Fuzzy differential equation explained

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.$$dx(t)/dt=\; F(t,x(t),\backslash alpha),$$ for all

\alpha \in [ 0 , 1 ] .

First order fuzzy differential equation

A first order fuzzy differential equation with real constant or variable coefficients

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

where

p(t) is a real continuous function and f(t) \colon [t _{0} , inf ty) → 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

$$x(t)\text{'}\_n\; =\; a\_n1(t)\; x\_1(t)\; +\; ......+\; a\_nn(t)\; x\_n(t)\; +\; f\_n(t)$$where

a _{ij} are real functions and f _{i} are fuzzy functions $$x\text{'}\_n(t)=\; \backslash sum\_^1\; a\_\; x\_i.$$

Fuzzy partial differential equations

A fuzzy differential equation with partial differential operator is $$\backslash nabla\; x(t)\; =\; F(t,x(t),\backslash alpha),$$for all

\alpha \in [ 0 , 1 ] .

Fuzzy fractional differential equation

A fuzzy differential equation with fractional differential operator is

$$\backslash frac\; =\; F(t,x(t),\backslash alpha),$$ for all

\alpha \in [ 0 , 1 ] where n is a rational number