MATHEMATICAL MODELING OF INFLATION PROCESSES IN THE ECONOMY USING DIFFERENTIAL EQUATIONS WITH FRACTIONAL DERIVATIVES
DOI:
https://doi.org/10.17721/1728-2667.2026/228-1/7Keywords:
inflationary (deflationary) processes in the economy, price index, unemployment rate, Caputo's fractional derivatives, and differential equations with these derivativesAbstract
B a c k g r o u n d . When studying rapidly changing inflationary processes in economics, theoretical methods based on ordinary differential equations or partial differential equations are often used. However, as demonstrated in this paper, in certain cases it is more appropriate to use the apparatus of differential equations with fractional derivatives. This is due to the presence of various types of nonlinearities in functional relationships within inflationary processes, the influence of parameter values from previous time points on current values, the existence of scaling relations, and so on. In fact, all these characteristics are inherent to fractional calculus.
M e t h o d s . The article is devoted to the application of differential equations with fractional derivatives of Caputo for the analysis of inflationary (deflationary) processes in the economy, based on the method of measuring inflation using the consumer price index, which takes into account changes in prices for a certain set of goods and services. This is demonstrated by changes in the specified index over finite time periods.
R e s u l t s . It is shown that the use of fractional order differential equations can be useful for building flexible tools for forecasting inflation/deflation processes. The relationship between the inflation rate and the unemployment rate is also investigated.
C o n c l u s i o n s . It has been established that the change in the fractional derivative index in theoretical models of economic processes allows describing different regimes of price dynamics – from moderate inflation to galloping and hyperinflation, as well as complex deflationary scenarios. The appearance of negative values of price indices for individual goods can be interpreted as a consequence of their excess production, which leads to a loss of market value. The proposed method of using differential equations with fractional values of the order of derivatives provides an expansion of the possibilities of modeling a wide range of economic processes.
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References
Afrouzi, H. (2024). Strategic inattention, inflation dynamics, and the nonneutrality of money. Journal of Political Economy, 132(10), 3378–3420. https://doi:10.1086/730201
Alam M. (2020). Analysis of partial differential equation models in macroeconomics. International Journal of Advanced Academic Studies, 2(3), 839–841. https://doi:10.33545/27068919.2020.v2.i3l.724
Ashish, R. (2025). System of fractal-fractional differential equations and Bernstein wavelets: A comprehensive study of environmental, epidemiological, and financial applications. Physica Scripta, 100, 025236. https://doi.org/10.1088/1402-4896/ada592
Awa, T., & Ndolane, S. (2020). Model of economic growth in the context of fractional derivative. Alexandria Engineering Journal, 59, 4843–4850. https://doi:10.1016/j.aej.2020.08.047
Badi′k, A., & Fečkan, M. (2021). Applying fractional calculus to analyze final consumption and gross investment influence on GDP. Journal of Applied Mathematics and Stochastic Analysis, 17(1), 65–72. https://doi:10.2478/jamsi-2021-0004
Bossone, B. (2019). The portfolio theory of inflation (and policy effectiveness). Economics Discussion Papers, 2019–29. Kiel Institute for the World Economy. https://doi.org/10.1515/ECON-2022-0032
Caputo, M., & Mainardi, F. (1971). Linear models of dissipation in inelastic solids. Rivista del Nuovo Cimento, 1, 161–198. https://doi:10.1007/BF02820620
Cheow, Y. H., Ng, K. H., Phang, C., & Ng, K. H. (2024). The application of fractional calculus in economic growth modelling: An approach based on regression analysis. Heliyon, 10(15), e35379. https://doi.org/10.1016/j.heliyon.2024.e35379
Conrad, C. A. (2022). Applied macroeconomics: A practical introduction. Springer. https://doi.org/10.1007/978-3-658-39315-1
Dąbrowski, M. A., Janus, J., & Mucha, K. (2025). Shades of inflation targeting: Insights from fractional integration (MPRA Paper No. 123455). University Library of Munich. https://doi.org/10.58116/UEK/KI15DF
Dhibyendu, B., & Chandra, R. (2025). Modern macroeconomics theory, application, and sustainability. Routledge. https://doi.org/10.4324/9781003501138
Dinh, D. V. (2020). Impulse response of inflation to economic growth dynamics: VAR model analysis. Journal of Asian Finance, Economics and Business, 7(9), 219–228. https://doi.org/10.13106/jafeb.2020.vol7.no9.219
Ferraris, L. (2025). Macroeconomic theory. Springer. https://doi.org/10.1007/978-3-031-88740-6
Ifeacho, O., & González-Parra, G. (2025). Mathematical modeling of economic growth, corruption, employment and inflation. Mathematics, 13, 1102. https://doi.org/10.3390/math13071102
Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations. Elsevier.
Kocapor, E., Valério, D., & Radonjić, L. (2025). The application of fractional calculus in modelling economic growth in Serbia. Fractal and Fractional, 9(6), 384. https://doi.org/10.3390/fractalfract9060384
Kubba, A. A. D. H., & Abdou, S. E. (2025). Developing mathematical models for analyzing financial market dynamics using partial differential equation. Journal of Research in Applied Mathematics, 11(5), 46–52. https://doi.org/10.35629/0743-11054652
Lemishovskyi, V., & Dumych, N. (2024). Features of inflation processes in Ukraine: Problems and causes. Bulletin of the National University "Lviv Polytechnic", Problems of Economics and Management, 8(1), 107–121. https://doi:10.23939/semi2024.01.107
Lester, R. (2023). Macroeconomics: Theory and applications (Loose-leaf version). Cengage Learning.
Luo, D., Wang, J. R., & Fečkan, M. (2018). Applying fractional calculus to analyze economic growth modelling. Journal of Applied Mathematics and Stochastic Analysis, 14(1), 25–36. https://doi:10.2478/jamsi-2018-0003
Maksyuta, N. V., Koshcheev, V. P., & Panina, T. A. (2012). Justification of the use of a fractional order diffusion equation in the theory of channeling using the Langevin approach. Poverkhnost, 12, 81–88. https://doi.org/10.1134/S1027451012040143
Md, A. (2020). Analysis of partial differential equation models in macroeconomics. International Journal of Advanced Academic Studies, 2(3), 839–841.
Ming, H., Wang, J., & Fečkan, M. (2019). The application of fractional calculus in Chinese economic growth models. Mathematics, 7(8), 665. https://doi.org/10.3390/math7080665
Moza, G., Brandibur, O., & Găină, A. (2023). Dynamics of a fourdimensional economic model. Mathematics, 11, 797. https://doi.org/10.3390/math11040797
Muhamad, D. J., Supriatna, A. K., Rusyaman, E., & Saputra, J. (2021). Application of fractional differential equation in economic growth model: A systematic review approach. AIMS Mathematics, 6(9), 10266–10280. https://doi:10.3934/math.2021594
Navarro, C. E. B., & Tomé, R. M. B. (2022). Structural stability analysis in a dynamic IS-LM-AS macroeconomic model with inflation expectations. International Journal of Differential Equations. Article ID 5026061.https://doi.org/10.1155/2022/5026061
Neumann, B. A. (2022). Essential stationary equilibria of mean field games with finite state and action space. Mathematical Social Sciences, 120, 85–91.https://doi:10.1016/j.mathsocsci.2022.09.006
Rasouli, S. M. M., de Oliveira Costa, E. W., Moniz, P., & Jalalzadeh, S. (2022). Inflation and fractional quantum cosmology. Fractal and Fractional, 6(11), 655. https://doi.org/10.3390/fractalfract6110655
Saeidi, H., Hejazi, S. R., & Mohammadi, S. (2025). Modeling and simulation of non-linear fractional-order chaotic system of supply chain and financial model. International Journal of Finance & Managerial Accounting, 12(44), 1–20. https://doi.org/10.30495/ijfma.2024.77500.2101
Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional integrals and derivatives: Theory and applications. Gordon & Breach.
Shumin, W., & Jianwei, W. (2024). Optimization study of microeconomics model under the integration of partial differential equations. Journal of Electrical Systems, 20(9s), 343–348.
Tamimi, O., & Orbán, I. (2020). Hyperinflation and its impact on the financial results. Intellectual Economics, 14(2), 5–16. https://doi.org/10.13165/IE-20-14-2-01
Tarasov, V. E. (2020a). Mathematical economics: Application of fractional calculus. MDPI. https://doi.org/10.3390/books9783039361199
Tarasov, V. E. (2020b). Non-linear macroeconomic models of growth with memory. Mathematics, 8, 2078. https://doi.org/10.3390/math8112078
Tarasova, V. V., & Tarasov, V. E. (2017). Economic interpretation of fractional derivatives. Progress in Fractional Differentiation and Applications, 3(1), 1–6. https://doi:10.18576/pfda/030101
Tsoularis, A. (2021). On some important ordinary differential equations of dynamic economics. In Advances in the solution of nonlinear differential equations. https://doi.org/10.5772/intechopen.97130
Vilenskyi, P. L., Lyvshits, V. N., & Smolyak, S. A. (2011). Evaluation of the effectiveness of investment projects: Theory and practice. Delo.
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