On the stability of solutions of fractional non conformable differential equations

Abstract

In this note we obtain sufficient conditions under which we can guarantee the stability of solutions of a fractional differential equations of non conformable type and we obtain some fractional analogous theorems of the direct Lyapunov method for a given class of equations of motion.

Note on the conformable boundary value problems: Sturm’s theorems and Green’s function

“…In addition, another new definition of local fractional derivative is introduced by P. M. Guzmán et al [25], and it is called the non-conformable fractional derivative which is considered as a natural extension of the usual derivative of a function a point. The difference between conformable fractional derivative and non-conformable fractional derivative is that the tangent line angle is conserved in the conformable one, while it is not conserved in the sense of nonconformable one [25] (see also [26,34,36,37] for more new related results about this newly proposed definition of non-conformable fractional derivative). The definition of non-conformable fractional derivative has been investigated and applied in various research studies and applications of physics and natural sciences such as the stability analysis, oscillatory character, and boundedness of fractional Liénard-type systems [27,28,33], analysis of the local fractional Drude model [29], Hermite-Hadamard inequalities [30], fractional Laplace transform [31], fractional logistic growth models [32], oscillatory character of fractional Emden-Fowler equation [35], asymptotic behavior of fractional nonlinear equations [38], and qualitative behavior of nonlinear differential equations [39].…”

The non-integer local order calculus

“…Proof. See 25 This derivative, and some variants, proved useful in various application problems (see [26][27][28][29][30][31][32][33][34][35] ).…”

On the generalized laplace transform and applications

“…On the other hand, although local differential operators have been known since the 1960s, it was not until 2014 that a complete formalization was achieved with the conformable derivative. 1 In 2018 we defined a new type of differential operator, called non conformable [2][3][4][5][6][7][8] and in 2020 we consolidate our ideas with a generalized derivative definition, 9-11 see also). 7,12 In this way, a new area has been formed in the Mathematical Sciences, which we call Generalized Calculus, with many applications and important theoretical results.Adding to this, integral transforms are also ground-breaking inventions in calculus.…”

“…1 In 2018 we defined a new type of differential operator, called non conformable [2][3][4][5][6][7][8] and in 2020 we consolidate our ideas with a generalized derivative definition, 9-11 see also). 7,12 In this way, a new area has been formed in the Mathematical Sciences, which we call Generalized Calculus, with many applications and important theoretical results.Adding to this, integral transforms are also ground-breaking inventions in calculus. The capability of integral transforms to manipulate several problems by altering the domain of the equation, have made it persistently important.…”

“…1 In 2018 we defined a new type of differential operator, called non conformable [2][3][4][5][6][7][8] and in 2020 we consolidate our ideas with a generalized derivative definition, 9-11 see also). 7,12 In this way, a new area has been formed in the Mathematical Sciences, which we call Generalized Calculus, with many applications and important theoretical results.…”