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.^[1][2][3] $${\displaystyle dx(t)/dt=F(t,x(t),\alpha ),}$$ for all ${\displaystyle \alpha \in [0,1]}$.

A first order fuzzy differential equation^[4] with real constant or variable coefficients

$${\displaystyle x'(t)+p(t)x(t)=f(t)}$$

where ${\displaystyle p(t)}$ is a real continuous function and ${\displaystyle f(t)\colon [t_{0},\infty )\rightarrow R_{F}}$ is a fuzzy continuous function $${\displaystyle y(t_{0})=y_{0}}$$ such that ${\displaystyle y_{0}\in R_{F}}$.

A system of equations of the form

$${\displaystyle x(t)'_{n}=a_{n}1(t)x_{1}(t)+......+a_{n}n(t)x_{n}(t)+f_{n}(t)}$$where ${\displaystyle a_{i}j}$ are real functions and ${\displaystyle f_{i}}$ are fuzzy functions $${\displaystyle x'_{n}(t)=\sum _{i=0}^{1}a_{ij}x_{i}.}$$

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

A fuzzy differential equation with fractional differential operator is

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