Numerical Schemes for Fractional Energy Balance Model of Climate Change with Diffusion Effects
Abstract
This study aims to propose numerical schemes for fractional time discretization of partial differential equations (PDEs). The scheme is comprised of two stages. Using von Neumann stability analysis, we ensure the robustness of the scheme. The energy balance model for climate change is modified by adding source terms. The local stability analysis of the model is presented. Also, the fractional model in the form of PDEs with the effect of diffusion is given and solved by applying the proposed scheme. The proposed scheme is compared with the existing scheme, which shows a faster convergence of the presented scheme than the existing one. The effects of feedback, deep ocean heat uptake, and heat source parameters on global mean surface and deep ocean temperatures are displayed in graphs. The current study is cemented by the fact-based popular approximations of the surveys and modeling techniques, which have been the focus of several researchers for thousands of years.
Mathematics Subject Classification:65P99, 86Axx, 35Fxx.
Doi: 10.28991/ESJ-2023-07-03-011
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References
Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K., Tignor, M., & Miller, H. (2007). IPCC fourth assessment report (AR4). Climate change 2007: The physical Science Basis. Cambridge University Press, Cambridge, United Kingdom.
Watson, R. T. (2001). Climate Change 2001: Synthesis Report. Contribution of Working Groups I, II, and III to the Third Assessment Report of the Intergovernmental Panel on Climate Change. IPCC 2001, Cambridge university Press, Cambridge, United Kingdom.
Aizebeokhai, A. P. (2009). Global warming and climate change: Realities, uncertainties and measures. International journal of physical sciences, 4(13), 868-879.
de Larminat, P. (2016). Earth climate identification vs. anthropic global warming attribution. Annual Reviews in Control, 42, 114–125. doi:10.1016/j.arcontrol.2016.09.018.
Mitchell, A. (2009). Sea sick: The global ocean in crisis. McClelland & Stewart Limited, Toronto, Canada.
Ogutu, B. K. Z. N. (2015). Energy balance mathematical model on climate change. Ph.D. Thesis, Université Pierre ET Marie University of Nairobi, Nairobi, Kenya.
Pierrehumbert, R. T. (2010). Principles of planetary climate. Cambridge University Press, New York, United States. doi:10.1017/CBO9780511780783.
Masih, A. (2018). An Enhanced Seismic Activity Observed Due To Climate Change: Preliminary Results from Alaska. IOP Conference Series: Earth and Environmental Science, 167, 012018. doi:10.1088/1755-1315/167/1/012018.
Malchow, H., Petrovskii, S. V., & Venturino, E. (2007). Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulation. CRC Press, New York, United States. doi:10.1201/9781482286137.
McCarty, J. P. (2001). Ecological Consequences of Recent Climate Change. Conservation Biology, 15(2), 320–331. doi:10.1046/j.1523-1739.2001.015002320.x.
Jonkers, L., Hillebrand, H., & Kucera, M. (2019). Global change drives modern plankton communities away from the pre-industrial state. Nature, 570(7761), 372–375. doi:10.1038/s41586-019-1230-3.
Rayner, N. A., Parker, D. E., Horton, E. B., Folland, C. K., Alexander, L. V., Rowell, D. P., Kent, E. C., & Kaplan, A. (2003). Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late nineteenth century. Journal of Geophysical Research: Atmospheres, 108(D14). doi:10.1029/2002jd002670.
Poloczanska, E. S., Brown, C. J., Sydeman, W. J., Kiessling, W., Schoeman, D. S., Moore, P. J., Brander, K., Bruno, J. F., Buckley, L. B., Burrows, M. T., Duarte, C. M., Halpern, B. S., Holding, J., Kappel, C. V., O’Connor, M. I., Pandolfi, J. M., Parmesan, C., Schwing, F., Thompson, S. A., & Richardson, A. J. (2013). Global imprint of climate change on marine life. Nature Climate Change, 3(10), 919–925. doi:10.1038/nclimate1958.
Houghton, E. (1996). Climate change 1995: The science of climate change: contribution of working group I to the second assessment report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, Unite Kingdom.
Metz, B., Davidson, O., Swart, R., & Pan, J. (2001). Climate change 2001: mitigation: contribution of Working Group III to the third assessment report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, Unite Kingdom.
Houghton, J. T., Ding, Y. D. J. G., Griggs, D. J., Noguer, M., Van der Linden, P. J., Dai, X., ... & Johnson, C. A. (2001). Climate Change 2001: The Scientific Basis. The Press Syndicate of the University of Cambridge, Cambridge, Unite Kingdom.
Rodenhouse, N. L., Matthews, S. N., McFarland, K. P., Lambert, J. D., Iverson, L. R., Prasad, A., Sillett, T. S., & Holmes, R. T. (2008). Potential effects of climate change on birds of the Northeast. Mitigation and Adaptation Strategies for Global Change, 13(5–6), 517–540. doi:10.1007/s11027-007-9126-1.
Contessa, G. M., Guardati, M., D’Arienzo, M., Poggi, C., Sandri, S., D’Auria, M. C., Genovese, E., & Cannatà, V. (2018). The impact of climate change on radiological practices in Italy. The European Physical Journal Plus, 133(9). doi:10.1140/epjp/i2018-12243-3.
Edwards, M., Johns, D. G., Licandro, P., John, A. W., & Stevens, D. P. (2008). Ecological Status Report: Results from the CPR Survey 2006/2007. SAHFOS Technical Report, 5, Plymouth, United Kingdom.
Worm, B., & Lotze, H. K. (2021). Marine biodiversity and climate change. Climate Change: Observed Impacts on Planet Earth (3rd Ed.), 2021, 445–464, Elsevier, Amsterdam, Netherlands. doi:10.1016/B978-0-12-821575-3.00021-9.
Din, A., Li, Y., Khan, T., & Zaman, G. (2020). Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China. Chaos, Solitons and Fractals, 141(110286). doi:10.1016/j.chaos.2020.110286.
Khan, F. M., Khan, Z. U., Lv, Y. P., Yusuf, A., & Din, A. (2021). Investigating of fractional order dengue epidemic model with ABC operator. Results in Physics, 24(103510). doi:10.1016/j.rinp.2021.104075.
Ghanbari, B., & Atangana, A. (2020). A new application of fractional Atangana–Baleanu derivatives: Designing ABC-fractional masks in image processing. Physica A: Statistical Mechanics and Its Applications, 542, 123516. doi:10.1016/j.physa.2019.123516.
Din, A., Li, Y., & Liu, Q. (2020). Viral dynamics and control of hepatitis B virus (HBV) using an epidemic model. Alexandria Engineering Journal, 59(2), 667–679. doi:10.1016/j.aej.2020.01.034.
Younis, M., Seadawy, A. R., Baber, M. Z., Husain, S., Iqbal, M. S., R. Rizvi, S. T., & Baleanu, D. (2021). Analytical optical soliton solutions of the Schrödinger-Poisson dynamical system. Results in Physics, 27(104369). doi:10.1016/j.rinp.2021.104369.
Sabatier, J. A. T. M. J., Agrawal, O. P., & Machado, J. T. (2007). Advances in fractional calculus. Springer, Dordrecht, Netherlands. doi:10.1007/978-1-4020-6042-7.
Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 73–85. doi:10.12785/pfda/010201.
Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model. Thermal Science, 20(2), 763–769. doi:10.2298/tsci160111018a.
Rahman, M. ur, Arfan, M., Shah, K., & Gómez-Aguilar, J. F. (2020). Investigating a nonlinear dynamical model of COVID-19 disease under fuzzy caputo, random and ABC fractional order derivative. Chaos, Solitons and Fractals, 140, 110232. doi:10.1016/j.chaos.2020.110232.
Kumar, T. S. (2021). Hybrid nanofluid slip flow and heat transfer over a stretching surface. Partial Differential Equations in Applied Mathematics, 4(100070). doi:10.1016/j.padiff.2021.100070.
Rahman, M. U., Arfan, M., Shah, Z., Kumam, P., & Shutaywi, M. (2021). Nonlinear fractional mathematical model of tuberculosis (TB) disease with incomplete treatment under Atangana - Baleanu derivative. Alexandria Engineering Journal, 60(3), 2845–2856. doi:10.1016/j.aej.2021.01.015.
Khan, M. A., & Atangana, A. (2020). Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alexandria Engineering Journal, 59(4), 2379–2389. doi:10.1016/j.aej.2020.02.033.
Liu, Y., Fan, E., Yin, B., & Li, H. (2020). Fast algorithm based on the novel approximation formula for the caputo - fabrizio fractional derivative. AIMS Mathematics, 5(3), 1729–1744. doi:10.3934/math.2020117.
Zhang, M., Liu, Y., & Li, H. (2019). High‐order local discontinuous Galerkin method for a fractal mobile/immobile transport equation with the Caputo–Fabrizio fractional derivative. Numerical Methods for Partial Differential Equations, 35(4), 1588–1612. doi:10.1002/num.22366.
Soldatenko, S., Bogomolov, A., & Ronzhin, A. (2021). Mathematical Modelling of Climate Change and Variability in the Context of Outdoor Ergonomics. Mathematics, 9(22), 2920. doi:10.3390/math9222920.
North, G. R. (1975). Analytical Solution to a Simple Climate Model with Diffusive Heat Transport. Journal of the Atmospheric Sciences, 32(7), 1301–1307. doi:10.1175/1520-0469(1975)032<1301:ASTASC>2.0.CO;2.
Budyko, M. I. (1969). The effect of solar radiation variations on the climate of the Earth. Tellus A: Dynamic Meteorology and Oceanography, 21(5), 611. doi:10.3402/tellusa.v21i5.10109.
Sellers, W. D. (1969). A global climatic model based on the energy balance of the earth-atmosphere system. Journal of Applied Meteorology and Climatology, 8(3), 392-400. doi:10.1175/1520-0450(1969)008<0392:AGCMBO>2.0.CO;2.
Podlubny, I. (1999). Fractional differential equations, mathematics in science and engineering. Academic Press, New York, United States.
Nawaz, Y., Arif, M. S., & Shatanawi, W. (2022). A New Numerical Scheme for Time Fractional Diffusive SEAIR Model with Non-Linear Incidence Rate: An Application to Computational Biology. Fractal and Fractional, 6(2), 78. doi:10.3390/fractalfract6020078.
Nawaz, Y., Arif, M. S., Shatanawi, W., & Ashraf, M. U. (2022). A new unconditionally stable implicit numerical scheme for fractional diffusive epidemic model. AIMS Mathematics, 7(8), 14299–14322. doi:10.3934/math.2022788.
Arif, M. S., Abodayeh, K., & Nawaz, Y. (2022). A Computational Approach to a Mathematical Model of Climate Change Using Heat Sources and Diffusion. Civil Engineering Journal (Iran), 8(7), 1358–1368. doi:10.28991/CEJ-2022-08-07-04.
Nawaz, Y., Arif, M. S., Abodayeh, K., & Bibi, M. (2022). Finite difference schemes for time-dependent convection q-diffusion problem. AIMS Mathematics, 7(9), 16407–16421. doi:10.3934/math.2022897.
DOI: 10.28991/ESJ-2023-07-03-011
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