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 $$ y_0 \in R_F $$.

Linear systems of fuzzy differential equations
A system of equations of the form

$$ x(t)'_n = a_n1(t) x_1(t) + ......+ a_nn(t) x_n(t) + f_n(t) $$where $$a_ij$$ 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

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