On Fractional Geometry of Curves

Abstract

Fractional Differential Geometry of curves is discussed, with the help of a new fractional derivative, the Λ-fractional derivative, with the corresponding Λ-fractional space. Λ-Fractional derivative completely conforms with the demands of Differential Topology, for the existence of a differential. Therefore Fractional Differential Geometry is established in that Λ-space. The results are pulled back to the initial space.

On Λ-Fractional Wave Propagation in Solids

“…The Λ-fractional analysis, proposed by Lazopoulos [16], is consistent with the prerequisites of differential topology, and Λ-fractional derivatives may correctly generate differential geometry. Λ-fractional analysis has already been applied to geometry, physics, mechanics, differential equations, etc., Lazopoulos [16][17][18][19][20].…”

“…Nevertheless, the application of fractional calculus is considered quite important in various scientific areas requiring the consideration of non-local procedures. Lazopoulos [16][17][18][19][20] introduced the Λ-fractional derivative, because it is a unique fractional derivative satisfying all the prerequisites of differential topology for being a mathematical derivative. Hence it is a unique fractional differential procedure formulating fractional differential geometry and correct differential equations with the existence and uniqueness theorem applied in physics and mechanics.…”

On Λ-Fractional fluid mechanics

“…The system of these sixteen equations contains 16 unknowns therefore it is determinate. Usually, the studies concerning the Newtonian fl uid, are restricted to the equations of conservation of mass (29), linear momentum (33), and constitutive equations (36). The solution in the -fractional space is transferred to the initial space.…”

“…Simple substitutions of the conventional differentials to fractional ones cannot express the realistic behavior of the various physical problems. Lately, Lazopoulos [36] proposed the fractional -derivative, a modifi cation of the fractional L-derivative, along with the conjugate fractional -space, where the fractional -derivative behaves with conventional derivative rules. Specifi cally, derivatives in Fractional Calculus are merely operators and not derivatives since none satisfi es the criteria of the differential topology of a derivative, as described in Chillingworth [21].…”

Deflection of Beams Modeled by Fractional Differential Equations

“…That is, the increment ∆y is no longer proportional to f (α) (x)∆x, where f (α) (x) is the α-order fractional derivative of f , in one of the three senses mentioned before. As is known, this is a major obstacle in the derivation of fractional models and there is a great variety of methods to solve this problem, see for example [14][15][16]. In our case, to derive the fractional differential equation we proceed, broadly speaking, from the following way.…”