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.[1][2][3] [math]\displaystyle{ dx(t)/dt= F(t,x(t),\alpha), }[/math] for all [math]\displaystyle{ \alpha \in [0,1] }[/math].
First order fuzzy differential equation
A first order fuzzy differential equation[4] with real constant or variable coefficients
[math]\displaystyle{ x'(t) + p(t) x(t) = f(t) }[/math]
where [math]\displaystyle{ p(t) }[/math] is a real continuous function and [math]\displaystyle{ f(t) \colon [t_0 , \infty) \rightarrow R_F }[/math] is a fuzzy continuous function [math]\displaystyle{ y(t_0) = y_0 }[/math] such that [math]\displaystyle{ 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.[citation needed]
Linear systems of fuzzy differential equations
A system of equations of the form
[math]\displaystyle{ x(t)'_n = a_n1(t) x_1(t) + ......+ a_nn(t) x_n(t) + f_n(t) }[/math]where [math]\displaystyle{ a_ij }[/math] are real functions and [math]\displaystyle{ f_i }[/math] are fuzzy functions [math]\displaystyle{ 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]\displaystyle{ \nabla x(t) = F(t,x(t),\alpha), }[/math]for all [math]\displaystyle{ \alpha \in [0,1] }[/math].
Fuzzy fractional differential equation
A fuzzy differential equation with fractional differential operator is
[math]\displaystyle{ d^n x(t)/{dt}^n= F(t,x(t),\alpha), }[/math] for all [math]\displaystyle{ \alpha \in [0,1] }[/math] where [math]\displaystyle{ n }[/math] is a rational number.
References
- ↑ "Theory of Fuzzy Differential Equations and Inclusions" (in en). https://www.routledge.com/Theory-of-Fuzzy-Differential-Equations-and-Inclusions/Lakshmikantham-Mohapatra/p/book/9780367395322.
- ↑ 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. 2112. Chennai, India. pp. 020169. doi:10.1063/1.5112354. http://aip.scitation.org/doi/abs/10.1063/1.5112354.
- ↑ 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.
- ↑ Keshavarz, M.; Allahviranloo, T.; Abbasbandy, S.; Modarressi, M. H. (2021). "A Study of Fuzzy Methods for Solving System of Fuzzy Differential Equations" (in en). New Mathematics and Natural Computation 17: 1–27. doi:10.1142/s1793005721500010. https://www.worldscientific.com/doi/epdf/10.1142/S1793005721500010. Retrieved 2022-10-15.
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