From Wikipedia, the free encyclopedia
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]
![{\displaystyle dx(t)/dt=F(t,x(t),\alpha ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/685c635468dd95517df245d514dcbf257bfc366a)
for all
![{\displaystyle \alpha \in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf3c62599ea71319c85f715c9e590d2bab2d036)
.
First order fuzzy differential equation
A first order fuzzy differential equation
[4] with real constant or variable coefficients
![{\displaystyle x'(t)+p(t)x(t)=f(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f76a3ada9dda29ab341094319b5a07309d7779)
where
is a real continuous function and
is a fuzzy continuous function
![{\displaystyle y(t_{0})=y_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba6cc4b1d5692b5d384b9e1f93705ae00f05cbc)
such that
![{\displaystyle y_{0}\in R_{F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc2dc972fbc2bf59a53db3ffb7437e22a89c2180)
.
Linear systems of fuzzy differential equations
A system of equations of the form
![{\displaystyle x(t)'_{n}=a_{n}1(t)x_{1}(t)+......+a_{n}n(t)x_{n}(t)+f_{n}(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50e3226f9116daec3373532da2df270b7b21d49b)
where
![{\displaystyle a_{i}j}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed4c0e1bf311a2c3ebcfed74a5548d3e97c87f5)
are real functions and
![{\displaystyle f_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79)
are fuzzy functions
![{\displaystyle x'_{n}(t)=\sum _{i=0}^{1}a_{ij}x_{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bef76144eb358abc24336c7dfec654a6cb3f4cff)
Fuzzy partial differential equations
A fuzzy differential equation with partial differential operator is
![{\displaystyle \nabla x(t)=F(t,x(t),\alpha ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c216f457ef6f8682d0370fd29ab187234d4def3)
for all
![{\displaystyle \alpha \in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf3c62599ea71319c85f715c9e590d2bab2d036)
.
Fuzzy fractional differential equation
A fuzzy differential equation with fractional differential operator is
![{\displaystyle d^{n}x(t)/{dt}^{n}=F(t,x(t),\alpha ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00c25d7fb2c15f01d15ba6d232058a4bef0516ab)
for all
![{\displaystyle \alpha \in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf3c62599ea71319c85f715c9e590d2bab2d036)
where
![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
is a
rational number.
References
-
^
"Theory of Fuzzy Differential Equations and Inclusions". Routledge & CRC Press. Retrieved 2022-10-15.
-
^ Devi, S. Sindu; Ganesan, K. (2019).
"Application of linear fuzzy differential equation in day to day life". The 11th National Conference on Mathematical Techniques and Applications. Vol. 2112. Chennai, India. p. 020169.
doi:
10.1063/1.5112354.
S2CID
198460805.
{{
cite book}}
: CS1 maint: location missing publisher (
link)
-
^ Qiu, Dong; Lu, Chongxia; Zhang, Wei; Zhang, Qinghua; Mu, Chunlai (2014-12-02).
"Basic theorems for fuzzy differential equations in the quotient space of fuzzy numbers". Advances in Difference Equations. 2014 (1): 303.
doi:
10.1186/1687-1847-2014-303.
ISSN
1687-1847.
S2CID
54172371.
-
^ Keshavarz, M.; Allahviranloo, T.; Abbasbandy, S.; Modarressi, M. H. (2021).
"A Study of Fuzzy Methods for Solving System of Fuzzy Differential Equations". New Mathematics and Natural Computation. 17: 1–27.
doi:
10.1142/s1793005721500010.
S2CID
225373837. Retrieved 2022-10-15.