Properties of Reflected Liu Process at the Origin

Uncertain delay differential equation is a class of functional differential equations driven by Liu process. It is an important model to describe the evolution process of uncertain dynamical system. In this paper, on the one hand, the analytic expression of a class of linear uncertain delay differential equations are investigated. On the other hand, the new sufficient conditions for uncertain delay differential equations being stable in measure and in mean are presented by using retarded-type Gronwall inequality. Several examples show that our stability conditions are superior to the existing results.

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Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

Fractal and Fractional ◽

10.3390/fractalfract5030112 ◽

2021 ◽

Vol 5 (3) ◽

pp. 112

Author(s):

Azmat Ullah Khan Niazi ◽

Naveed Iqbal ◽

Rasool Shah ◽

Fongchan Wannalookkhee ◽

Kamsing Nonlaopon

Keyword(s):

Differential Equations ◽

Evolution Equations ◽

Numerical Solutions ◽

Initial Conditions ◽

Exact Controllability ◽

Fractional Evolution Equations ◽

Liu Process ◽

Bounded Space ◽

Fractional Differential ◽

Derivatives Of

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order β∈(1,2), 0CDtβu(t,ζ)=Au(t,ζ)+f(t,u(t,ζ))dCt+Bx(t)Cx(t)dt with initial conditions u(0)=u0,u′(0)=u1, where u(t,ζ) takes values from U(⊂EN),V(⊂EN) is the other bounded space, and EN represents the set of all upper semi-continuously convex fuzzy numbers on R. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

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Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201428 ◽

2020 ◽

pp. 1-10

Author(s):

Ziqiang Lu ◽

Yuanguo Zhu ◽

Jiayu Shen

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Fractional Differential Equation ◽

Fractional Differential Equations ◽

Continuous Dependence ◽

Liu Process ◽

Uncertain Dynamic Systems ◽

Caputo’S Derivative ◽

Fractional Differential ◽

Lipschitz Conditions

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

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Least squares estimation of high-order uncertain differential equations

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-202522 ◽

2021 ◽

pp. 1-10

Author(s):

Jing Zhang ◽

Yuhong Sheng ◽

Xiaoli Wang

Keyword(s):

Differential Equations ◽

Least Squares ◽

Estimation Method ◽

Sum Of Squares ◽

High Order ◽

Least Squares Estimation ◽

Uncertain Differential Equation ◽

Diffusion Term ◽

First Order ◽

Liu Process

Parameter estimation of high-order uncertain differential equations is an inevitable problem in practice. In this paper, the equivalent equations of high-order uncertain differential equations are obtained by transformation, and the parameters of the first-order uncertain differential equation including Liu process are estimated. Based on the least squares estimation method, this paper proposes a means to minimize the residual sum of squares to obtain an estimate of the parameters in the drift term, and make the noise sum of squares equal to the residual sum of squares to obtain an estimate of the parameters in the diffusion term. In addition, some numerical examples are given to illustrate the proposed method. Finally, applications of the high-order uncertain spring vibration equations verify the viability of our method.

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Complex uncertain differential equations with application to time integral

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-211030 ◽

2021 ◽

pp. 1-15

Author(s):

Zhifu Jia ◽

Xinsheng Liu

Keyword(s):

Differential Equations ◽

Uniqueness Theorem ◽

Existence And Uniqueness ◽

Uncertainty Theory ◽

Numerical Algorithms ◽

Existence And Uniqueness Theorem ◽

Uncertainty Distribution ◽

Time Integral ◽

Liu Process ◽

Homogeneous Linear

In this paper, we propose complex uncertain differential equations (CUDEs) based on uncertainty theory. In order to describe the evolution of complex uncertain phenomenon related to belief degrees, we apply the complex Liu process to CUDEs. Firstly, we pose a concept of a linear CUDE and prove that homogeneous linear CUDE and general linear CUDE have solutions. Then, we prove existence and uniqueness theorem of a special CUDE. Further, we design a numerical algorithm to obtain inverse uncertainty distribution of the solution. Finally, as an application, we analyse the inverse uncertainty distributions of time integral of CUDEs and design numerical algorithms to obtain inverse uncertainty distributions of time integral.

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Stability in measure for uncertain delay differential equations based on new Lipschitz conditions

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-210089 ◽

2021 ◽

pp. 1-13

Author(s):

Yin Gao ◽

Lifen Jia

Keyword(s):

Control System ◽

Differential Equations ◽

Lipschitz Condition ◽

Delay Time ◽

Delay Differential Equations ◽

Special Class ◽

Liu Process ◽

The Stability ◽

Delay Differential ◽

Lipschitz Conditions

Uncertain delay differential equations (UDDEs) charactered by Liu process can be employed to model an uncertain control system with a delay time. The stability of its solution always be a significant matter. At present, the stability in measure for UDDEs has been proposed and investigated based on the strong Lipschitz condition. In reality, the strong Lipschitz condition is so strictly and hardly applied to judge the stability in measure for UDDEs. For the sake of solving the above issue, the stability in measure based on new Lipschitz condition as a larger scale of applications is verified in this paper. In other words, if it satisfies the strong Lipschitz condition, it must satisfy the new Lipschitz conditions. Conversely, it may not be established. An example is provided to show that it is stable in measure based on the new Lipschitz conditions, but it becomes invalid based on the strong Lipschitz condition. Moreover, a special class of UDDEs is verified to be stable in measure without any limited condition. Besides, some examples are investigated in this paper.

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Runge-Kutta Nystrom Method of Order Three for Solving Fuzzy Differential Equations

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2010-0011 ◽

2010 ◽

Vol 10 (2) ◽

pp. 195-203 ◽

Cited By ~ 3

Author(s):

K. Kanagarajan ◽

M. Sambath

Keyword(s):

Differential Equations ◽

Numerical Method ◽

Numerical Algorithm ◽

Nyström Method ◽

Nystrom Method ◽

Fuzzy Process ◽

Runge Kutta

AbstractIn this paper we present a numerical algorithm for solving fuzzy differential equations based on Seikkala’s derivative of a fuzzy process. We discuss in detail a numerical method based on a Runge-Kutta Nystrom method of order three. The algorithm is illustrated by solving some fuzzy differential equations.

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