THERMAL SCIENCE

International Scientific Journal

Tian Qi Zhang

,

Yin-Shan Yun

,

Yan Hong Bai

LIE SYMMETRY CLASSIFICATION AND REDUCTION OF THE RIEMANN-LIOUVILLE FRACTIONAL RDCE WITH FOUR ARBITRARY FUNCTIONS

ABSTRACT
The Lie symmetry classification of the fractional order reaction-diffusion-convection equation (RDCE) with four arbitrary functions in the sense of Riemann-Liouville fractional derivative is carried out by using the Lie symmetry analysis method. It is noteworthy that the equation retains two symmetries when it contains four arbitrary functions. When four arbitrary functions are substituted for concrete functions, the resulting equation exhibits enhanced symmetry. However, it is imperative to acknowledge that the Lie algebra space that encompasses fractional order RDCE is a sub-space of the Lie algebra of the integer order RDCE. In summary, it has been demonstrated that the initial equation is converted into a fractional ODE by the corresponding symmetry when four arbitrary functions are substituted with specific functions. This provides a foundation for further research, which may enhance our understanding of certain phenomena in life.
KEYWORDS
fractional order RDCE, Riemann-Liouville fractional derivative, Lie symmetry, symmetry classifications
PAPER SUBMITTED: 2024-08-10
PAPER REVISED: 2025-05-01
PAPER ACCEPTED: 2025-05-15
PUBLISHED ONLINE: 2026-04-12
DOI REFERENCE: https://doi.org/10.2298/TSCI2602169Z
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2026, VOLUME 30, ISSUE No. 2, PAGES [1169 - 1177]
REFERENCES
[1] Feng, G. Q.. A Circular Sector Vibration System in a Porous Medium: A Fractal-Fractional Model and He's Frequency Formulation, Facta Universitatis Series: Mechanical Engineering, 23 (2025), 2, pp. 377- 385
[2] Hendy, M. H., et al., A Problem in Fractional Order Thermo-Viscoelasticity Theory for a Polymer Micro-Rod with and without Energy Dissipation, Advances in Differential Equations and Control Processes, 31 (2024), 4, pp. 583-607
[3] He, C.-H., et al., A Fractal-Based Approach to the Mechanical Properties of Recycled Aggregate Concretes, Facta Universitatis, Series: Mechanical Engineering, 22 (2024), 2, pp. 329-342
[4] Lei, X. H., He, J.-H., Frontiers in Thermal Science Driven by Artificial Intelligence, Thermal Science, 29 (2025), 3A, pp. 1671-1677
[5] Rahioui, M., et al., Lie Symmetry Analysis and Conservation Laws for the Time Fractional Generalized Advection-Diffusion Equation, Computational and Applied Mathematics, 42 (2023), 50
[6] Ren, R. C., Zhang, S. L., Invariant Analysis, Conservation Laws, and Some Exact Solutions for (2+1)- Dimension Fractional Long-Wave Dispersive System, Computational and Applied Mathematics, 39 (2020), 4, pp. 4125-4135
[7] Sun, H., et al., A New Collection of Real World Applications of Fractional Calculus in Science and Engineering, Communications in Nonlinear Science and Numerical Simulation, 64 (2018), Nov., pp. 213- 231
[8] Guo, B. L., Fractional Partial Differential Equations and Their Numerical Solutions, Science Press, Beijing, China, 2015
[9] Wang, G. W., Xu, T. Z., Invariant Analysis and Exact Solutions of Nonlinear Time Fractional Sharma-Tasso-Olver Equation by Lie Group Analysis, Nonlinear Dynamics, 76 (2014), 1, pp. 571-580
[10] San, S., Yasar, E., On the Lie Symmetry Analysis, Analytic Series Solutions, and Conservation Laws of the Time Fractional Belousov-Zhabotinskii System, Nonlinear Dynamics, 109 (2022), 4, pp. 2997-3008
[11] Olver, P. J., Applications of Lie Groups to Differential Equations, Springer, New York, USA, 1993
[12] Bluman, G. W., et al., Applications of Symmetry Methods to Partial Differential Equations, Springer New York, USA, 2010
[13] Ovsiannikov, L. V., Group Analysis of Differential Equations, Academic Press, New York, USA, 1982
[14] Tian, Y., Symmetry Reduction a Promising Method for Heat Conduction Equations, Thermal Science, 23 (2019), 4, pp. 2219-2227
[15] Wang, Y., Dong, Z., Symmetry Analysis of a (2+1)-D System, Thermal Science, 22 (2018), 4, pp. 1811-1822
[16] Cherniha, R., et al., A Complete Lie Symmetry Classification of a Class of (1+2)-Dimensional Reaction-Diffusion-Convection Equations, Communications in Nonlinear Science and Numerical Simulation, 92 (2021), 105466
[17] Zuhal, K., et al., On Exact Solutions for New Coupled Nonlinear Models Getting Evolution of Curves in Galilean Space, Thermal Science, 23 (2019), Suppl. 1, pp. S227-S233
[18] Temuer, C. L., et al., A Mechanized Algorithm for Determining the Structural Constants of Symmetric Lie Algebras of Differential Equations Based on Wu's Method (in Chinese), Scientia Sinica (Mathematica), 49 (2019), 5, pp. 751-764
[19] Tian, Y., Wang, L. K., Polynomial Characteristic Method an Easy Approach to Lie Symmetry, Thermal Science, 24 (2020), 4, pp. 2629-2635
[20] Gaur, M., Singh, K., On Group Invariant Solutions of Fractional Order Burgers-Poisson Equation, Applied Mathematics and Computation, 244 (2014), Oct., pp. 870-877
[21] Dorjgotov, K., et al., Lie Symmetry Analysis of a Class of Time Fractional Nonlinear Evolution Systems, Applied Mathematics and Computation, 329 (2018), July, pp. 105-117
[22] Gazizov, K. R., et al., Symmetry Properties of Fractional Diffusion Equations, Physica Scripta, 136 (2009), 5
[23] Zhang, Z. Y., Symmetry Determination and Nonlinearization of a Nonlinear Time-Fractional Partial Differential Equation, Proceedings of the royal society A-Mathematical Physical and Engineering Sciences, 476 (2020), 2233, 20190564
[24] Zhang, Z. Y., Liu C. B., Leibniz-Type Rule of Variable-Order Fractional Derivative and Application to Build Lie Symmetry Framework, Applied Mathematics and Computation, 430 (2022), 127268
[25] Hejazi, S. R., et al., Anisotropic Nonlinear Time-Fractional Diffusion Equation with a Source Term: Classification via Lie Point Symmetries, Analytic Solutions and Numerical Simulation, Applied Mathematics and Computation, 391 (2021), 125652
[26] Prakash, P., et al., Initial Value Problem for the (2+1)-Dimensional Time-Fractional Generalized Convection-Reaction-Diffusion Wave Equation: Invariant Subspaces and Exact Solutions, Computational and Applied Mathematics, 41 (2022), 30
[27] Zhang, Z. Y., Zheng, J., Symmetry Structure of Multi-Dimensional Time-Fractional Partial Differential Equations, Nonlinearity, 34 (2021), 8, pp. 5186-5212
PDF VERSION [DOWNLOAD]
Lie symmetry classification and reduction of the Riemann-Liouville fractional RDCE with four arbitrary functions

© 2026 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence