Page 78 - IJOCTA-15-1

E.M. Rabei et al. / IJOCTA, Vol.15, No.1, pp.71-81 (2025)
self-similarity of fractals. In other words, all frac- The conformable derivative is straightforward in
tal dimensions are fractional dimensions, but not that it fulfills the Leibniz and chain principles as
all fractional dimensions are fractal dimensions. well as the general characteristics of the ordinary
derivative. In. 30,31
In these frameworks, fractional dimensionality.
Refers to an additional effective environment used Properties 29
to represent the real system rather than the phys-
ical space itself. However, some authors have also 1-T β (af +bg) = aT β (f)+bT β (g) for all real con-
considered the possibility that genuine spacetime stant a, b
has a dimension slightly different from four. 18,19 2-T β (fg) = fT β (g) + gT β (f)

p
3-T β (t ) = pt p−β for all p
Observations suggest that spacetime dimension- f gT β (f)−fT β (g)
ality deviates from four in a minor manner. 18,19 4-T β ( ) = g 2
g
Nonetheless, the question of whether the dimen- 5- T β (c) = 0 with c is constant.
sion of spacetime is an integer or a fractional
number is crucial due to its conceptual signifi- β
In this paper, we adopt D f to denote the con-
cance. Moreover, intriguing implications arise if
formable derivative (CD) of f of order β, T β (f)(t).
spacetime has a dimension other than four. For
instance, it is well-known that, even with a tiny
A strong mathematical tool that improves com-
deviation of elements, the logarithmic divergences
plex system modeling in physics and engineering,
of quantum electrodynamics once the value of
four is exceeded. 20 Furthermore, the deformation among other domains, is the conformable deriva-
tive. The reason for its usefulness is that it can
of quantum mechanics in fractional-dimensional
space is explored in Ref. 21 overcome the drawbacks of classical derivatives,
especially when nonlocal events and discontinu-
ities are involved. Below are references highlight-
One of the most fascinating recent topics in var- 32–45
ing its usefulness.
ious physical scientific fields is the application
of fractional calculus. In 1695, L’Hospital and
2. Conformable N-dimensional polar
Leibniz corresponded regarding the meaning of
n
1 22
non-integer order derivatives d f n when n = . coordinates
dx 2
Hence, the concept of non-integer order deriva- In this section, we aim to derive a concise expres-
tives first emerged from those exchanges. Frac- sion for the conformable scalar Laplacian in N
tional derivatives have been defined in various dimensions. We rely on N-dimensional polar co-
ways throughout history, including Riemann- ordinates as outlined in Ref, 46,47 along with the
Louville, 22,23 Caputo. 24 Grunwald-Latnikov, 25 conformable trigonometric function described in
Riesz 26 Weyl 27 and Riesz-Caputo. 28 As most of Ref, 32 and conformable spherical coordinates de-
these definitions involve fractional integrals, they tailed in Ref. 33 The conformable N-dimensional
all inherit nonlocal properties from integrals.
polar coordinates are defined as follows
Khalil et al. 29 introduced a novel derivative con- β
β
cept called the conformable derivative a few years x = r cos θ 1 ,
β
1
ago, building upon the definition of the deriva- β
tive. Therefore, the conformable derivative can be β β θ β θ 2 β
1
x = r sin cos ,
viewed as an extension of the original definition 2 β β
of a derivative. The term conformable derivative β θ β θ β θ β
β
(CD) is one definition used to describe the non- x = r sin 1 sin 2 cos 3 ,
3
β β β
integer derivative, employing the limit definition · · ·
of the derivative. 29
β β β β β
Definition 2.1. Given a function f ∈ [0, ∞) → β β θ 1 θ 2 θ 3 θ N−2 θ N−1
x = r sin sin sin · · · sin cos ,
R. The conformable derivative of f with order β N−1 β β β β β
is defined by 29 β β β β
β β θ 1 θ 2 θ 3 θ N−1
x = r sin sin sin · · · sin .
N β β β β
f(t + ϵt 1−β ) − f(t)
T β (f)(t) = lim (2)
ϵ→0 ϵ Where 0 ≤ θ a ≤ π, (a = 1, 2, 3, · · · , N − 2), 0 ≤
for all t > 0, β ∈ (0.1). θ N−1 ≤ 2π.
72