An Enhanced Collocation Method for Multi-Term Fractional Differential Equations: Development and Application to Disease Dynamics
DOI:
https://doi.org/10.62054/ijdm/0203.02Keywords:
Fractional Differential Equations, Collocation Method, Caputo Derivative, Epidemiological Modeling, Monkeypox, Tuberculosis, Numerical Solution.Abstract
This paper presents an enhanced collocation-based numerical framework for solving mixed-order fractional differential equations (FDEs) by transforming them into an algebraic system using polynomial expansion and Gauss-Legendre collocation points. The method's high accuracy and robustness are validated through its application to fractional-order models of Monkeypox transmission and Tuberculosis dynamics, achieving perfect agreement with exact solutions and yielding zero error. The results demonstrate that this approach is a powerful and efficient computational tool for handling the memory-dependent characteristics inherent in complex scientific problems, especially in epidemiological modeling.
References
Abbas, S., & Mehdi, D. (2010). A new operational matrix for solving fractional order differential equations. Computers and Mathematics with Applications, 59(4), 1326–1336.
Ajileye, G., & James, A. A. (2023). Collocation method for the numerical solution of multi-order fractional differential equations. Journal of the Nigerian Society of Physical Sciences, 5, 1075.
Atkinson, K. E. (2008). The numerical solution of integral equations of the second kind. Cambridge University Press. https://doi.org/10.1016/j.apm.2006.10.025
Bhraway, A. H., Tohidi, E., & Soleymani, F. (2012). A new Bernoulli matrix method for solving high-order linear and nonlinear Fredholm integro-differential equations with piecewise intervals. Applied Mathematics and Computation, 219(9), 482–497.
Bolandtalat, A., Babolian, E., & Jafari, H. (2016). Numerical solution of multi-order fractional differential equations by Boubakar polynomial. Open Physics, 14(2), 226–230.
Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer-Verlag. https://doi.org/10.1007/978-3-642-14574-2
Ercan, C., & Kharerah, T. (2013). Solving a class of Volterra integral systems by the differential transform method. International Journal of Nonlinear Sciences, 16, 87–91.
El-Kady, M., & Biomy, M. (2010). Efficient Legendre pseudospectral method for solving integral and integro-differential equations. Communications in Nonlinear Science and Numerical Simulation, 15(7), 1724–1739.
Fadugba, S. E. (2019). Solution of fractional-order equations in the domain of the Mellin transform. Journal of the Nigerian Society of Physical Sciences, 4, 138–142. https://doi.org/10.46481/jnsps.2019.31
Gegele, D. A., Evans, O. P., & Akoh, D. (2014). Numerical solution of higher-order linear Fredholm integro-differential equations. American Journal of Engineering Research, 3(3), 243–247.
Issa, H. A., & Saleh, R. (2017). Numerical solutions of fractional differential equations by a fractional Taylor method. Journal of Fractional Calculus and Applications, 8(1), 1–10.
James, A., Ojobo, S. O., & Danjuma, A. M. (2025). A collocation-based framework for the computational solution of mixed-order fractional differential equations. International Journal of Development Mathematics, 2(1), 60–74. https://doi.org/10.62054/ijdm/0201.05
Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier. https://doi.org/10.1016/S1874-5717(06)80002-1
Miller, K. S., & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley-Interscience.
Peter, O. J., Kumar, S., Kumari, N., Oguntolu, F. A., Oshinubi, K., & Musa, R. (2022). Transmission dynamics of Monkeypox virus: a mathematical modelling approach. Modeling Earth Systems and Environment, 8(3), 3423–3434. https://doi.org/10.1007/s40808-021-01313-2
Podlubny, I. (1999). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. San Diego: Academic Press.
Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers.
Shahooth, K. M., Hasan, F. A., & Fadhil, M. S. (2016). Analytical and numerical solutions of fractional differential equations by using Adomian decomposition method. International Journal of Applied Mathematical Research, 5(2), 54–63.
Uwaheren, O. A., Adebisi, A. F., & Taiwo, O. A. (2020). Perturbed collocation method for solving singular multi-order fractional differential equations of Lane-Emden type. Journal of the Nigerian Society of Physical Sciences, 3, 141–148. https://doi.org/10.46481/jnsps.2020.69
Wazwaz, A. M., & El-Sayed, S. M. (2001). A new modification of the Adomian decomposition method for linear and nonlinear operators. Applied Mathematics and Computation, 181(1), 393–404
Downloads
Published
Issue
Section
License
Copyright (c) 2025 International Journal of Development Mathematics (IJDM)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors are solely responsible for obtaining permission to reproduce any copyrighted material contained in the manuscript as submitted. Any instance of possible prior publication in any form must be disclosed at the time the manuscript is submitted and a
copy or link to the publication must be provided.
The Journal articles are open access and are distributed under the terms of the Creative
Commons Attribution-NonCommercial-NoDerivs 4.0 IGO License, which permits use,
distribution, and reproduction in any medium, provided the original work is properly cited.
No modifications or commercial use of the articles are permitted.
