Total and Directional Fractional Derivatives

Abstract

Vector calculus is an important subject in mathematics with applications in all areas of applied sciences. Till now researchers deal with the partial fractional derivative as the fractional derivative with respect to x, y,... . In this paper we shall define total and directional fractional derivative of functions of several variables, we set some basics about fractional vector calculus then we use our definition to modify the definition of conformal fractional derivative obtained by R. Khalil et al [6].

Conformable Euler's theorem on homogeneous functions

“…Finally, the conformable partial derivative of a real valued function with several variables is defined as follows.…”

“…For instance, definitions of left and right conformable fractional derivatives and fractional integrals of higher order (ie, of order > 1), the fractional power series expansion and the fractional transform Laplace definition, fractional integration by parts formulas, chain rule, and Gronwall inequality are also provided by him. Moreover, the conformable partial derivative of the order ∈ of the real value of several variables and conformable gradient vector are defined 11,12 ; and a conformable version of Clairaut's theorem for partial derivatives of conformable fractional orders is proved. 11,12 In short time, many studies 13-21 about theory and application of the fractional differential equations are based on this new fractional derivative definition.On the other hand, Euler's theorem on homogeneous functions is used to solve many problems in engineering, science, and finance.…”

“…Moreover, the conformable partial derivative of the order ∈ of the real value of several variables and conformable gradient vector are defined 11,12 ; and a conformable version of Clairaut's theorem for partial derivatives of conformable fractional orders is proved. 11,12 In short time, many studies 13-21 about theory and application of the fractional differential equations are based on this new fractional derivative definition.On the other hand, Euler's theorem on homogeneous functions is used to solve many problems in engineering, science, and finance. Hiwarekar 22 discussed the extension and applications of Euler's theorem for finding the values of higher-order expressions for two variables.…”

Fractional Euler-Bernoulli Beam Theory Based on the Fractional Strain-Displacement Relation and its Application in Free Vibration, Bending and Buckling Analyses of Micro/Nanobeams

“…Modified CFDD is a definition, which has been presented by Khalil et al [15] and Tallafha et al [16]…”

A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

“…In a very short period of time, many mathematicians became interested and provided mathematical models related to conformable derivatives, for the details we refer reader to see [6][7][8][9]. In [10,11], the conformable derivatives were applied to some problems in mechanics, and in [12] total frational derivative and directional fractional derivative of functions of several variables were studied.…”