Fuzzy differential equation

Fuzzy differential equation are general concept of ordinary differential equation in mathematics defined as differential inclusion for non-uniform upper hemicontinuity convex set with compactness in fuzzy set. [math]\displaystyle{ dx(t)/dt= F(t,x(t),\alpha), }[/math] for all [math]\displaystyle{ \alpha \in [0,1] }[/math].

First order fuzzy differential equation

A first order fuzzy differential equation with real constant or variable coefficients

[math]\displaystyle{ x'(t) + p(t) x(t) = f(t) }[/math]

where [math]\displaystyle{ p(t) }[/math] is a real continuous function and [math]\displaystyle{ f(t) \colon [t_0 , \infty) \rightarrow R_F }[/math] is a fuzzy continuous function [math]\displaystyle{ y(t_0) = y_0 }[/math] such that [math]\displaystyle{ y_0 \in R_F }[/math].

Application

It is useful for calculating Newton's law of cooling, compartmental models in epidemiology and multi-compartment model.

Linear systems of fuzzy differential equations

A system of equations of the form

[math]\displaystyle{ x(t)'_n = a_n1(t) x_1(t) + ......+ a_nn(t) x_n(t) + f_n(t) }[/math]where [math]\displaystyle{ a_ij }[/math] are real functions and [math]\displaystyle{ f_i }[/math] are fuzzy functions [math]\displaystyle{ x'_n(t)= \sum_{i=0}^1 a_{ij} x_i. }[/math]

Fuzzy partial differential equations

A fuzzy differential equation with partial differential operator is [math]\displaystyle{ \nabla x(t) = F(t,x(t),\alpha), }[/math]for all [math]\displaystyle{ \alpha \in [0,1] }[/math].

Fuzzy fractional differential equation

A fuzzy differential equation with fractional differential operator is

[math]\displaystyle{ d^n x(t)/{dt}^n= F(t,x(t),\alpha), }[/math] for all [math]\displaystyle{ \alpha \in [0,1] }[/math] where [math]\displaystyle{ n }[/math] is a rational number.