Page 163 - IJOCTA-15-3

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.3, pp.535-548 (2025)
https://doi.org/10.36922/IJOCTA025110048

RESEARCH ARTICLE

A numerical method for solving distributed-order multi-term
time-fractional telegraph equations involving Caputo and Riesz
fractional derivatives

1
1*
Safar Irandoust Pakchin , Mohammad Hossein Derakhshan , and Shahram Rezapour 2,3*
1
Department of Applied Mathematics, Faculty of Mathematics, Statistics and Computer Sciences,
University of Tabriz, Tabriz, East Azerbaijan, Iran
2
Department of Mathematics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, Tabriz,
East Azerbaijan, Iran
3
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung,
Taiwan
s.irandoust@tabrizu.ac.ir, m.h.derakhshan.20@gmail.com, sh.rezapour@azaruniv.ac.ir

ARTICLE INFO ABSTRACT
Article History:
This paper introduces a robust distributed-order time-fractional telegraph
Received: March 10, 2025
model, incorporating Caputo time- and Riesz space-fractional derivatives.
Revised: April 27, 2025
The spatial Riesz derivative is discretized using an optimized finite difference
Accepted: April 30, 2025
Published Online: July 23, 2025 method. For the distributed-order fractional operator, the midpoint rule was
first used to approximate the integral with respect to the order distribution,
Keywords: followed by the application of a finite difference scheme to approximate the
Distributed-order Caputo time-fractional derivative. The method’s flexibility and high accuracy
Finite difference method make it a valuable tool for modeling and simulating these systems, providing
Fractional derivative insights into the behavior of fractional-order systems with both temporal and
Riesz fractional derivative spatial fractional effects. Additionally, the proposed approach outperforms
Stability analysis existing numerical methods in terms of both precision and computational effi-
Telegraph equations ciency, making it highly applicable for real-world problems requiring accurate
Subject Classification: and efficient solutions. A comprehensive analysis of convergence and stability
26A33; 65M06; 65M12; 35R11; was conducted to validate the proposed numerical method. To demonstrate
47B06; 65R20 its effectiveness, several numerical simulations were performed, revealing the
method’s exceptional accuracy and computational efficiency. Furthermore, a
comparison with existing numerical approaches from the literature is provided,
highlighting the proposed method’s superior performance in both precision and
practical applicability.

1. Introduction processes, especially those involving memory ef-
fects and spatial heterogeneity. 8–15 Many com-
Over the past few decades, there has been plex physical systems, such as diffusion in multi-
growing interest among researchers in model- fractal media, cannot be effectively described
ing and analyzing complex physical phenom- by single-order differential equations. There-
ena using fractional-order operators, particu- fore, it becomes essential to explore fractional-
larly in the fields of mathematical sciences and order and, particularly, distributed-order differ-
engineering. 1–7 The significant advantage of frac- ential models to account for the inherent com-
tional models, viewed as generalizations of clas- plexities of such phenomena. The study of
sical integer-order models, lies in their ability distributed-order time-fractional models began
to more accurately capture anomalous transport
*Corresponding Author
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