Fuzzy differential equation

A first order fuzzy differential equation{{r|keshavarz}} with real constant or variable coefficients

$$x\text{'}(t)\; +\; p(t)\; x(t)\; =\; f(t)$$

where $p(t)$ is a real continuous function and $f(t)\; \backslash colon\; [t\_0\; ,\; \backslash infty)\; \backslash rightarrow\; R\_F$ is a fuzzy continuous function

$$y(t\_0)\; =\; y\_0$$ such that $y\_0\; \backslash in\; R\_F$.

A system of equations of the form

$$x(t)\text{'}\_n\; =\; a\_n1(t)\; x\_1(t)\; +\; ......+\; a\_nn(t)\; x\_n(t)\; +\; f\_n(t)$$where $a\_ij$ are real functions and $f\_i$ are fuzzy functions

$$x\text{'}\_n(t)=\; \backslash sum\_\{i=0\}^1\; a\_\{ij\}\; x\_i.$$

A fuzzy differential equation with partial differential operator is

$$\backslash nabla\; x(t)\; =\; F(t,x(t),\backslash alpha),$$for all $\backslash alpha\; \backslash in\; [0,1]$.

A fuzzy differential equation with fractional differential operator is

$$\backslash frac\; \{d^n\; x(t)\}\; \{dt^n\}=\; F(t,x(t),\backslash alpha),$$ for all $\backslash alpha\; \backslash in\; [0,1]$ where $n$ is a rational number.

{{reflist|refs=

{{Cite web

|title = Theory of Fuzzy Differential Equations and Inclusions

|url = https://www.routledge.com/Theory-of-Fuzzy-Differential-Equations-and-Inclusions/Lakshmikantham-Mohapatra/p/book/9780367395322

|access-date = 2022-10-15

|website = Routledge & CRC Press

|language = en}}

{{Cite book

|last1 = Devi |first1 = S. Sindu

|last2 = Ganesan |first2 = K.

|title = The 11th National Conference on Mathematical Techniques and Applications

|date = 2019

|chapter = Application of linear fuzzy differential equation in day to day life

|volume = 2112

|chapter-url = http://aip.scitation.org/doi/abs/10.1063/1.5112354

|location = Chennai, India

|pages = 020169

|doi = 10.1063/1.5112354

|s2cid = 198460805}}

{{Cite journal

|last1 = Qiu |first1 = Dong

|last2 = Lu |first2 = Chongxia

|last3 = Zhang |first3 = Wei

|last4 = Zhang |first4 = Qinghua

|last5 = Mu |first5 = Chunlai

|date = 2014-12-02

|title = Basic theorems for fuzzy differential equations in the quotient space of fuzzy numbers

|journal = Advances in Difference Equations

|volume = 2014

|issue = 1

|pages = 303

|doi = 10.1186/1687-1847-2014-303

|s2cid = 54172371

|issn = 1687-1847

|doi-access = free}}

{{Cite journal

|last1 = Keshavarz |first1 = M.

|last2 = Allahviranloo |first2 = T.

|last3 = Abbasbandy |first3 = S.

|last4 = Modarressi |first4 = M. H.

|title = A Study of Fuzzy Methods for Solving System of Fuzzy Differential Equations

|url = https://www.worldscientific.com/doi/epdf/10.1142/S1793005721500010

|access-date = 2022-10-15

|journal = New Mathematics and Natural Computation

|year = 2021

|language = en

|doi = 10.1142/s1793005721500010

|volume = 17

|pages = 1–27

|s2cid = 225373837}}

}}

Category: Fuzzy logic

Category: Differential equations