Английская Википедия:Fuzzy differential equation

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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.Шаблон:R <math display="block"> dx(t)/dt= F(t,x(t),\alpha),</math> for all <math> \alpha \in [0,1] </math>.

First order fuzzy differential equation

A first order fuzzy differential equationШаблон:R with real constant or variable coefficients

<math display="block"> x'(t) + p(t) x(t) = f(t) </math>

where <math>p(t)</math> is a real continuous function and <math> f(t) \colon [t_0 , \infty) \rightarrow R_F </math> is a fuzzy continuous function <math display="block"> y(t_0) = y_0 </math> such that <math> y_0 \in R_F </math>.

Application

It is useful for calculating Newton's law of cooling, compartmental models in epidemiology and multi-compartment model.Шаблон:Citation needed

Linear systems of fuzzy differential equations

A system of equations of the form

<math display="block"> x(t)'_n = a_n1(t) x_1(t) + ......+ a_nn(t) x_n(t) + f_n(t) </math>where <math>a_ij</math> are real functions and <math> f_i</math> are fuzzy functions <math display="block"> x'_n(t)= \sum_{i=0}^1 a_{ij} x_i.</math>

Fuzzy partial differential equations

A fuzzy differential equation with partial differential operator is <math display="block"> \nabla x(t) = F(t,x(t),\alpha),</math>for all <math> \alpha \in [0,1] </math>.

Fuzzy fractional differential equation

A fuzzy differential equation with fractional differential operator is

<math display="block"> d^n x(t)/{dt}^n= F(t,x(t),\alpha),</math> for all <math> \alpha \in [0,1] </math> where <math>n</math> is a rational number.