Fuzzy Conformable Fractional Differential Equations

Abstract

In this study, fuzzy conformable fractional differential equations are investigated. We study conformable fractional differentiability, and we define fractional integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy fractional differential equation by using the concept of conformable differentiability. This concept is based on the enlargement of the class of differentiable fuzzy mappings; for this, we consider the lateral Hukuhara derivatives of order q ∈ 0,1 .

Representation of solutions to fuzzy linear conformable differential equations

“…, where the integral t c f α i s 1−q (s)ds, for i = 1, 2 is the usual Riemann improper integral. Lemma 2.8 (see [10], Theorem 7). Let q ∈ (0, 1] and F, G : I → R F be q-differentiable and λ ∈ R. Then (i) (F + G) (q) (t) = F (q) (t) + G (q) (t) and (ii) (λF) (q) (t) = λF (q) (t).…”

“…Let q ∈ (0, 1] and F, G : I → R F be q-differentiable and λ ∈ R. Then (i) (F + G) (q) (t) = F (q) (t) + G (q) (t) and (ii) (λF) (q) (t) = λF (q) (t). Lemma 2.9 (see [10], Theorem 8). Let F : I → R F be any continuous function on I.…”

“…Let x(t) : I → R F be a fuzzy function such that x q (1) (t) or x q (2) (t) exists. If x(t) and x q (1) (t) satisfy problem (10), we say x(t) is a q (1) -solution of problem (10). Similarly if x(t) and x q (2) (t) satisfy problem (10), we say x(t) is a q (2) -solution of problem (10).…”

“…In [3] the authors introduced the notion of a fuzzy fractional derivative and the theory was developed in [4][5][6][7][8][9]. In [10] a new definition of a fuzzy fractional derivative called the fuzzy conformable fractional derivative is given, some useful results on fuzzy conformable fractional derivatives and fractional integrals are given and a class of linear fuzzy conformable fractional differential equations is considered. For first order linear fuzzy differential equations we refer the reader to [11].…”

Generalized Hukuhara conformable fractional derivative and its application to fuzzy fractional partial differential equations

“…This new definition seems to be a natural extension of the usual derivative. Researchers started to combine this new definition with fuzzy calculus [22,23,29]. They used the concept of H-differentiability or strongly generalized differentiability.…”