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Examples and Counterexamples
journal homepage: www.elsevier.com/locate/exco
An example of Disease-induced fractal–fractional dynamic system for potato
spots
Shaher Momani
a,b
, Rabha W. Ibrahim
c ,∗
a
Department, of Mathematics, The University of Jordan, Amman, Jordan
b
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, United Arab Emirates
c
Information and Communication Technology Research Group, Scientific Research Center, Al-Ayen University, Thi-Qar, Iraq
A R T I C L E I N F O
Communicated by F. Nestler
Keywords:
Fractional calculus
Fractal–fractional calculus
Fractional differential operator
Fractional dynamic system
Fractal–fractional differential equation
A B S T R A C T
The biological significance of fractal–fractional calculus resides to analyzing and describing complex biological
systems (like the dynamic system of potato spots). These systems frequently have erratic patterns or processes
with memory and long-term dependencies. We seek an example to enhance models of dynamic system of
potato spots.
1. Introduction
The transmission and repercussions of a disease on potato crops
necessitates a number of mathematical and biological factors. The
majority of models have traditionally been based on approaches for
connecting plant virus disease dynamics to large data sets of variables
(popularly referred or figured to have any impact on the disease cycle)
and datasets of variables which include host growth and development
of plants, damage to crops, cropping practices, and vector biology. The
easy accessibility of observing and laboratory data acquired in crop
species characterizes the development and application of these models,
which are frequently used to anticipate or historical study disease
dynamics (Taylor et al. [1] and Jeger et al. [2]). Another sort of model
is based on an abstract representation of what drives disease dynamics,
which is then mathematically expressed and studied using typical math-
ematical or analytic approaches. These models, which are the major
emphasis other, are conceptual in character and are not immediately
tied to the instant accessibility of necessary data, but serve a variety of
functions and may thereafter be evaluated versus information (a review
effort is given in [3]). In this effort, we present a generalization of the
dynamic system of potato spots using the concept of fractal–fractional
operators. Analysis of the suggested system is presented with examples.
2. Methods and models
Consider the differentiable function () over (0,b). The following
is the definition of the Riemann–Liouville fractal–fractional differential
∗
Corresponding author.
E-mail addresses: s.momani@ajman.ac.ae (S. Momani), rabhaibrahim@yahoo.com (R.W. Ibrahim).
operator: for , ∈( −1,], (see [4–6])
,
() ∶=
1
( − )
∫
0
( )( − )
−−1
,
where
()
= lim
→
()− ( )
−
. Considering three fractal–fractional
parameters, the following generalization is proposed: for ,, ∈( −
1,]
,,
()
=
∫
0
( )( − )
−−1
( − )
,
where
()
= lim
→
()−
( )
−
. The mathematical equation for
the fractal–fractional integral operators of order , > 0 and ,, > 0
are as described below: for , > 0,
,
( )=
() ∫
0
−1
( )( − )
−1
and for ,, > 0
,,
( ) ∶=
() ∫
0
−1
( )( − )
−1
.
A model of potato spots is shown by
()
=−()() (2.1)
https://doi.org/10.1016/j.exco.2025.100204
Received 8 October 2024; Received in revised form 8 April 2025; Accepted 7 October 2025
Examples and Counterexamples 8 (2025) 100204
Available online 10 October 2025
2666-657X/© 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-
nc-nd/4.0/).