Document Type : Research Article

Authors

1 Department of Mathematics, Urmia Branch, Islamic Azad University, Urmia, Iran

2 Faculty of Science, Department of Mathematics, University of Zakho, Iraq

3 Faculty of Science, Department of Mathematics, Firat University, Elazig, Turkey

4 Department of Mathematics and Statistics, Washington State University, Pullman, WA, USA

Abstract

In this article, the price adjustment equation has been proposed and studied in the frame of fractal calculus which plays an important role in market equilibrium. Fractal time has been recently suggested by researchers in physics due to the self-similar properties and fractional dimension. We investigate the economic models from the viewpoint of local and non-local fractal Caputo derivatives. We derive some novel analytical solutions via the fractal Laplace transform. In fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard computational sense, and the non-local fractal Caputo fractal derivative is a generalization of the non-local fractional Caputo derivative. The economic models involving fractal time provide a new framework that depends on the dimension of fractal time. The suggested fractal models are considered as a generalization of standard models that present new models to economists for fitting the economic data. In addition, we carry out a comparative analysis to understand the advantages of the fractal calculus operator on the basis of the additional fractal dimension of time parameter, denoted by \$alpha\$, which is related to the local derivative, and we also indicate that when this dimension is equal to \$1\$, we obtain the same results in the standard fractional calculus as well as when \$alpha\$ and the nonlocal memory effect parameter, denoted by \$gamma\$, of the nonlocal fractal derivative are both equal to \$1\$, we obtain the same results in the standard calculus.

Keywords

References
[1] B. B. Mandelbrot, The fractal geometry of nature, New York: WH freeman,1983.
[2] K. Falconer, Fractal geometry: mathematical foundations and applications, John Wiley & Sons, 2004.
[3] A. A. Kirillov, A tale of two fractals, Birkhauser Basel, 2013.
[4] J. M. Blackledge, A. K. Evans, M. J. Turner, Fractal Geometry: Mathematical Methods, Algorithms, Applications, Elsevier, 2002.
[5] Y. B. Pesin, Dimension theory in dynamical systems: contemporary views and applications, University of Chicago Press, 2008.
[6] E. E. Peters, Fractal market analysis: applying chaos theory to investment and economics (Vol. 24), John Wiley & Sons 1994.
[7] G. Edgar, Measure, topology, and fractal geometry, Springer Science & Business Media, 2007.
[8] T. G. Dewey, Fractals in molecular biophysics, Oxford University Press, 1998.
[9] L. Pietronero, E. Tosatti, Fractals in physics, Elsevier; 2012.
[10] M. Fernandez-Martnez, M. A. Sanchez-Granero, Fractal dimension for fractal structures, Topology and its Applications, 163, (2014), 93-111.
[11] F. Family, T. Vicsek, Dynamics of fractal surfaces, World Scienti c Publishing Company, 1991.
[12] P. R. Massopust, Fractal functions, fractal surfaces, and wavelets, Academic Press, 2016.
[13] A. D. Ieva, ed., The fractal geometry of the brain, New York: Springer, 2016.
[14] G. A. Losa, D. Merlini, T. F. Nonnenmacher, E. R. Weibel, Fractals in Biology and Medicine, Birkhauser Basel, 2005.
[15] J. A. Kaandorp, P. Prusinkiewicz, Fractal Modelling: Growth and Form in Biology, Springer, 1994.
[16] U. Freiberg, M. Zahle, Harmonic calculus on fractals-a measure geometric approach I., Potential Anal, 16(2002), 265-77.
[17] R.S. Strichartz, Differential Equations on Fractals: A Tutorial; Princeton University Press: Princeton, NJ, USA, 2006.
[18] K. Falconer, Techniques in Fractal Geometry; John Wiley and Sons: Hoboken, NJ, USA, 1997.
[19] M. T. Barlow, E. A. Perkins, Brownian motion on the Sierpinski gasket, Probab.Theory Relat. Fields, 79,(1988), 543623.
[20] F. H. Stillinger, Axiomatic basis for spaces with noninteger dimension, J. MathPhys., 18,(1977), 12241234.
[21] A. S. Balankin, A continuum framework for mechanics of fractal materials I: From fractional space to continuum with fractal metric, Eur. Phys. J. B, 88,(2015), 90.
[22] M. Zubair, M. J. Mughal, Q.A. Naqvi, Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer: New York, NY, USA, 2012.
[23] L. Nottale, J. Schneider, Fractals and nonstandard analysis, J. Math. Phys.25,(1998), 12961300.
[24] T. G. Dewey, Fractals in molecular biophysics, Oxford University Press, 1998.
[25] M. Czachor, Waves along fractal coastlines: Fromfractal arithmetic to wave equations, Acta Phys. Pol. B 50 (2019), 813831.
[26] J. Kigami, Analysis on Fractals, Cambridge University Press, 2001.
[27] V.E. Tarasov, Fractional dynamics: applications of fractional calculus to dynamics of particles,  elds and media, Springer Science Business Media, 2011.
[28] T. Sandev, Z. Tomovski, Fractional Equations and Models: Theory and Applications, Springer Nature, 2019.
[29] S. Das, Functional fractional calculus, Springer Science Business Media, 2011.
[30] V. V. Uchaikin, Fractional derivatives for physicists and engineers Vol. 1 Background and theory Application, vol. 2, Springer, Berlin (2013)
[31] A. A. Iliasov, M. I. Katsnelson, S. Yuan, Hall conductivity of a Sierpinski carpetPhy. Rev. B, 101(4), (2020), 045413.
[32] J. Wu, C. Wang, Fractal Stokes' Theorem Based on Integration on Fractal Manifolds, Fractals, 2019 https://doi.org/10.1142/S0218348X20500103
[33] M. Bohner, A.C. Peterson, eds., Advances in dynamic equations on time scales. Springer Science & Business Media, 2002.
[34] A. Parvate, A.D. Gangal, Calculus on fractal subsets of real line-I: formulation, Fractals, 17 (2009), 53-81.
[35] A. Parvate, A.D. Gangal, Calculus on fractal subsets of real line II: Conjugacy with ordinary calculus, Fractals 19(03) (2011), 271-290.
[36] A. K. Golmankhaneh, A. Fernandez, A. K. Golmankhaneh, D. Baleanu, Diffusionon middle- Cantor sets, Entropy, 20(504), (2018), 1-13.
[37] A. K. Golmankhaneh, D. Baleanu, Non-local Integrals and Derivatives on Fractal Sets with Applications, Open Physics, 14(1) (2016), 542-548.
[38] A. K. Golmankhaneh, A review on application of the local fractal calculus, Num.Com. Meth. Sci. Eng., 1(2) (2019), 57-66.
[39] A. K. Golmankhaneh, D. Baleanu, Diffraction from fractal grating Cantor sets, Journal of Modern Optics, 63(14) (2016), 1364-9.
[40] A. K. Golmankhaneh, A.S. Balankin, Sub-and super-diffusion on Cantor sets: Beyond the paradox, Phys. Lett. A., 382(14) (2018), 960-7.
[41] A. K. Golmankhaneh, C. Tunc, Stochastic differential equations on fractal sets, Stochastics, 4 (2019), 1-7, https://doi.org/10.1080/17442508.2019.1697268.
[42] A. K. Golmankhaneh, A. Fernandez, Random Variables and Stable Distributions on Fractal Cantor Sets, Fractal Fract. 3(2) 31,(2019), 1-13, https://doi.org/10.3390/fractalfract3020031.
[43] A. K. Golmankhaneh, C. Cattani, Fractal Logistic Equation, Fractal Fract , 3(3) 41,(2019) https://doi.org/10.3390/fractalfract3030041
[44] A. K. Golmankhaneh, C. Tunc, Sumudu transform in fractal calculus, Appl.Math. Comput., 350, (2019), 386-401.
[45] A. Parvate, A. D. Gangal, Fractal differential equations and fractal-time dynamical systems, Pramana, 64(3), (2005), 389-409.
[46] M. F. Shlesinger, Fractal time in condensed matter, Annu. Rev. Phys. Chem., 39(1),(1988), 269-290.
[47] M. F. Shlesinger, Williams-Watts dielectric relaxation: a fractal time stochastic process, J. Stat. Phys., 36(5-6), (1984), 639-648.
[48] G. Braden, Fractal Time: The Secret of 2012 and a New World Age: Hay House Inc, (2010).
[49] K.Welch, A Fractal Topology of Time: Deepening into Timelessness, Fox Finding Press; 2nd Edition, 2020.
[50] A. K. Golmankhaneh, S. Ashra , D. Baleanu, A. Fernandez, Brownian Motion on Cantor Sets, Int.J. Nonlin. Sci. Num, 2020, DOI: https://doi.org/10.1515/ijnsns-2018-0384.
[51] A. C. Chiang, Fundamental methods of mathematical economics, 1984.
[52] A. Takayama, Mathematical economics, Cambridge University Press, 1985.
[53] D. Cohen-Vernik, A. Pazgal, Price adjustment policy with partial refunds, J. Retail, 93(4)(2017), 507-26.
[54] V.E. Tarasov, On History of Mathematical Economics: Application of Fractional Calculus, Mathematics,7 (2019) 509.
[55] I. Tejado, E. Perez, D. Valerio, Fractional Derivatives for Economic Growth Modelling of the Group of Twenty: Application to Prediction, Mathematics, 8(1), (2020) 50.
[56] B. Acay, E. Bas, T. Abdeljawad, Fractional economic models based on market equilibrium in the frame of different type kernels, Chaos, Solitons & Fractals, 130 (2020), 109438.
[57] A. Giusti, A comment on some new de nitions of fractional derivative. Nonlinear Dyn., 93(3),(2018) 1757-1763.
[58] K. Diethelm, R. Garrappa, A. Giusti, M. Stynes, Why fractional derivatives with nonsingular kernels should not be used, Fract. Calc. Appl. Anal., 23(3),(2020) 610-634.
[59] V.E. Tarasov, No nonlocality. No fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 62,(2018) 157-163.
[60] V.E. Tarasov, V.V. Tarasova, Macroeconomic models with long dynamic memory: Fractional calculus approach, Appl. Math. Comput., 338,(2018) 466-486.
[61] R. DiMartino, W. Urbina, On Cantor-like sets and Cantor-Lebesgue singular functions, arXiv preprint arXiv:1403.6554 (2014).
[62] V.E. Tarasov, Mathematical Economics: Application of Fractional Calculus, Mathematics, 8 (2020) 660.
[63] V.V. Tarasova, V.E. Tarasov, Concept of Dynamic Memory in Economics, Commun. Nonlinear Sci. Numer. Simul., 55 (2018) 127-145.
[64] V.E. Tarasov, Fractional econophysics: Market price dynamics with memory effects, Physica A: 557(1) (2020) 124865.
[65] M. Takayasu, H. Takayasu, Fractals and economics. Complex Systems in Finance and Econometrics, (2009) 444-463.
[66] H. Takayasu, M., Takayasu, M.P. Okazaki, K. Marumo, T. Shimizu, Fractal properties in economics. arXiv preprint (2000) cond-mat/0008057.
[67] La Torre, D., Marsiglio, S., Privileggi, F. Fractal attractors in economic growth models with random pollution externalities, Chaos, 28 (2018) 055916.