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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. 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 } .

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 } .

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 } where n {\displaystyle n} is a rational number.

References

  1. "Theory of Fuzzy Differential Equations and Inclusions". Routledge & CRC Press. Retrieved 2022-10-15.
  2. 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)
  3. 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.
  4. 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.
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