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]

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

First order fuzzy differential equation

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

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

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

Application

It is useful for calculating Newton's law of cooling, compartmental models in epidemiology and multi-compartment model.^{[citation needed]}

Linear systems of fuzzy differential equations

A system of equations of the form

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

Fuzzy partial differential equations

A fuzzy differential equation with partial differential operator is

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

Fuzzy fractional differential equation

A fuzzy differential equation with fractional differential operator is

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

References