Existence and Uniqueness of Mild Solution for Fractional-Order Controlled Fuzzy Evolution Equation

In this article, we investigated the existence and uniqueness of mild solutions for fractional-order controlled fuzzy evolution equations with Caputo derivatives of the controlled fuzzy nonlinear evolution equation of the form 0 c D I γ x I = α x I + P I , x I + A I W I , I ∈ 0 , T , x I 0 = x 0 , in which γ ∈ 0 , 1 , E 1 is the fuzzy metric space and I = 0 , T is a real line interval. With the help of few conditions on functions P : I × E 1 × E 1 ⟶ E 1 , W I is control and it belongs to E 1 , A ∈ F I , L E 1 , and α stands for the highly continuous fuzzy differential equation generator. Finally, a few instances of fuzzy fractional differential equations are shown.

The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations

In many aspects of people's production and life, artistic images have been widely used. Because the image has the function of transmitting information, it can provide necessary space environment information for people. However, there are many problems in the design of stylised art images, and hence the usability of images is affected. Due to its unique advantages, the study of artistic eye movement has gradually become a research hotspot. The fuzzy differential equation is an important branch of differential equation theory, which can be used to study eye movement experiments in the field of the art research. In the process of observation, experiment and maintenance, errors cannot be avoided, and the variables and parameters obtained are often fuzzy, incomplete and inaccurate. And fuzzy differential equations can deal with these uncertainties well. At first, this paper studies the migration-image-study-related theory and art image, based on the study of an artistic image that can be divided into instructions image and symbol and image, with the help of eye movement experiment method to investigate the effects of two types of image on people read mechanism. This research mainly uses the fuzzy differential equation for the visual search experimental paradigm to identify the influence of the difference of the effect.

A New Technique for solving Picture Fuzzy Differential Equation

Picture Fuzzy set (PFS) is an extension of fuzzy set (FS) and intuitionistic fuzzy set (IFS) that can model the uncertainty by integrating the concept of positive, negative and neutral membership degree of an element. In this paper, the solution of Picture Fuzzy ordinary differential equation of first order by means of picture fuzzy number is exemplified and intend to define the picture fuzzy number for (∝, δ, β)-cut. Finally, we illustrate the numerical example for drug distribution in human body for different drug levels is discussed for determining its effectiveness and practicality of the first order differential equation involving picture fuzzy numbers.

Revisiting albedo from a fuzzy perspective

PurposeThe aim of this paper is to revisit the albedo for uncertainty. The albedo is considered as a fuzzy value due to some realistic reasons which they will be discussed in details. After defining an appropriate uncertain albedo by using fuzzy set theory, the related energy balance model is also redefined as a fuzzy differential equation by using the concept of fuzzy derivative.Design/methodology/approachThe well-known Earth energy balance model is redefined as a fuzzy differential equation by using the concept of fuzzy derivative. Thus, instead of an ordinary differential equation, a fuzzy differential equation arises which it's solution procedure will be discussed in details.FindingsResults indicate that the fuzzy uncertainty for albedo causes more real results after solving the fuzzy energy balance equation. Considering albedo as a fuzzy number is more realistic than considering a single certain number for albedo of a surface. This is due to this fact that the Earth's surface coverage is not crisp and the boundaries of different types of lands are not consistent. The proposed approach of this paper can help us to provide more realistic climate models and construct dynamical models which can model the albedo based on its variability.Originality/valueIn this paper, we defined fuzzy energy balance model as a fuzzy differential equation for the first time. We also, considered albedo as a fuzzy number which is another novel approach.

Fuzzy form of Euler Method to Solve Fuzzy Differential Equations

: This paper proposes fuzzy form of Euler method to solve fuzzy initial value problems. By this method, fuzzy differential equations can be solved directly using fuzzy arithmetic. The solution by this method is readily available in a form of fuzzy-valued function. The method does not require to re-write fuzzy differential equation into system of two crisp ordinary differential equations. Algorithm of the method and local error expression are discussed. An illustration and solution of fuzzy Riccati equation are provided for the applicability of the method.