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RESEARCH ARTICLE

Fractional order convection-diffusion systems show chaos under exponential cubic B-spline discretization

Carlos Lizama1 ,  Majeed Ahmad Yousif2 ,  Rahel Qadir3 ,  Dumitru Baleanu4 ,  Pshtiwan Othman Mohammed5*
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1 Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencia, Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago , Chile
2 Department of Mathematics, College of Education, University of Zakho, Duhok , Iraq
3 Department of Mathematical Science, College of Basic Education, University of Sulaimani, Sulaymaniyah , Iraq
4 Department of Computer Science and Mathematics, Lebanese American University, Beirut , Lebanon
5 Department of Mathematics, College of Education, University of Sulaimani, Sulaymaniyah , Iraq
IJOCTA, 026350230 https://doi.org/10.36922/IJOCTA026350230
Received: 25 August 2026 | Revised: 30 September 2026 | Accepted: 1 October 2026 | Published online: 4 November 2026
© 2026 by the Author(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

The dynamical behavior of nonlinear fractional convection-diffusion equations (CDEs) is very rich and extends beyond their applicability as transport models, but it has yet to be explored numerically. In this paper, it is demonstrated that a time- and space-fractional nonlinear convection-diffusion equation exhibits a transition from stable equilibrium to deterministic chaos its nonlinear convection parameter increases by employing an exponential cubic B-spline scheme for this said problem. The Caputo fractional derivative is discretized using the L1 formula, whereas the fractional spatial operators are approximated using the exponential spline method. Using phase portraits, bifurcation diagrams, and Lyapunov exponent spectra, we identify three different dynamical behaviors: a stable regime, periodic behavior, and finally chaos, arising from the increasing value of the nonlinear convection parameter. Therefore, this work establishes the nonlinear convection parameter as a true bifurcation parameter and demonstrates the possibility of complex nonlinear dynamics in fractional convection-diffusion systems.

Graphical abstract
Keywords
Time-space fractional convection-diffusion equation
Exponential cubic B-spline
Stability
Convergence
Bifurcation
Lyapunov exponents
Funding
None.
Conflict of interest
The authors declare they have no competing interests.
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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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