Variation of parameters for local fractional nonhomogenous lineardifferential equations

Abstract

In this paper we study the method of variation of parameters to find a particular solution of a nonhomogenous linear fractional differential equations. A formula similar to that for usual ordinary differential equations is obtained.

Conformable Euler's theorem on homogeneous functions

“…Moreover, the conformable partial derivative of the order α ∈ of the real value of several variables and conformable gradient vector are defined; and a conformable version of Clairaut's theorem for partial derivatives of conformable fractional orders is proved . In short time, many studies about theory and application of the fractional differential equations are based on this new fractional derivative definition.…”

Electrical circuits described by fractional conformable derivative

“…However, properties, such as the product rule, quotient rule, chain rule, Rolle theorem, mean value theorem, and composition rule, are lacking in almost all fractional derivatives. [27][28][29][30][31] Motivated by this new conformable derivative, we apply it to the well-known series RC, LC, and RLC electric circuits and analyse their behaviour. To avoid these difficulties, in Khalil et al, 24 it was proposed an interesting idea that extends the ordinary limit definitions of the derivatives of a function called conformable fractional derivative.…”

“…Therefore, there is a large number of works carried out using this new definition. [27][28][29][30][31] Motivated by this new conformable derivative, we apply it to the well-known series RC, LC, and RLC electric circuits and analyse their behaviour. The solutions depend on time and on the fractional order parameter 0 < γ ≤ 1.…”

The fractional annihilator technique for solving nonhomogeneous fractional linear differential equations

“…Indeed, they employed the same methods of the ordinary case with a bit adaptation to finding the solution of the conformable fractional differential equations. Unal et al [9] and Al Horani et al [2] used the variation of parameters method and Unal et al [10] utilized the operator method for giving a particular solution of nonhomogeneous sequential linear conformable fractional differential equations. Also, these same fractional differential equations have been solved with the help of undetermined coefficient method in [5].…”