method. The differential equations is converted to a crisp function system of linear equations according to the operations of fuzzy numbers. The fuzzy approximate solution of the fuzzy linear differential equation is obtained by solving the crisp linear equations. Some numerical examples are given to illustrate the proposed method. It is an extension of Allahviranloo's results.

On Solutions of Linear Fractional Differential Equations with Uncertainty

The solutions of linear fuzzy fractional differential equations (FFDEs) under the Caputo differentiability have been investigated. To this end, the fuzzy Laplace transform was used to obtain the solutions of FFDEs. Then, some new results regarding the relation between some types of differentiability have been obtained. Finally, some applicable examples are solved in order to show the ability of the proposed method.

Numerical solutions of fuzzy linear system differential equations and application of a Radioactivity decay model

In this paper, homotopy analysis method is presented for linear system of first-order fuzzy differential equations which involves a radioactivity decay model. We give the illustrative example to demonstrate the efficiency of the method. Finally, homotopy analysis method (HAM) is applied to obtain series solutions to the the radioactivity decay model.

On Boundary Value Problems for Second-order Fuzzy Linear Differential Equations with Constant Coefficients

In this paper we investigate the solutions of boundary value problems for second-order fuzzy linear differential equations with constant coefficients. There are four different solutions for the problems by using a generalized differentiability. Solutions and several comparison results are presented. Some examples are provided for which the solutions are found.

On the existence of solutions to periodic boundary value problems for fuzzy linear differential equations

In this work, we provide sufficient conditions which guarantee the existence of solutions to periodic boundary value problems for first-order linear fuzzy differential equations by using generalized differentiability and switching points. In comparison with some previous works, we consider equations whose coefficient may change its sign a finite number of times in the interval of interest. We also study the existence of solutions which are crisp (or real) at the switching points where the diameter of the level sets changes from nonincreasing to nondecreasing character.

Periodic boundary value problems for first-order linear differential equations with uncertainty under generalized differentiability

We study the existence of solutions to a class of first-order linear fuzzy differential equations subject to periodic boundary conditions from the point of view of generalized differentiability. The objective of this paper is to show that fuzzy differential equations under generalized differentiability can be used in the study of periodic phenomena, by considering a combination of two types of derivatives with a switching point. We provide sufficient conditions which guarantee that the piecewise-defined solutions match adequately and illustrate, through some examples, the process of construction and calculation of solutions.

A new method to solve time-dependent intuitionistic fuzzy differential equations and its application to analyze the intuitionistic fuzzy reliability of industrial systems

To the best of our knowledge, there is no method in the literature for solving such differential equations in which all the parameters except independent variables are represented by intuitionistic fuzzy numbers. In this article, a new method is proposed for solving such nth-order time-dependent intuitionistic fuzzy linear differential equations. To show the application of the proposed method in real-life problems, the time-dependent intuitionistic fuzzy Kolmogorov’s differential equations, obtained by Markov model of condensate system, are solved by the proposed method, and the obtained solution is used to analyze the intuitionistic fuzzy reliability of condensate system.

Minimal solution of fuzzy linear system of differential equations

In this paper, a new approach to solve the fuzzy linear system of differential equations based on pseudo-inverse is presented. In this work, we discuss the minimal solution of a system of linear fuzzy differential equations such as $$A\dot{x}(t)=B\dot{x}(t)+Cx(t), x(0)=x_0$$ where A, B, and C are three real m × n matrices and the initial condition x 0 is described by a vector made up of n fuzzy numbers. In this paper, we investigated a necessary and sufficient conditions for the existence fuzzy derivative $$\dot{x}(t)$$ of a fuzzy process x(t) and a necessary and sufficient conditions for the minimal solution vector to be a fuzzy vector, given arbitrary input fuzzy vector x 0.

A Note on Fuzzy Initial Value Problem for N th-Order Fuzzy Linear Differential Equations

Buckley and Feuring [1] solved a fuzzy linear differential equation and on the basis of the obtained solution, they introduced a new result. The main aim of this note is to show that the solution of the fuzzy linear differential equation, obtained by Buckley and Feuring [1], is incorrect and due to which the result, introduced by Buckley and Feuring [1], is also incorrect.

Fuzzy Modeling and Control of HIV Infection

The present study proposes a fuzzy mathematical model of HIV infection consisting of a linear fuzzy differential equations (FDEs) system describing the ambiguous immune cells level and the viral load which are due to the intrinsic fuzziness of the immune system's strength in HIV-infected patients. The immune cells in question are considered CD4+ T-cells and cytotoxic T-lymphocytes (CTLs). The dynamic behavior of the immune cells level and the viral load within the three groups of patients with weak, moderate, and strong immune systems are analyzed and compared. Moreover, the approximate explicit solutions of the proposed model are derived using a fitting-based method. In particular, a fuzzy control function indicating the drug dosage is incorporated into the proposed model and a fuzzy optimal control problem (FOCP) minimizing both the viral load and the drug costs is constructed. An optimality condition is achieved as a fuzzy boundary value problem (FBVP). In addition, the optimal fuzzy control function is completely characterized and a numerical solution for the optimality system is computed.

Solution of the first order linear fuzzy differential equations by some reliable methods

Fuzzy differential equations are used in modeling problems in science and engineering. For instance, it is known that the knowledge of dynamical systems modeled by ordinary differential equations is often incomplete or vague. While, fuzzy differential equations represent a proper way to model dynamical systems under uncertainty and vagueness. In this paper, two methods for solving first order linear fuzzy differential equations under generalized differentiability are proposed and compared. These methods are variational iteration method (VIM) and Adomian decomposition method (ADM). The comparison of the exact solutions with solutions obtained by VIM and ADM are in details. The comparison shows that solutions are excellent agreement.

The variational iteration method for solving n-th order fuzzy differential equations

Abstract The variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

Existence of global solutions to nonlinear fuzzy Volterra integro-differential equations

In this paper, we introduce new solutions for fuzzy differential equations as mixed solutions, and prove the existence and uniqueness of global solutions for fuzzy initial value problems involving integro-differential operators of Volterra type. One example is also given by applying mixed solution concept to fuzzy linear differential equations for obtaining their global solutions.

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green's function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.

The Averaging of Control Linear Fuzzy Differential Equations

The Averaging of Control Linear Fuzzy Differential Equations

Variation of constant formula for first order fuzzy differential equations

In this paper, we study first order linear fuzzy differential equations by using the generalized differentiability concept and we present the general form of their solutions. We also correct and complete some previous results. Finally, some examples are given to illustrate our results.

A new method for solving fuzzy linear differential equations

In this paper, a novel operator method is proposed for solving fuzzy linear differential equations under the assumption of strongly generalized differentiability. To this end, the equivalent integral form of the original problem is obtained then by using its lower and upper functions the solutions in the parametric forms are determined. The proposed method is illustrated with numerical examples.

Numerical Methods for Fuzzy Linear Partial Differential Equations under new Definition for Derivative

In this paper difference methods to solve partial differential (FPDE) such as fuzzy hyperbolic and fuzzy parabolic equations are considered. The existence of the solution and stability of the method are examined in detail. Finally examples are presented to show that the Hausdorff distance between the exact solution and approximate solution tends to zero.