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.

$$dx(t)/dt=F(t,x(t),\alpha ),$$

for all $\alpha \in [0,1]$.

First order fuzzy differential equation

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

$$x'(t)+p(t)x(t)=f(t)$$

where $p(t)$ is a real continuous function and $f(t)\colon [t_{0},\infty )\rightarrow R_{F}$ is a fuzzy continuous function

$$y(t_{0})=y_{0}$$

such that $y_{0}\in R_{F}$.

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

$$x(t)'_{n}=a_{n}1(t)x_{1}(t)+......+a_{n}n(t)x_{n}(t)+f_{n}(t)$$

where $a_{i}j$ are real functions and $f_{i}$ are fuzzy functions

$$x'_{n}(t)=\sum _{i=0}^{1}a_{ij}x_{i}.$$

Fuzzy partial differential equations

A fuzzy differential equation with partial differential operator is

$$\nabla x(t)=F(t,x(t),\alpha ),$$

for all $\alpha \in [0,1]$.

Fuzzy fractional differential equation

A fuzzy differential equation with fractional differential operator is

$$d^{n}x(t)/{dt}^{n}=F(t,x(t),\alpha ),$$

for all $\alpha \in [0,1]$ where $n$ is a rational number.