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

$$x^{\prime }(t)+p(t)x(t)=f(t)$$

where $p(t)$ is a real continuous function and $f(t):[t_0,\mathrm{\infty })\to R_F$ 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

$$x(t){\text{'}}_n=a_{n}1(t)x_1(t)+......+a_{n}n(t)x_n(t)+f_n(t)$$where $a_{i}j$ are real functions and $f_i$ are fuzzy functions $$x{\text{'}}_n(t)=\sum _{i=0}^1{a}_{ij}{x}_i.$$

Fuzzy partial differential equations

A fuzzy differential equation with partial differential operator is $$\mathrm{\nabla }x(t)=F(t,x(t),\alpha ),$$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.