Soft Numerical Computing in Uncertain Dynamic Systems

3.6.1.21: gH-partial differentiability

The fuzzy number valued function of two variables $f\left(t,x\right)\in {\mathbb{F}}_R$ is called gH-partial differentiable (gH − p) at the point $\left(t_0,x_0\right)\in \mathbb{D}$ with respect to t and x and denoted by ∂_tgHf(t₀, x₀) and ∂_xgHf(t₀, x₀) if:

${\partial }_{\mathit{tgH}}f\left(t_0,x_0\right)=\underset{h\to 0}{lim}\frac{f\left(t_0+h,x_0\right){\ominus }_{\mathit{gH}}f\left(t_0,x_0\right)}{h}$

${\partial }_{\mathit{xgH}}f\left(t_0,x_0\right)=\underset{k\to 0}{lim}\frac{f\left(t_0,x_0+k\right){\ominus }_{\mathit{gH}}f\left(t_0,x_0\right)}{k}$

Provided that both derivatives ∂_tgHf(t₀, x