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),\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[t0,infty)RF

is a fuzzy continuous function y(t_0) = y_0 such that

y0\inRF

.

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

aij

are real functions and

fi

are fuzzy functions x'_n(t)= \sum_^1 a_ 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 = F(t,x(t),\alpha), for all

\alpha\in[0,1]

where

n

is a rational number