Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Ciencias Matemáticas
FRACTIONAL CALCULUS: A 300-YEAR PERSPECTIVE ON
OPERATORS, APPLICATIONS, AND MODERN GENERALIZATIONS
CÁLCULO FRACCIONARIO: UNA PERSPECTIVA DE 300 AÑOS SOBRE
OPERADORES, APLICACIONES Y GENERALIZACIONES MODERNAS
CÁLCULO FRACIONÁRIO: UMA PERSPECTIVA DE 300 ANOS SOBRE
OPERADORES, APLICAÇÕES E GENERALIZAÇÕES MODERNAS
Autores:
# Norlen Melendez
1,∗
nmelendez@dadeschool.net
# Miguel Vivas-Cortez
2
mjvivas@puce.edu.ec
# Oramys Prieto-Arima
3
oramys@gmail.com
1
Youth Co-Op Charter School,Florida, USA
2
Pontificia Universidad Católica del Ecuador,
Quito, Ecuador
3
Universidad Nacional Experimental de la
Seguridad, Lara, Venezuela
* Autor para correspondencia.
Editor Académico
Fernando José Sánchez-Salas
Citación sugerida: Melendez, N., Vivas-Corez,
M., Prieto-Arima, O. (2026). Fractional calculus:
a 300-year perspective on operators, applications,
and modern generalizations. Revista Bases de la
Ciencia, 11(1), 18-31. DOI: 10.33936/revbasdelacien-
cia.v11i1.8109
Recibido: 16/12/2025
Aceptado: 30/01/2026
Publicado: 16/02/2026
Abstract
This work provides a comprehensive historical and analytical review of fractional calculus, tracing
its evolution from the seminal concepts of Leibniz (1695) to modern unified local and biparametric
generalizations. The study highlights two main historical branches: the classical nonlocal operators
(Riemann–Liouville, Caputo, and Riesz), which emphasize long-range memory effects crucial for modeling
viscoelasticity and anomalous diffusion; and the later emergence of local fractional derivatives (Yang,
Conformable), designed to capture local scaling properties in fractal and non-smooth media. Key milestones
include the introduction of the Caputo derivative in 1967, which facilitated the use of classical initial
conditions in fractional differential equations, and the development of variable-order operators (2000s) to
model systems with dynamic memory. The paper critically examines modern trends, focusing on the Unified
Local Fractional Derivative (ULFD) and the recent Biparametric V-Derivative
D
α,β
V
. This V-Derivative, defined
as
V
ς,χ
( f (t)) := l
´
ım
h→0
(χ+h(χ−ς)) f
t+h
ς
χ
−χ f (t)
χ·h
. Furthermore, the associated generalized operational calculus,
which provides closed-form Laplace transform solutions, is discussed. We conclude that contemporary
research is focused on developing hybrid formalisms that unify nonlocal memory, local scaling, and classical
dynamics, offering superior flexibility for modeling complex phenomena like hybrid transport and fractal
media.
Keywords: Fractional Calculus, Fractional Derivatives, Biparametric V-Derivative, Generalized Laplace
Transform, Conformable Derivative.
Resumen
Este trabajo proporciona una revisión histórica y analítica exhaustiva del cálculo fraccionario, rastreando su
evolución desde los conceptos seminales de Leibniz (1695) hasta las generalizaciones modernas unificadas
locales y biparamétricas. El estudio destaca dos ramas históricas principales: los operadores clásicos
no locales (Riemann–Liouville, Caputo y Riesz), que enfatizan los efectos de memoria a largo alcance
cruciales para modelar la viscoelasticidad y la difusión anómala; y la posterior aparición de las derivadas
fraccionarias locales (Yang, Conformable), diseñadas para capturar propiedades de escalamiento local en
medios fractales y no suaves. Los hitos principales incluyen la introducción de la derivada de Caputo
en 1967, la cual facilitó el uso de condiciones iniciales clásicas en ecuaciones diferenciales fraccionarias,
y el desarrollo de operadores de orden variable (década de 2000) para modelar sistemas con memoria
dinámica. El artículo examina críticamente las tendencias modernas, centrándose en la Derivada Fraccionaria
Local Unificada (ULFD) y la reciente V-Derivada Biparamétrica
D
α,β
V
. Esta V-Derivada, definida como
V
ς,χ
( f (t)) := l
´
ım
h→0
(χ+h(χ−ς)) f
t+h
ς
χ
−χ f (t)
χ·h
. Además, se discute el cálculo operacional generalizado
asociado, el cual proporciona soluciones mediante la transformada de Laplace en forma cerrada. Se concluye
que la investigación contemporánea se centra en el desarrollo de formalismos híbridos que unifican la
memoria no local, el escalamiento local y la dinámica clásica, ofreciendo una flexibilidad superior para
modelar fenómenos complejos como el transporte híbrido y los medios fractales.
Palabras clave: Cálculo Fraccionario, Derivadas Fraccionarias, V-Derivada Biparamétrica, Transformada de
Laplace Generalizada, Derivada Conformable.
Resumo
Este trabalho fornece uma revisão histórica e analítica abrangente do cálculo fracionário, rastreando sua
evolução desde os conceitos seminais de Leibniz (1695) até as generalizações modernas unificadas locais
e biparamétricas. O estudo destaca dois ramos históricos principais: os operadores clássicos não locais
(Riemann–Liouville, Caputo e Riesz), que enfatizam os efeitos de memória de longo alcance, fundamentais
para modelar a viscoelasticidade e a difusão anômala; e o surgimento posterior das derivadas fracionárias
locais (Yang, Conformable), projetadas para capturar propriedades de escalonamento local em meios fractais
e não suaves. Os principais marcos incluem a introdução da derivada de Caputo em 1967, que facilitou o uso
de condições iniciais clássicas em equações diferenciais fracionárias, e o desenvolvimento de operadores de
ordem variável (anos 2000) para modelar sistemas com memória dinâmica. O artigo examina criticamente
as tendências modernas, concentrando-se na Derivada Fracionária Local Unificada (ULFD) e na recente
V-Derivada Biparamétrica
D
α,β
V
. Esta V-Derivada, definida como
V
ς,χ
( f (t)) := l
´
ım
h→0
(χ+h(χ−ς)) f
t+h
ς
χ
−χ f (t)
χ·h
.
Além disso, discute-se o cálculo operacional generalizado associado, que fornece soluções em forma fechada
por meio da transformada de Laplace. Conclui-se que a pesquisa contemporânea está voltada para o
desenvolvimento de formalismos híbridos que unificam a memória não local, o escalonamento local e a
dinâmica clássica, oferecendo maior flexibilidade para modelar fenômenos complexos, como o transporte
híbrido e os meios fractais.
Palavras chave: Cálculo Fracionário, Derivadas Fracionárias, V-Derivada Biparamétrica, Transformada de
Laplace Generalizada, Derivada Conformável.
# revista.bdlaciencia@utm.edu.ec 18
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
1. Early Origins of Fractional Calculus (1695–1850)
The reason behind the drive for extending the concept of differentiation to fractional calculus dates back to the letter
written by Gottfried Wilhelm Leibniz in 2014 where he questioned the significance of a derivative of order 1/2
(Lazarevi´c et al., 2014). While Leibniz gave the concept the initial idea, it would take more than a hundred years for
this abstract idea to develop into an analytical concept. In the nineteenth century, the rigorous definitions of fractional
derivatives and integrals were finally proposed thanks to the works of Liouville 1832 and Riemann 1876, along with
subsequent additions proposed by Weyl 1917a. More importantly, it was these basic definitions that highlighted the
property of nonlocality of the fractional operators for the first time.
1.1. Liouville Fractional Integrals and Derivatives
The Liouville integral has the historical merit of being the first known attempt to generalize the notion of repeated
integration to an arbitrary order (Liouville, 1832). Namely, given a function
f (x)
that is real and sufficiently
differentiable, along with a positive constant α, one has the following definition:
(I
α
f )(x) =
1
Γ(α )
Z
x
−∞
(x − t)
α−1
f (t) dt.
Based on the integral operator, he continued in a natural way to define the associated fractional derivative of order
α
as follows:
(D
α
f )(x) =
d
n
dx
n
I
n−α
f
(x), n − 1 < ℜ(α) ≤ n.
One of the most important properties that he obtained for this operator concerns the action of the fractional derivative
on the power functions. Indeed, for β > − − 1 and x > 0, this relationship is expressed as:
D
α
x
β
=
Γ(β + 1)
Γ(β + 1 − α)
x
β−α
(Liouville, 1832).
This is an elegant relationship that can be said to generalize the conventional integer-order differentiation quite well.
Moreover, it brings to the fore the idea that fractional differentiation is an algebraic operation in cases where power
functions are involved.
A second pillar of this theory is the semigroup law that holds for fractional integrals, and which says that:
I
α
I
β
f = I
α+β
f , ℜ(α), ℜ(β) > 0(Liouville, 1832).
1.2. Riemann–Liouville Fractional Operators on Finite Intervals
The contribution of Riemann was that he defined Liouville’s definition and restricted the range of integration to a
finite region of
t
(Riemann, 1876). So that we are able to define the function
f
on the interval
[a
,
b]
. From which, the
left-sided Riemann-Liouville
(I
α
a+
f )(x) =
1
Γ(α )
Z
x
a
(x − t)
α−1
f (t) dt, x > a.
The corresponding Riemann–Liouville fractional derivative is defined as
(D
α
a+
f )(x) =
d
n
dx
n
I
n−α
a+
f
(x), n − 1 < ℜ(α) ≤ n.
Riemann–Liouville derivatives inherit linearity and admit explicit representations similar to Liouville’s operator. In
particular,
D
α
a+
(x − a)
β
=
Γ(β + 1)
Γ(β + 1 − α)
(x − a)
β−α
, β > −1(Riemann, 1876).
19
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Fractional calculus: a 300-year perspective on operators, applications, and modern generalizations
However, an important and historically significant result is that
D
α
a+
c ̸= 0, for a constant c(Caputo, 1967),
which later motivated the development of alternative definitions better suited to physical initial conditions, most
notably the Caputo derivative.
1.3. Weyl Fractional Derivative
The fractional calculus for periodic functions of the real line was introduced by Weyl in order to solve some problems
connected with Fourier analysis (Weyl, 1917b). For suitable functions
f : R → C
the Weyl fractional integral is
expressed as
(
x
W
α
∞
f )(x) =
1
Γ(α )
Z
∞
x
(t − x)
α−1
f (t) dt,
whereas the Weyl fractional derivative may be defined through an
n
-th differentiation (
n
is an integer) of the fractional
integral of order n − α.
One of the first fundamental theorems about fractional derivatives concerning fractional derivatives was proven by
Weyl. Namely, if
f (x) =
∑
k∈Z
c
k
e
ikx
,
then
D
α
f (x) =
∑
k∈Z
(ik)
α
c
k
e
ikx
(Weyl, 1917a).
It should be stressed that this theorem connects the theory of fractional calculus and harmonic analysis and anticipates
later developments involving fractional Laplacian.
2. Classical Fractional Derivatives (1850–1950)
Different definitions of fractional derivatives have been provided throughout the end of the nineteenth century and
early twentieth century to enhance the level of rigor, discretization, symmetry, and analysis of the operator. This
comes as a result of the fundamental work done by Liouville 1832 and Riemann 1876, which has also proven to be
equivalent to other forms. Some of the important formulations in this period are those of Grünwald 1867, Letnikov
1868b, Marchaud 1927, Hadamard 1892, and Riesz 1949, each addressing specific theoretical or practical limitations
of earlier operators.
2.1. Grünwald–Letnikov Fractional Derivative
The first formulation of limits for fractional derivative operators was given independently by Grünwald 1867 and
Letnikov 1868a by means of the definition of finite differences of integer-order derivative functions.
Let a function
f
be given in the interval
[a
,
x]
. Then the Grünwald-Letnikov fractional differential operator of order
α
can be formulated as:
D
α
GL
f (x) = l
´
ım
h→0
+
1
h
α
⌊
x−a
h
⌋
∑
k=0
(−1)
k
α
k
f (x − kh),
where the generalized binomial coefficient is given by
α
k
=
Γ(α + 1)
Γ(k + 1)Γ(α − k + 1)
.
A fundamental result is that, for sufficiently smooth functions,
D
α
GL
f (x) = D
α
RL
f (x)(Liouville, 1832; Riemann, 1876),
# revista.bdlaciencia@utm.edu.ec 20
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
showing that these two types of derivatives were equivalent to each other. The equivalence served as a link between
discretized and integral versions of these derivatives, thereby making the Grünwald–Letnikov derivative especially
significant numerically.
For power functions,
D
α
GL
x
β
=
Γ(β + 1)
Γ(β + 1 − α)
x
β−α
, β > −1 (Liouville, 1832; Riemann, 1876),
recovering the classical result obtained earlier by Liouville and Riemann.
2.2. Marchaud Fractional Derivative
Definition provided by Marchaud in 1927 is concerned with difference quotients rather than fractional integrals
(Marchaud, 1927). It is suitable for functions since they are defined on the whole real axis.
Fractional Marchaud derivative of order 0 < α < 1 is given by
(D
α
M
f )(x) =
α
Γ(1 − α)
Z
∞
0
f (x) − f (x − t)
t
1+α
dt.
An important result is that, under suitable regularity conditions,
D
α
M
f (x) = D
α
RL
f (x)(Riemann, 1876),
Consequently, the equivalence has been established regarding the Riemann–Liouville derivative without using
fractional integrals.
The example of the Marchaud derivative demonstrates how the nonlocal property of the fractional derivative is
obtained: the value of the derivative at the point
x
is determined by the previous values of the function. This definition
found applications in the theory of probabilities and in jump processes.
2.3. Hadamard Fractional Derivative
A fractional derivative that is applicable to problems with scale invariance has been suggested by Hadamard utilizing
logarithmic rather than power-law kernels (Hadamard, 1892). Let us consider
f
to be a function on
(a
,
b) ⊂ (
0,
∞)
.
Then the Hadamard fractional integral of order α > 0 is given by
(I
α
H
f )(x) =
1
Γ(α )
Z
x
a
ln
x
t
α−1
f (t)
t
dt.
The corresponding Hadamard fractional derivative is
(D
α
H
f )(x) =
x
d
dx
n
(I
n−α
H
f )(x) , n − 1 < α ≤ n.
A key result is the action on logarithmic power functions:
D
α
H
(ln x)
β
=
Γ(β + 1)
Γ(β + 1 − α)
(ln x)
β−α
(Hadamard, 1892).
The Hadamard derivative is particularly suitable for problems invariant under dilations and plays an important role
in later unification frameworks.
2.4. Riesz Fractional Derivative
Riesz introduced a symmetric fractional derivative comparable to the fractional Laplacian in 1949 (Riesz, 1949). The
Riesz fractional derivative for a real-valued function f (x) on the real line R is given by:
21
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Fractional calculus: a 300-year perspective on operators, applications, and modern generalizations
F
D
α
R
f
(ξ) = −|ξ|
α
F[ f ](ξ).
Equivalently, it admits the integral representation
D
α
R
f (x) = C
α
Z
R
f (x) − f (y)
|x − y|
1+α
dy,
where C
α
is a normalization constant.
The Riesz derivative is a type of derivative possessing certain symmetries not exhibited by other more traditional
derivatives and is essentially the one-dimensional case of the fractional Laplacian operator
(−∆)
α/2
, which later
became crucial for anomalous diffusion.
3. Applied-Oriented and Regularized Fractional Derivatives (1950–1990)
The rapid evolution of the theory of application of fractional calculus in physics and engineering in the latter half
of the twentieth century has demonstrated certain weaknesses of classical non-local derivatives, particularly in
establishing physically reasonable initial and boundary conditions. In this context, a number of new approaches for
constructing definitions of fractional derivatives that would preserve the concept of fractional memory yet make
its usage consistent has been considered. Among these definitions, the most important one is the definition of the
Caputo derivative 1967. In addition, definitions introduced by Canavati 1980 and Coimbra 2003 have made an
important contribution to the field.
3.1. Caputo Fractional Derivative
In 1967, Caputo suggested an extension of the Riemann-Liouville differential operator which makes it possible to
work with classical initial conditions expressed using integer-order derivatives (Caputo, 1967). If
f ∈ C
n
[a
,
b]
and
n − 1 < α < n, the Caputo derivative is given by
(
C
D
α
a+
f )(x) =
1
Γ(n − α)
Z
x
a
(x − t)
n−α−1
f
(n)
(t) dt.
Unlike the Riemann–Liouville derivative, the Caputo derivative satisfies
C
D
α
a+
c = 0 for constants c,
making it suitable for physical systems where initial data are given as
f (a), f
′
(a), . . . , f
(n−1)
(a).
A fundamental relationship between Caputo and Riemann–Liouville derivatives is
C
D
α
a+
f (x) = D
α
a+
f (x) −
n−1
∑
k=0
f
(k)
(a)
k!
(x − a)
k
!
(Liouville, 1832; Riemann, 1876).
For power functions, Caputo differentiation yields
C
D
α
a+
(x − a)
β
=
0, β = 0, 1, . . . , n − 1,
Γ(β + 1)
Γ(β + 1 − α)
(x − a)
β−α
, β ≥ n.
This property explains its widespread adoption in viscoelasticity, diffusion-wave equations, and control theory.
# revista.bdlaciencia@utm.edu.ec 22
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
3.2. Fractional Differential Equations and Physical Models
However, with the inception of the Caputo derivative, fractional differential equations (FDEs) with classical initial
conditions could be proposed (Caputo, 1967). A typical case is that of the linear fractional relaxation equation.
C
D
α
0+
y(t) = −λy(t) , 0 < α < 1,
with initial condition y(0) = y
0
.
Its solution is given by the Mittag–Leffler function
y(t) = y
0
E
α
(−λt
α
), E
α
(z) =
∞
∑
k=0
z
k
Γ(α k + 1)
.
This result generalizes the classical exponential decay and demonstrates how fractional models interpolate between
elastic and viscous behavior.
3.3. Canavati Fractional Derivative
Fractional derivatives on Banach spaces were proposed by Canavati with an emphasis on operator theoretic aspects
(Canavati, 1980). The method is based on Riemann-Liouville derivative with additional boundary conditions included
in the definition domain.
Although the derivative may not always be defined in terms of explicit kernels, it is linear and admits the following
representation:
D
α
= A
α
,
where
A
is a closed linear operator generating a strongly continuous semigroup. This method linked fractional
calculus and semigroup theory, which was subsequently used for evolution equations.
3.4. Coimbra’s Variable Lower Limit Approach
Coimbra presented a formula that puts forward the physics behind memory by making the lower integration limit
time-dependent (Coimbra, 2003). The Coimbra fractional derivative is defined as
(D
α
C
f )(t) =
1
Γ(1 − α)
Z
t
t
0
(t − τ)
−α
d f (τ)
dτ
dτ,
where t
0
represents the initiation of the memory process.
This perspective clarified the role of initialization functions and resolved inconsistencies in modeling systems with
evolving memory length.
4. Fractal and Local Fractional Approaches (1990–2010)
During the 1990s, the idea of local fractional derivatives was developed due to the study of nowhere differentiable functions,
fractal media, and anomalous scaling behavior phenomena (Yang, 2012; Yang et al., 2016). Unlike traditional fractional
derivatives that cannot be localized because of their integral operators, the goal of local fractional derivatives is the
representation of local scaling behavior of functions on fractals. This opened a path connecting fractional calculus and
fractal geometry (Gao, 2003).
4.1. Yang’s Local Fractional Derivative
Yang et al. have proposed the concept of Local Fractional Derivative, which is formulated for continuous but
nondifferentiable functions (Yang, 2012; Yang et al., 2016). Consider a function
f (x)
defined on a fractal subset
[a, b]
fractal
of dimension 0 < α ≤ 1, the Local Fractional Derivative is defined by
23
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Fractional calculus: a 300-year perspective on operators, applications, and modern generalizations
f
(α)
(x) = l
´
ım
x→x
0
∆
α
f (x)
(x − x
0
)
α
,
where
∆
α
f (x)
is a local fractional difference, which expresses the change of the function
f
taking into account its local
fractality. As an illustration, for the power function f (x) = (x − x
0
)
β
, we get
f
(α)
(x) =
Γ(β + 1)
Γ(β + 1 − α)
(x − x
0
)
β−α
, β > 0.
This derivative reduces to the classical derivative when α = 1, ensuring consistency with standard calculus.
4.2. Local Fractional Integrals
The respective local fractional integral is defined by
a
I
α
x
f (x) =
1
Γ(α )
Z
x
a
(x − t)
α−1
f (t) (dt)
α
, (Gao, 2003; Yang, 2012)
where
(dt)
α
stands for the fractal measure, taking into account the local dimension of the domain. This definition
enables the solving of local fractional differential equations (LFDEs) such as
f
(α)
(x) = λ f (x), f (a) = f
0
,
whose solution is expressed in terms of the local fractional exponential function:
f (x) = f
0
E
α
(λ(x − a)
α
)(Yang, 2012).
These results generalize classical ODE solutions to fractal domains, capturing anomalous growth and self-similar
dynamics.
4.3. Applications to Fractal Media
The theory of local fractional calculus was quickly applied to fractal heat conduction, diffusion in fractals, and analysis of
signals on fractal sets (Gao, 2003; Yang, 2012). For instance, the local fractional diffusion equation
∂
α
u(x, t)
∂t
α
= D
∂
2α
u(x, t)
∂x
2α
,
where 0
< α ≤
1, describes the process of diffusion in fractal media and predicts subdiffusive behavior consistent with
experimental observations in disordered materials. The exact solution for an initial delta distribution is given by
u(x, t) =
1
2
t
−α/2
W
α
|x|
t
α/2
,
where W
α
is a fractal Gaussian-like function.
4.4. Yang–Gao Deformable Local Fractional Operators
Further work by Yang and Gao generalizes the concept of local fractional derivatives into deformable operators 2018,
where θ ∈ [0, 1] is an adjustable parameter such that:
D
α
θ
f (x) = θ f
(α)
(x) + (1 − θ)D
α
R-L
f (x),
offering a hybrid model which allows incorporation of memory effects but keeps the fractal locality property intact.
The model made possible the application to control systems, fractured materials, and financial modeling on fractal time
(Gao, 2003).
# revista.bdlaciencia@utm.edu.ec 24
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
5. Variable-Order and Generalized Nonlocal Operators (2000–2017)
The early years of the 21st century witnessed the introduction of the concept of variable-order (VO) fractional derivatives
due to the need for models of complex phenomena, where the order of differentiation changes in accordance with time,
space or system state (Katugampola, 2014). This concept represents an extension of classical fractional calculus via the
introduction of the notion of dynamic memory effects, allowing more accurate modeling of viscoelasticity, diffusion,
and control systems.
5.1. Variable-Order Fractional Derivatives
Let
α(t)
denote a time-dependent order, with 0
< α(t) <
1. The variable-order Riemann–Liouville derivative is defined as:
D
α(t)
0+
f (t) =
1
Γ(1 − α(t))
d
dt
Z
t
0
(t − τ)
−α(t)
f (τ) dτ.
Similarly, the variable-order Caputo derivative is given by:
C
D
α(t)
0+
f (t) =
1
Γ(1 − α(t))
Z
t
0
(t − τ)
−α(t)
f
′
(τ) dτ.
These definitions generalize constant-order derivatives, reducing to classical Caputo or Riemann–Liouville derivatives
when α(t) is constant.
5.2. Generalized Fractional Operators
A generalized fractional integral and derivative that can interpolate between the Riemann-Liouville and Hadamard
operators has been proposed by Katugampola (2014). The generalized fractional integral of order
α >
0 is given by:
ρ
I
α
a+
f (t) =
ρ
1−α
Γ(α )
Z
t
a
τ
ρ−1
f (τ)
(t
ρ
− τ
ρ
)
1−α
dτ, ρ > 0.
Its corresponding generalized derivative is:
ρ
D
α
a+
f (t) =
t
1−ρ
d
dt
n
ρ
I
n−α
a+
f (t), n = ⌈ℜ(α)⌉(Katugampola, 2014).
This operator satisfies key properties:
• Linearity:
ρ
D
α
(a f + bg) = a
ρ
D
α
f + b
ρ
D
α
g
• Inverse relationship:
ρ
D
α ρ
I
α
f (t) = f (t)
• Interpolation: Reduces to Riemann–Liouville (ρ = 1) or Hadamard (ρ → 0) derivatives.
5.3. Results from Applications
5.3.1. Variable-Order Diffusion Equation
A variable-order diffusion equation takes the form:
C
D
α(t)
0+
u(x, t) = D
∂
2
u(x, t)
∂x
2
, u(x, 0) = δ(x),
where
D
denotes the diffusion coefficient. For
α(t) =
0
,
5
+
0
,
5
sin
πt
T
, numerical studies revealed dynamic
subdiffusion, characterized by a time-dependent mean square displacement obeying:
⟨x
2
(t)⟩ ∼ t
¯
α(t)
,
¯
α(t) =
1
t
Z
t
0
α(τ) dτ.
25
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Fractional calculus: a 300-year perspective on operators, applications, and modern generalizations
5.3.2. Generalized Katugampola Operators
In the case of scale invariant viscoelastic materials, the Katugampola operators have been employed to solve the
fractional relaxation equations 2014:
ρ
D
α
0+
f (t) + λ f (t) = 0, f (0) = f
0
.
The exact solution is:
f (t) = f
0
E
α
−λ
t
ρ
ρ
α
,
recovering classical Mittag–Leffler relaxation when
ρ =
1 (Riemann–Liouville) or Hadamard-type relaxation for
logarithmic scaling (ρ → 0).
5.4. Advantages and Implications
The advantages that variable-order and generalized operators provide include the following:
1. Versatility: Modeling processes involving memory varying with time or space.
2. Unified approaches: Obtaining traditional operators as particular cases.
3.
Improved physical meaning: Greater agreement with experiments in viscoelasticity, diffusion, and control problems.
4.
Computational convenience: Can be used in finite differences, finite elements, and spectral methods due to
smooth variation of order.
In recent years, there have been developments of unified theories of local fractional derivatives that are based on the
requirement to describe phenomena which are scale-invariant locally, non-differentiable, or fractal in nature (Anderson &
Ulness, 2016; Khalil et al., 2014; Yang et al., 2018).
5.5. Conformable Fractional Derivative
Khalil et al. (2014) introduced the conformable derivative, defined for 0 < α ≤ 1 as:
T
α
( f )(t) = l
´
ım
ϵ→0
f (t + ϵt
1−α
) − f (t)
ϵ
= t
1−α
f
′
(t), t > 0.
Properties:
• Linearity: T
α
(a f + bg) = aT
α
( f ) + bT
α
(g)
• Product Rule: T
α
( f g) = f T
α
(g) + gT
α
( f )
• Chain Rule: T
α
( f ◦ g)(t) = f
′
(g(t))T
α
(g)(t)
Applications: Solving differential equations with fractional order while maintaining classical initial conditions. For
instance, for the conformable fractional differential equation:
T
α
y(t) + λy(t) = 0, y(0) = y
0
,
the solution is:
y(t) = y
0
e
−λt
α
/α
.
5.6. Deformable and Extended Local Derivatives
Subsequent extensions have proposed deformable derivatives
D
α
µ
Anderson y Ulness (2016) that allow a deformation
parameter µ, so as to interpolate between the classical derivative (α = 1) and purely local fractional behavior (α < 1):
D
α
µ
f (t) = l
´
ım
ϵ→0
f (t + µϵt
1−α
) − f (t)
ϵ
.
Results from numerical simulations indicated that controlling
µ
allows fine-tuning of local scaling properties, critical
for fractured media and anomalous transport.
# revista.bdlaciencia@utm.edu.ec 26
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
6. Unified Local Fractional Derivatives and Modern Developments (2018–2025)
6.1. Generalized Unified Local Fractional Derivatives
Yang et al. (2018) developed a generalized unified local fractional derivative (ULFD), denoted as D
α,β
UL
, defined via:
D
α,β
UL
f (t) = l
´
ım
t→t
0
f (t) − f (t
0
)
(t − t
0
)
α
+ β f
′
(t
0
), 0 < α ≤ 1, β ∈ R.
Keys Results:
• For β = 0, it becomes classic local fractional derivative.
• For α = 1, it becomes the derivative.
• Offers an interpolation for local and classic derivatives that is continuous.
Applications: ULFDs were applied to local fractional diffusion equations:
D
α,β
UL
u(x, t) = k
∂
2
u(x, t)
∂x
2
, u(x, 0) = u
0
(x),
in which there was more accuracy in describing fractal transport than that provided by classical fractional models. In
particular, the mean square displacement behaved
⟨x
2
(t)⟩ ∼ t
α
+ βt,
demonstrating a hybrid scaling behavior combining local fractal and classical contributions.
6.2. Biparametric V-Derivative and Two-Parameter Generalizations
In recent years, Vivas-Cortez, Jarrín et al. (2025) y Vivas-Cortez, Velasco-Velasco y Jarrín (2025) have provided the
notion of biparametric V-derivative
V
ς,χ
, which represents an evolution from the ULFD concept. The novelty about
this operator resides in the fact that it makes possible to modify two parameters at once, namely the scaling exponent
locally (ς), and the interpolation with classical dynamics (χ).
The definition of the V−derivative operator for a real-value function f : R → R, with ς ≥ 0 and χ > 0, is given by:
V
ς,χ
( f (t)) := l
´
ım
h→0
(χ + h(χ − ς)) f
t + h
ς
χ
− χ f (t)
χ · h
.
Key Features:
•
Allows for the development of a versatile method of analysis of processes which exhibit local fractality and are
subject to effects from classical mechanics.
•
Preserves important properties including linearity, product rule, and chain rule, which make the process of
analysis easier.
•
Becomes an identity operator when
ς =
0 and adjusts to the scaled classical differentiation operator based on
parameter changes.
Applications: The V-derivative was used by the authors on generalized diffusion problems and provided enhanced
precision in modeling the hybrid scaling effects. Specifically, the mean squared displacement (MSD) for such systems
is described by:
⟨x
2
(t)⟩ ∼ t
ς
+ χt,
exhibiting a superposition between local fractional dynamics and classical dynamics. This provides greater flexibility
in characterizing the anomalous transport processes than the usual local fractional models.
27
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Fractional calculus: a 300-year perspective on operators, applications, and modern generalizations
6.3. Operational Calculus and Analytical Tools
To support ULFDs and V-derivatives, generalized Laplace transforms and operational calculi were introduced
Vivas-Cortez, Jarrín et al., 2025; Yang et al., 2018:
L{V
ς,χ
f (t)} = s
ς
F(s) −
n−1
∑
k=0
s
ς−1−k
f
(k)
(0) + χsF(s).
The research carried out by Vivas-Cortez et al. regarding the generalized Laplace transform Vivas-Cortez, Jarrín
et al., 2025, which formulates this transformation, is undergoing the process of review and provides a basic change
in the understanding of the derivative in order to reconcile both fractional and classic approaches. The described
above techniques enable the direct solution of linear differential equations using local fractional derivatives, classic
elements, and biparametric generalizations. The presented techniques are the generalization of the classic Laplace
technique.
7. Discussion and Perspectives
Fractional Calculus – a brief history of which – shows an interplay between abstraction and application. The first nonlocal
operators, such as Riemann-Liouville and Caputo derivatives, have incorporated memory through
RL
D
α
f (t) =
1
Γ(n − α)
d
n
dt
n
Z
t
0
f (τ)
(t − τ)
α−n+1
dτ, n − 1 < α < n,
C
D
α
f (t) =
1
Γ(n − α)
Z
t
0
f
(n)
(τ)
(t − τ)
α−n+1
dτ,
that succeeded in modeling viscoelastic materials, anomalous diffusion, and electric circuits. Numerical simulations based on
the three articles indicate that Caputo derivatives lead to better agreement with experiment because of their treatment of
initial conditions. This is demonstrated by solving the fractional relaxation equation:
C
D
α
y(t) + λy(t) = 0, y(0) = y
0
,
yields
y(t) = y
0
E
α
(−λt
α
),
where
E
α
represents the Mittag–Leffler function. The findings of Vivas-Cortez et al. (2021) reveal that an increase in
the value of
α
from 0.5 to 0.9 leads to a gradual shift in the decay process from subdiffusion to near-classical exponential
behavior.
7.1. Local Fractional Derivatives
The more recent ones such as local fractional derivatives presented in Yang et al. (2018) are centered on local scaling as
opposed to non-local memory:
D
α
loc
f (x) = l
´
ım
x→x
0
∆
α
f (x)
∆x
α
,
enabling the modeling of processes on fractal sets. In Katugampola (2014) the author shows that the solution of the local
fractional diffusion equation:
D
α
loc
u(x, t) = k
∂
2
u(x, t)
∂x
2
, u(x, 0) = u
0
(x),
produce numerical solutions that accurately capture anomalous dispersion behaviors for fractal domains with greater local
precision than conventional nonlocal derivative methods.
# revista.bdlaciencia@utm.edu.ec 28
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
7.2. Variable-Order and Generalized Operators
Flexibility is enhanced through the use of variable-order and generalized derivatives, whereby the order
α = α(t)
or
α = α(x, t):
RL
D
α(t)
f (t) =
1
Γ(n − α(t))
d
n
dt
n
Z
t
0
f (τ)
(t − τ)
α(t) −n+1
dτ.
Examples of how adaptively changing fractional orders provide better fits for complex diffusion processes are shown in
Vivas-Cortez, Jarrín et al. (2025). Numerical simulations show that variable-order differential equations can describe
transitions from subdiffusion to superdiffusion, emphasizing the necessity of choosing flexible orders.
7.3. Key Trends
The main trends in fractional calculus include:
1.
Trend 1: The memory of classical nonlocal operators is contrasted by the locality of local derivatives, which
reflect fractal properties at small scales.
2.
Trend 2: The use of operational calculus, Laplace transforms, and Mittag-Leffler functions allows analytical or
semi-analytical solutions.
3. Trend 3: Numerical techniques are also required, especially for the variable order and local operators.
4.
Trend 4: Contemporary fractional calculus aims to combine nonlocal and local operators into hybrid models
suitable for problems including global memory and local fractals.
5.
Trend 5: A wide variety of applications ranging from mechanical materials and transport phenomena in porous
media to electrical circuits and biology shows the versatility of fractional calculus.
8. Conclusions
In this survey, the historical development of fractional and fractal derivatives is described by mentioning:
•
Historical background: From the question raised by Leibniz in 1695 to the present day unification of classical
and variable-order derivatives.
•
Mathematical preliminaries: Essential operators such as Riemann–Liouville, Caputo, Grünwald–Letnikov,
Weyl, Hadamard, Marchaud, and Riesz derivatives, as well as explicit formulas and solutions of problems.
•
Practical implications: Classical and local operators can describe physical systems due to their ability to include
memory, anomalous diffusion and fractal effects within the framework.
•
Modern development: Unifying formulations, variable-order and operational approaches extend the
applicability and possibilities of the fractional calculus.
With historical approach in the literature review and mathematical formulas and numerical findings, this survey helps to
understand relations between operators and provides an easy way for further research. The next steps could be:
1. Combination of local and nonlocal derivatives for complicated media.
2. Estimation of fractional order from data in real time modeling.
3. Extension of operational calculus to multidimensional and stochastic fractional systems.
The combination of mathematical rigor, historical perspective, and application-driven results positions fractional calculus as
a continuously evolving field with broad interdisciplinary relevance.
9. Conflict of Interest
All authors contributed equally in the preparation of the present work taking into account articles and book cited,
writing draft preparation, writing review and editing.
29
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Fractional calculus: a 300-year perspective on operators, applications, and modern generalizations
10. References
Anderson, D. R., & Ulness, D. J. (2016). Deformable local fractional derivatives and applications. Fractional Calculus
and Applied Analysis, 19, 1050-1072.
Canavati, F. (1980). Fractional operators on Banach spaces. Journal of Mathematical Analysis and Applications, 76,
150-167.
Caputo, M. (1967). Linear models of dissipation whose Q is almost frequency independent. Geophysical Journal
International, 13, 529-539.
Coimbra, C. F. M. (2003). Mechanics with variable-order fractional derivatives. Annalen der Physik, 12, 692-703.
Gao, J. (2003). Local fractional differential equations and applications. Mathematical Problems in Engineering, 2003, 1
-
15.
Grünwald, A. K. (1867). Über die Potenzreihen mit beliebigen Exponenten. Journal für die reine und angewandte
Mathematik, 66, 253-264.
Grünwald, A. (1867). Über die Ableitung der Functionen einer reellen Veränderlichen beliebiger Ordnung. Zeitschrift
für Mathematik und Physik, 12, 441-480.
Hadamard, J. (1892). Essai sur l’étude des fonctions données par leur développement de Taylor. Journal de
Mathématiques Pures et Appliquées, 8, 101-186.
Katugampola, U. N. (2014). New approach to generalized fractional derivatives. Journal of Mathematical Analysis and
Applications, 399, 779-792.
Khalil, R., Horani, M. A., Yousef, A., & Sababheh, M. (2014). A new definition of fractional derivative. Journal of
Computational and Applied Mathematics, 264, 65-70.
Lazarevi´c, M., Rapai´c, M., & Šekara, T. (2014, enero). Introduction to Fractional Calculus with Brief Historical
Background.
Letnikov, V. (1868a). Mémoire sur les différentielles d’ordre fractionnaire. Matematicheskii Sbornik, 3(2), 197-234.
Letnikov, V. (1868b). Theory of differentiation of fractional order. Matematicheskii Sbornik, 3, 325-371.
Liouville, J. (1832). Sur le calcul des différentielles d’ordre quelconque. Journal de Mathématiques Pures et Appliquées,
17, 1-30.
Marchaud, A. (1927). Sur les dérivées et intégrales des fonctions quelconques. Journal de Mathématiques Pures et
Appliquées, 6, 337-389.
Riemann, B. (1876). Versuch einer allgemeinen Definition der Integrale von
x
α
dx
(H. Weber & R. Dedekind, Eds.)
[Escrito en 1847], 331-344.
Riesz, M. (1949). L’intégrale de Riemann-Liouville et le problème de Dirichlet. Acta Mathematica, 81, 1-223.
Vivas-Cortez, M., Guzmán, P. M., Lugo, L. M., & V., J. E. N. (2021). Fractional Calculus: Historical Notes (Apuntes
Históricos del Cálculo Fraccionario). Revista MATUA, 8(2), 35
-
45. https://revistas.uniatlantico.edu.co/index.
php/MATUA/article/view/3358
Vivas-Cortez, M., Jarrín, H. D., Moreno, F. O., & Velasco-Velasco, J. (2025). A Laplace transform approach to the
biparametric V-derivative with applications in generalized calculus. Applied Mathematics & Information
Sciences.
Vivas-Cortez, M., Velasco-Velasco, J., & Jarrín, H. D. (2025). A new generalized local derivative of two parameters.
Applied Mathematics & Information Sciences, 19(3), 713-723. https://doi.org/10.18576/amis/190319
# revista.bdlaciencia@utm.edu.ec 30
ISNN 2588-0764 Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
https: //revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
Weyl, H. (1917a). Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung. Vierteljahresschrift der
Naturforschenden Gesellschaft in Zürich, 62, 10-27.
Weyl, H. (1917b). Theorie der Darstellung der Funktionen durch Fourierreihen. Mathematische Annalen, 77, 313-370.
Yang, X.
-
J. (2012). Local Fractional Calculus and Its Applications (1.
a
ed.) [Monograph]. World Science Publisher, New
York.
Yang, X.
-
J., Baleanu, D., & Machado, J. A. T. (2018). Unified local fractional derivatives and applications. Advances in
Difference Equations, 2018, 1-18.
Yang, X.
-
J., Machado, J. A. T., Baleanu, D., & Gao, F. (2016). A new numerical technique for local fractional diffusion
equation in fractal heat transfer. Journal of Numerical Simulation in Applied Sciences, 9(10), 5621
-
5628. https:
//doi.org/10.22436/jnsa.009.10.09
11. Contributions from authors
Author Contribution
Norlen Melendez,
Miguel Vivas-Cortez,
Oramys Prieto-Arima
Conceptualization, manuscript preparation, and manuscript review,
Experimental section, literature review, data processing, and
manuscript review.
31
# revista.bdlaciencia@utm.edu.ec Vol. 11. Núm. 1 (18-31): Enero-Abril, 2026 DOI: 10.33936/revbasdelaciencia.v11i1.8109
