Mathematical modeling and simulation of traffic flow dynamics using nonlinear differential equations
Supriya Kumbharsupriyashirke679@gmail.comDepartment of Mechanical EngineeringIndira College of Engineering & ManagementPune, Maharashtra, 410506, IndiaView full profile → , Shriraghavan Madbushim.shriraghavan@gmail.comDepartment of Industrial Internet of ThingsSt. Vincent Pallotti College of Engineering & TechnologyNagpur, Maharashtra, 441108, IndiaView full profile → , Shraddha Gugulothushraddha26sangewar@gmail.comDepartment of Electronics EngineeringYeshwantrao Chavan College of Engineering (YCCE)Nagpur, Maharashtra, 441110, IndiaView full profile → , *B. YamunaCorresponding authoryamunab-maths@dayanandasagar.eduDepartment of MathematicsDayananda Sagar College of EngineeringBangalore, Karnataka, 560111, IndiaView full profile → , Varsha Yadavvarsha.yadav@rungta.orgDepartment of Computer Science and EngineeringRungta College of Engineering and TechnologyRungta International Skills UniversityBhilai, Chhattisgarh, 490024, IndiaView full profile → , Jyotsna Suryavanshijyotsna.suryavanshi@vit.eduDepartment of Engineering Science and HumanitiesVishwakarma Institute of TechnologyPune, Maharashtra, 411037, IndiaView full profile → , M. Sreenivasa Reddydypc@adityauniversity.inDepartment of Mechanical EngineeringAditya UniversitySurampalem, Andhra Pradesh, 533437, IndiaView full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Apr 2026
- Published Online:
- 27 Oct 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2832
- Pages:
- 3219–3229
Abstract
Interactions occur between vehicle density, velocity, traffic demand and roadway capacity, creating complex, nonlinear traffic flow dynamics. This research proposes a mathematical modelling and simulation approach to traffic flow analysis based on non-linear differential equations. Basic relations between traffic density, flow and velocity are derived together with the fundamental conservation law for traffic moving along road networks. Different initial and boundary conditions are used to simulate traffic states numerically, using solution techniques. The model goes on to study the propagation of a shock wave and stop and go traffic conditions resulting from obstruction and congestion and changes in traffic density. The simulation results presented are used to show the temporal and spatial changes of traffic characteristics, and to identify the change from free-flow to congested traffic. The framework proposed here is an effective mathematical basis for traffic prediction, traffic congestion analysis, and transportation management applications.
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References
[1] K. Aldwoah, F. A. Alqarni, O. Osman, L. M. Abdalgadir, A. Touati, and W. Adel, “Mathematical modeling and dynamical analysis of a nonlinear coupled stress-mitigation system with signed threshold-relative policy feedback and physics-informed neural network simulation,” Mathematics, vol. 14, no. 12, Art. no. 2231 (2026), doi: 10.3390/math14122231.
[2] A. Paliathanasis, “Symmetry analysis for the 2D Aw-Rascle traffic-flow model of multi-lane motorways in the Euler and Lagrange variables,” Symmetry, vol. 15, no. 8, Art. no. 1525 (2023), doi: 10.3390/sym15081525.
[3] R. M. Soliby and S. S. Jamaian, “Non-linearity flux of fractional transport density equation in traffic flow with solutions,” Smart Cities, vol. 5, no. 4, pp. 1655–1669 (2022), doi: 10.3390/smartcities5040084.
[4] S. V. Sadavarte, S. B. Sarwe, and R. V. Saraykar, “A special class of solutions for three dimensional steady state magnetohydrodynamics equations with nonzero boundary velocity,” Journal of Dynamical Systems and Geometric Theories, vol. 22, nos. 1–2, pp. 87–98 (2025), doi: 10.47974/JDSGT-2024-010.
[5] A. Kantaros, T. Ganetsos, E. Pallis, and M. Papoutsidakis, “From mathematical modeling and simulation to digital twins: Bridging theory and digital realities in industry and emerging technologies,” Applied Sciences, vol. 15, no. 16, Art. no. 9213 (2025), doi: 10.3390/app15169213.
[6] G. Nastasi, “Mathematical models and simulations,” Axioms, vol. 13, no. 3, Art. no. 149 (2024), doi: 10.3390/axioms13030149.
[7] T. Alghamdi, S. Mostafi, G. Abdelkader, and K. Elgazzar, “A comparative study on traffic modeling techniques for predicting and simulating traffic behavior,” Future Internet, vol. 14, no. 10, Art. no. 294 (2022), doi: 10.3390/fi14100294.
[8] M. L. Scutaru and C.-I. Pruncu, “Mathematical modeling and simulation in mechanics and dynamic systems, 2nd edition,” Mathematics, vol. 12, no. 2, Art. no. 341 (2024), doi: 10.3390/math12020341.
[9] L. Hong and L. Zhang, “Nonlinear dynamical model and analysis of emotional propagation based on Caputo derivative,” Mathematics, vol. 13, no. 13, Art. no. 2044 (2025), doi: 10.3390/math13132044.
[10] S. Y. Barykin, I. V. Kapustina, S. M. Sergeev, and V. K. Yadykin, “Algorithmic foundations of economic and mathematical modeling of network logistics processes,” Journal of Open Innovation: Technology, Market, and Complexity, vol. 6, no. 4, Art. no. 189 (2020), doi: 10.3390/joitmc6040189.
[11] M. Khachumov and V. Khachumov, “Modeling the solution of the pursuit–evasion problem based on the intelligent–geometric control theory,” Mathematics, vol. 11, no. 23, Art. no. 4869 (2023), doi: 10.3390/math11234869.
[12] E. Sebastian and R. C. Poonia, “Entropy diagnostics for cryptographic key material from random circuit sampling,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 29, no. 2-B, pp. 1023–1032 (2026), doi: 10.47974/JDMSC-2641.
[13] S. Cheon and J. Lee, “An enhanced hybrid-level interface-reduction method combined with an interface discrimination algorithm,” Mathematics, vol. 11, no. 23, Art. no. 4867 (2023), doi: 10.3390/math11234867.
[14] I. Fernández-Irisarri and M. Mañas, “Laguerre–Freud equations for the Gauss hypergeometric discrete orthogonal polynomials,” Mathematics, vol. 11, no. 23, Art. no. 4866 (2023), doi: 10.3390/math11234866.
[15] V. Tsybulin and P. Zelenchuk, “Predator–prey dynamics and ideal free distribution in a heterogeneous environment,” Mathematics, vol. 12, no. 2, Art. no. 275 (2024), doi: 10.3390/math12020275.
[16] C.-L. Chiu, P. Huang, M.-Y. Day, Y. Ni, and Y. Chen, “Mastery of ‘monthly effects’: Big data insights into contrarian strategies for DJI 30 and NDX 100 stocks over a two-decade period,” Mathematics, vol. 12, no. 2, Art. no. 356 (2024), doi: 10.3390/math12020356.
