AccScience Publishing / IJOCTA / Volume 15 / Issue 3 / DOI: 10.36922/IJOCTA025110048
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RESEARCH ARTICLE

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

Safar Irandoust Pakchin1* ,  Mohammad Hossein Derakhshan1 ,  Shahram Rezapour2,3*
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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
IJOCTA 2025, 15(3), 535–548; https://doi.org/10.36922/IJOCTA025110048
Received: 10 March 2025 | Revised: 27 April 2025 | Accepted: 30 April 2025 | Published online: 23 July 2025
© 2025 by the Auhor(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

This paper introduces a robust distributed-order time-fractional telegraph model, incorporating Caputo time- and Riesz space-fractional derivatives. The spatial Riesz derivative is discretized using an optimized finite difference method. For the distributed-order fractional operator, the midpoint rule was first used to approximate the integral with respect to the order distribution, followed by the application of a finite difference scheme to approximate the Caputo time-fractional derivative. The method’s flexibility and high accuracy make it a valuable tool for modeling and simulating these systems, providing insights into the behavior of fractional-order systems with both temporal and spatial fractional effects. Additionally, the proposed approach outperforms existing numerical methods in terms of both precision and computational efficiency, making it highly applicable for real-world problems requiring accurate and efficient solutions. A comprehensive analysis of convergence and stability was conducted to validate the proposed numerical method. To demonstrate 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.

Keywords
Distributed-order
Finite difference method
Fractional derivative
Riesz fractional derivative
Stability analysis
Telegraph equations
Funding
The work was supported by the University of Tabriz, Iran (Grant No. 2070).
Conflict of interest
The authors declare that they have no competing interests.
References
  1. Azroul E, Kamali N, Ragusa MA, Shimi M. Variational methods for a p(x,.)-fractional bi-nonlocal problem of elliptic type. Rend Circ Mat Palermo, II. Ser. 2025;74(1):1-21. https://doi.org/10.1007/s12215-024-01156-7

 

  1. Alotaibi M, Jleli M, Ragusa MA, Samet B. On the absence of global weak solutions for a nonlinear time-fractional SchrN˜†dinger equation. Appl Anal. 2024;103(1):1-15. https://doi.org/10.1080/00036811.2022.2036335

 

  1. Kilbas AA, Srivastave HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam: Elsevier B.V.; 2006. ISBN:0444518320, 9780444518323

 

  1. Podlubny I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of their Applications, vol. 198. New York: Academic Press; 1998. ISBN:0080531989, 9780080531984

 

  1. Mollahasani N, Moghadam MM, Afrooz K. A new treatment based on hybrid functions to the solution of telegraph equations of fractional order. Appl Math Model. 2016; 40(4):2804-2814. https://doi.org/10.1016/j.apm.2015.08.020

 

  1. Wang R, Shi L, Wang B. Asymptotic behavior of fractional nonclassical diffusion equations driven by nonlinear colored noise on RN . Nonlinearity. 2019; 32(11): 4524-4556. https://doi.org/10.1088/1361-6544/ab32d7

 

  1. Yasin F, Afzal Z, Saleem MS, Jahangir N, Shang Y. Hermite–Hadamard type inequality for non- convex functions employing the Caputo–Fabrizio fractional integral. Res Math. 2024;11(1):2366164. https://doi.org/10.1080/27684830.2024.2366164

 

  1. Mohammed AA, Marasi HR, Derakhshan MH, Kumar P. An efficient numerical method for the time-fractional distributed order nonlinear Klein-Gordon equation with shifted fractional Gegenbauer multi-wavelets method. Phys Scr. 2023;98:084001. https://doi.org/10.1088/1402-4896/accedb

 

  1. Umarov S. Continuous time random walk models associated with distributed order diffusion equations. Fract Cal Appl Anal. 2015;18:821-837. https://doi.org/10.1515/fca-2015-0049

 

  1. Guo S, Mei L, Zhang Z. Time-fractional Gardner equation for ion-acoustic waves in negative- ion-beam plasma with negative ions and non- thermal nonextensive electrons. Phys Plasmas. 2015;22:052306. https://doi.org/10.1063/1.4919264

 

  1. Ansari A, Derakhshan MH, Askari H. Distributed order fractional diffusion equation with fractional Laplacian in axisymmetric cylindrical configuration. Commun Nonlinear Sci Numer Simul. 2022;113:106590. https://doi.org/10.1016/j.cnsns.2022.106590

 

  1. Derakhshan MH. Existence, uniqueness, Ulam-Hyers stability and numerical simulation of so- lutions for variable order fractional differential equations in fluid mechanics. J Appl Math Com- put. 2022;68(1):403-429. https://doi.org/10.1007/s12190-021-01537-6

 

  1. Irandoust-pakchin S, Abdi-mazraeh S, Fahimi- khalilabad I. Higher order class of finite difference method for time-fractional Liouville-Caputo and space-Riesz fractional diffusion equation. Filomat. 2024;38(2):505-521. https://doi.org/10.2298/FIL2402505I

 

  1. Irandoust-Pakchin S, Abdi-Mazraeh S, Khani A. Numerical solution for a variable-order fractional nonlinear cable equation via Chebyshev cardinal functions. Comput Math and Math Phys. 2017;57:2047-2056. https://doi.org/10.1134/S0965542517120120

 

  1. Fahimi-khalilabad I, Irandoust-pakchin S, Abdi- mazraeh S. High-order finite difference method based on linear barycentric rational interpolation for Caputo type sub-diffusion equation. Math Comput Simulat. 2022;199:60-80. https://doi.org/10.1016/j.matcom.2022.03.008

 

  1. Caputo M. Mean fractional-order-derivatives differential equations and filters. Ann Univ Ferrara. 1995;41:73-84. https://doi.org/10.1007/BF02826009

 

  1. Caputo M. Distributed order differential equations modelling dielectric induction and diffusion. Fract Calc Appl Anal. 2001;4:421-442. math.bas.bg/fcaa, www.diogenes.bg/fcaa.

 

  1. Caputo M. Diffusion with space memory modelled with distributed order space fractional differential equations. Ann Geophys. 2003;46(2). https://doi.org/10.4401/ag-3395

 

  1. Chechkin A, Gorenflo R, Sokolov I. Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys Rev E. 2002;66:046129. https://doi.org/10.1103/PhysRevE.66.046129

 

  1. Sokolov I, Chechkin A, Klafter J. Distributed- order fractional kinetics. Acta Physica Polonica B. 2004;35:1323-1341. https://doi.org/10.48550/arXiv.cond- mat/0401146

 

  1. Vieira N, Rodrigues MM, Ferreira M. Time- fractional telegraph equation of distributed order in higher dimensions. Commun Nonlinear Sci Numer Simul. 2021;102:105925. https://doi.org/10.1016/j.cnsns.2021.105925

 

  1. Ratner V, Zeevi YY. Denoising-enhancing images on elastic manifolds. IEEE Trans Image Process. 2011;20(8):2099-2109. https://doi.org/10.1109/TIP.2011.2118221

 

  1. Gorenflo R, Luchko Y, Stojanovi Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract Calc Appl Anal. 2013;16(2):297-316. https://doi.org/10.2478/s13540-013-0019-6

 

  1. Mainardi F, Pagnini G. The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. J Comput Appl Math. 2007;207(2):245-257. https://doi.org/10.1016/j.cam.2006.10.014

 

  1. Ahmed HF, Moubarak MRA, Hashem WA. Gegenbauer spectral tau algorithm for solving fractional telegraph equation with convergence analysis. Pramana - J Phys. 2021;95(2):1-16. https://doi.org/10.1007/s12043-021-02113-0

 

  1. Zhang Y, Qian J, Papelis C, Sun P, Yu Z. Im- proved understanding of bimolecular reactions in deceptively simple homogeneous media: from laboratory experiments to Lagrangian quantification. Water Resour Res. 2014;50(2):1704-1715. https://doi.org/10.1002/2013WR014711

 

  1. Sun HG, Chen W, Li C, Chen YQ. Fractional differential models for anomalous diffusion. Physica A: Stat Mech Appl. 2010;389(14):2719-2724. https://doi.org/10.1016/j.physa.2010.02.030

 

  1. Gao GH, Sun ZZ. Two difference schemes for solving the one-dimensional time distributed-order fractional wave equations. Numer Algorithms. 2017;74:675-697. https://doi.org/10.1007/s11075-016-0167-y

 

  1. Wang Z, Vong S. Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation. J Comput Phys. 2014;277:1-15. https://doi.org/10.1016/j.jcp.2014.08.012

 

  1. Ye H, Liu F, Anh V, Turner I. Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains. IMA J Appl Math. 2015;80(3):825-838. https://doi.org/10.1093/imamat/hxu015

 

  1. Jian HY, Huang TZ, Gu XM, Zhao YL. Fast second-order implicit difference schemes for time distributed–order and Riesz X.L. space fractional diffusion–wave equations. Comput Math Appl. 2021;94:136-154. https://doi.org/10.1016/j.camwa.2021.05.003

 

  1. Kharazmi E, Zayernouri M, Karniadakis G. Petrov-Galerkin and spectral collocation methods for distributed order differential equations. SIAM J Sci Comput. 2017;39:A1003-A1037. https://doi.org/10.1137/16M1073121

 

  1. Gao G, Sun Z. Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations. Comput Math Appl. 2015;69: 926-948. https://doi.org/10.1016/j.camwa.2015.02.023

 

  1. Duong P, Kwok E, Lee M. Deterministic analysis of distributed order systems using operational matrix. Appl Math Model. 2016;40:1929-1940. https://doi.org/10.1016/j.apm.2015.09.035

 

  1. Mashayekhi S, Razzaghi M. Numerical solution of distributed order fractional differential equations by hybrid functions. J Comput Phys. 2016;315:169-181. https://doi.org/10.1016/j.jcp.2016.01.041

 

  1. Abbaszadeh M, Dehghan M. Meshless upwind local radial basis function-finite difference technique to simulate the time-fractional distributed- order advection-diffusion equation. Eng Comput. 2021;37:873-889. https://doi.org/10.1007/s00366-019-00861-7

 

  1. Ye H, Liu F, Anh V. Compact difference scheme for distributed-order time-fractional diffusion- wave equation on bounded domains. J Comput Phys. 2015;298:652-660. https://doi.org/10.1016/j.jcp.2015.06.025

 

  1. Chen H, LA˜¼ S, Chen W. Finite difference/spectral approximations for the distributed order time fractional reaction-diffusion equation on an unbounded domain. J Comput Phys. 2016;315:84-97. https://doi.org/10.1016/j.jcp.2016.03.044

 

  1. Qiu W, Xu D, Chen H, Guo J. An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile- immobile equation in two dimensions. Comput Math Appl. 2020;80(12):3156-3172. https://doi.org/10.1016/j.camwa.2020.11.003

 

  1. Yang Z, Zheng X, Wang H. A variably distributed-order time-fractional diffusion equation: analysis and approximation. Comput Methods Appl Mech Eng. 2020;367:113118. https://doi.org/10.1016/j.cma.2020.113118

 

  1. Abdelkawy MA, Lopes AM, Zaky MA. Shifted fractional Jacobi spectral algorithm for solving distributed order  time-fractional reaction- diffusion    Comput  Appl Math. 2019;38:1-21. https://doi.org/10.1007/s40314-019-0845-1

 

  1. Ansari A, Derakhshan MH. On spectral polar fractional Laplacian. Math Comput Simul. 2023;206:636-663. https://doi.org/10.1016/j.matcom.2022.12.008

 

  1. Derakhshan MH, Aminataei A. A numerical method for finding solution of the distributed- order time-fractional forced Kortewegde Vries equation including the Caputo fractional derivative. Math Methods Appl Sci. 2022;45(5):3144- 3165. https://doi.org/10.1002/mma.7981

 

  1. Ahmed HF, Hashem WA. Improved Gegenbauer spectral tau algorithms for distributed- order time-fractional telegraph models in multi- dimensions. Numer Algorithms. 2023; 93(3):1013- 1043. https://doi.org/10.1007/s11075-022-01452-2

 

  1. Ilic M, Liu F, Turner I, Anh V. Numerical approximation of a fractional-in-space diffusion equation, I. Fract Calc Appl Anal. 2005;8(3):323-341. http://eudml.org/doc/11303

 

  1. Morgado ML, Rebelo M. Numerical approximation of distributed order reaction-diffusion equations. J Appl Math Comput. 2015; 275:216-227. https://doi.org/10.1016/j.cam.2014.07.029

 

  1. Moghaddam BP, Tenreiro Machado JA, Morgado ML. Numerical approach for a class of distributed order time fractional partial differential equations. Appl Numer Math. 2019;136:152-162. https://doi.org/10.1016/j.apnum.2018.09.019
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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