- Source: 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.
d
x
(
t
)
/
d
t
=
F
(
t
,
x
(
t
)
,
α
)
,
{\displaystyle dx(t)/dt=F(t,x(t),\alpha ),}
for all
α
∈
[
0
,
1
]
{\displaystyle \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
)
{\displaystyle x'(t)+p(t)x(t)=f(t)}
where
p
(
t
)
{\displaystyle p(t)}
is a real continuous function and
f
(
t
)
:
[
t
0
,
∞
)
→
R
F
{\displaystyle f(t)\colon [t_{0},\infty )\rightarrow R_{F}}
is a fuzzy continuous function
y
(
t
0
)
=
y
0
{\displaystyle y(t_{0})=y_{0}}
such that
y
0
∈
R
F
{\displaystyle y_{0}\in R_{F}}
.
Linear systems of fuzzy differential equations
A system of equations of the form
x
(
t
)
n
′
=
a
n
1
(
t
)
x
1
(
t
)
+
.
.
.
.
.
.
+
a
n
n
(
t
)
x
n
(
t
)
+
f
n
(
t
)
{\displaystyle x(t)'_{n}=a_{n}1(t)x_{1}(t)+......+a_{n}n(t)x_{n}(t)+f_{n}(t)}
where
a
i
j
{\displaystyle a_{i}j}
are real functions and
f
i
{\displaystyle f_{i}}
are fuzzy functions
x
n
′
(
t
)
=
∑
i
=
0
1
a
i
j
x
i
.
{\displaystyle x'_{n}(t)=\sum _{i=0}^{1}a_{ij}x_{i}.}
Fuzzy partial differential equations
A fuzzy differential equation with partial differential operator is
∇
x
(
t
)
=
F
(
t
,
x
(
t
)
,
α
)
,
{\displaystyle \nabla x(t)=F(t,x(t),\alpha ),}
for all
α
∈
[
0
,
1
]
{\displaystyle \alpha \in [0,1]}
.
Fuzzy fractional differential equation
A fuzzy differential equation with fractional differential operator is
d
n
x
(
t
)
d
t
n
=
F
(
t
,
x
(
t
)
,
α
)
,
{\displaystyle {\frac {d^{n}x(t)}{dt^{n}}}=F(t,x(t),\alpha ),}
for all
α
∈
[
0
,
1
]
{\displaystyle \alpha \in [0,1]}
where
n
{\displaystyle n}
is a rational number.
References
Kata Kunci Pencarian:
- Fuzzy differential equation
- Fuzzy differential inclusion
- Fuzzy set
- Schrödinger–Newton equation
- Black–Scholes model
- Discrete mathematics
- Differential inclusion
- Fuzzy sphere
- Fuzzy concept
- Aboodh transform
