In collaboration with Payame Noor University and the Iranian Society of Instrumentation and Control Engineers

Document Type : Research Article


1 Department of Mathematics‎, ‎Faculty of Sciences‎, ‎University of Gonabad‎, ‎Gonabad‎, ‎Iran.

2 Department of Mathematics, Fasa Branch, Islamic Azad university, Fasa, Iran.


‎This paper aims to demonstrate the flexibility of mathematical models in analyzing carbon dioxide emissions and account for memory effects. ‎The use of real data amplifies the importance of this study‎. ‎This research focuses on developing a mathematical model utilizing fractional-order differential equations to represent carbon dioxide emissions stemming from the energy sector. By comparing simulation results with real-world data, it is determined that the fractional model exhibits superior accuracy when contrasted with the classical model‎. ‎Additionally‎, ‎an optimal control strategy is proposed to minimize the levels of carbon dioxide, CO2, and associated implementation costs‎. ‎The fractional optimal control problem is addressed through the utilization of an iterative algorithm‎, ‎ and the effectiveness of the model is verified by presenting comparative results.


Main Subjects

[1] Agrawal, O.P., Defterli, O., Baleanu, D. (2016). “Fractional optimal control problems with several state and control variables”, Journal of Vibration and Control, 16, 1967-1976.
[2] Akhavan Ghassabzadeh, F., Tohidi, E., Singh, H., Shateyi, S. (2021). “RBF collocation approach to calculate numerically the solution of the nonlinear system of qFDEs”, Journal of King Saud University - Science, 33(2), 101288.
[3] Bagherpoorfard, M., Akhavan Ghassabzade, F. (2023). “Analysis and optimal control of a fractional MSD model”, Iranian Journal of Numerical Analysis and Optimization, 13(3), 481-499.
[4] Baleanu, D., Akhavan Ghassabzade, F., Nieto, J.J., Jajarmi, A. (2022). “On a new and generalized fractional model for a real Cholera outbreak”, Alexandria Engineering Journal, 61, 9175-9186.
[5] Baleanu, D., Arshad, S., Jajarmi, A., Shokat, W., Akhavan Ghassabzade, F., Wali, M. (2023). “Dynamical behaviours and stability analysis of a generalized fractional model with a real case study”, Journal of Advanced Research, 48, 157-173.
[6] BP Statistical Review of World Energy, (2019), 68th Edition, London: BP. Accessed 14 March 2020.
[7] Ge, W., Li, H., Wen, X., Li, Ch., Cao, H., Xing, B. (2023). “Mathematical modelling of carbon emissions and process parameters optimisation for laser welding cell”, International Journal of  Production Research, 61, 15, 5009-5028.
[8] Han, L., Sui, H., Ding, Y. (2022). “Mathematical modeling and stability analysis of a delayed carbon absorption-emission model associated with China’s adjustment of industrial structure”, Mathematics, 10(17), 3089.
[9] IEA. (2019). “Global energy & CO2 status report 2019”,, Accessed 14 March, 2020.
[10] Khan, Q., Suen, A., Khan, H., Kumam, P. (2023). “Comparative analysis of fractional dynamical systems with various operators”, AIMS Mathematics, 8(6), 13943-13983.
[11] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. (2006). “Theory and applications of fractional differential equations”, Elsevier Science, BV, Amsterdam.
[12] Lin, B., Zhu, J. (2019). “The role of renewable energy technological innovation on climate change: Empirical evidence from China”, Science of the Total Environment, 659, 1505-1512.
[13] Prakash, D.V, Sarvesh, D., Devendra, K., Jagdev, S. (2021). “A computational study of fractional model of atmospheric dynamics of carbon dioxide gas”, Chaos Solitons Fractals, 142, 110375.
[14] Srivastava, H.M., Dubey, V.P., Kumar, R., Singh, J., Kumar, D., Baleanu, D. (2020). “An efficient computationalapproachforafractional-orderbiologicalpopulationmodelwithcarryingcapacity”, Chaos, Solitons and Fractals, 138, 109880.
[15] Traore, A., Sene, N. (2020). “Model of economic growth in the context of fractional derivative”, Alexandria Engineering Journal, 59(6), 4843-4850.
[16] Verma, M., Misra, A.K. (2018). “Optimal control of anthropogenic carbon dioxide emissions through technological options: a modeling study”, Computational and Applied Mathematics, 37, 605-626.
[17] Verma, M., Verma, A.K., Misra, A.K. (2021). “Mathematical modeling and optimal control of carbon dioxide emissions from energy sector”, Environment Development and Sustainability, 23, 13919-13944.