Page 98 - IJOCTA-15-1

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.1, pp.92-102 (2025)
https://doi.org/10.36922/ijocta.1702

RESEARCH ARTICLE

On a robust stability criterion in the Cattaneo–Hristov diffusion
equation

´
1
Ra´ul Temoltzi-Avila , Javier Temoltzi-Avila 2*
1 ´
Area Acad´emica de Matem´aticas y F´ısica, Universidad Aut´onoma del Estado de Hidalgo,
Pachuca–Tulancingo Km 4.5, Mineral de la Reforma, 42184, Hidalgo, M´exico
2 ´
Area Industrial El´ectrica y Electr´onica, Universidad Tecnol´ogica del Norte de Guanajuato, Avenida
Educaci´on Tecnol´ogica, 34, Fracc. Universidad, Dolores Hidalgo C. I. N., 37800, Guanajuato, M´exico
temoltzi@uaeh.edu.mx, javier.avila@utng.edu.mx

ARTICLE INFO ABSTRACT

Article History: The aim of this paper is to establish a robust stability criterion in the Cattaneo–
Received: 7 October 2024 Hristov diffusion equation moving over an interval under the influence of heat
Accepted: 19 December 2024 sources. The robust stability criterion arises as a generalization of the definition
Available Online: 24 January 2025 of stability under constant-acting perturbations that is employed in systems of
differential equations. The criterion obtained allows to ensure that the solution
Keywords:
of the Cattaneo–Hristov diffusion equation and its first partial derivatives with
Cattaneo–Hristov diffusion equation
respect to the longitudinal axis and with respect to time can be bounded by
Caputo–Fabrizio fractional derivative
a constant whose value is defined a priori. The criterion is illustrated by a
Reachability tube
numerical example.
Robust stability
AMS Classification 2010:
35A09; 42A16; 34A08; 34D10; 93D09

1. Introduction and gradually its importance is emerging in var-
ious perspectives of our daily life: most of the
It is well known that the use of traditional inte- complex natural phenomena are bounded by frac-
5–7
ger calculus may be insufficient to adequately cap- tional order systems; see e.g.
ture certain phenomena that occur in some math-
8
ematical models, and as an alternative the use of In particular, Caputo and Fabrizio claim in that
fractional calculus has been adopted to provide fractional differential equations with non-singular
more adequate tools to overcome such complex- kernel contribute to the study of the behavior of
ity; see e.g. 1–3 In this sense, fractional calculus dissipative phenomena in materials that appear
has generated interest in different applications in in modern technology. In general, the Caputo–
mathematics, mechanics, physics, electronics, en- Fabrizio fractional derivative is useful to study
gineering, etc., due to its ability to describe inter- some real-world problems; see e.g. 9–12
mediate processes, memory phenomena, heredi-
tary properties and complex phenomena; see e.g. 4 The relation between the Caputo–Fabrizio frac-
The methods and techniques of fractional calculus tional derivative and the Cattaneo heat diffusion
have been used to model various practical prob- equation was established by Hristov in 14 using a
lems. Some real-world phenomena are explained non-singular Jeffrey kernel (exponential memory
by fractional order differential equations, espe- kernel), which gave rise to the Cattaneo–Hristov
cially in the fields of applied sciences and engi- diffusion equation in which the concept of fading
neering. In the last decades, several mathemati- memory is used. This diffusion equation describes
cians have contributed to this area of research, transient heat conduction with a damping term
*Corresponding Author
92