Abstract
Establishing the accurate relationship between fractional calculus and fractals is an important research content of fractional calculus theory. In the present paper, we investigate the relationship between fractional calculus and fractal functions, based only on fractal dimension considerations. Fractal dimension of the Riemann–Liouville fractional integral of continuous functions seems no more than fractal dimension of functions themselves. Meanwhile fractal dimension of the Riemann–Liouville fractional differential of continuous functions seems no less than fractal dimension of functions themselves when they exist. After further discussion, fractal dimension of the Riemann–Liouville fractional integral is at least linearly decreasing and fractal dimension of the Riemann–Liouville fractional differential is at most linearly increasing for the Hölder continuous functions. Investigation about other fractional calculus, such as the Weyl-Marchaud fractional derivative and the Weyl fractional integral has also been given elementary. This work is helpful to reveal the mechanism of fractional calculus on continuous functions. At the same time, it provides some theoretical basis for the rationality of the definition of fractional calculus. This is also helpful to reveal and explain the internal relationship between fractional calculus and fractals from the perspective of geometry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barkai, E., Metzler, R., Klafter, J.: From continuous time random walks to the fractional Fokker–Planck equation. Phys. Rev. E (3), 61, 132–156 (2000)
Bush, K. A.: Continuous functions without derivatives. Am. Math. Mon., 59, 222–225 (1952)
Butera, S., Paola, M. D.: A physically based connection between fractional calculus and fractal geometry. Ann. Physics, 350, 146–158 (2014)
Butzer, P. L., Kilbas, A. A., Trujillo, J. J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl., 270, 1–15 (2002)
Cui, X. X., Xiao, W.: What is the effect of the Weyl fractional integral on the Hölder continuous functions? Fractals, 29, 2150026 (2021)
Falconer, K. J.: Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Chicheste, 1990
Ferrari, F.: Weyl and Marchaud derivatives: A forgotten history. Mathematica, 6, 1–25 (2018)
Fu, H., Wu, G. C., Yang, G.: Continuous-time random walk to a general fractional Fokker–Planck equation on fractal media. Eur. Phys. J.-Spec. Top., 230, 3927–3933 (2021)
Hadamard, J.: Essai sur l’étude des fonctions données par leur développement de Taylor. Journal de Mathématiques Pures et Appliquées, 8, 101–186 (1892)
Hu, T. Y., Lau, K. S.: The sum of Radamacher functions and Hausdorff dimension. Math. Proc. Cambridge, 108, 91–103 (1990)
Hu, T. Y., Lau, K. S.: Fractal dimensions and singularities of the Weierstrass type functions. Trans. Amer. Math. Soc., 335, 649–665 (1993)
Katugampola, U. N.: New approach to a generalized fractional integral. Appl. Math. Comput., 218, 860–865 (2011)
Katugampola, U. N.: New approach to generalized fractional derivatives. J. Math. Anal. Appl., 6, 1–15 (2014)
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and applications of fractional differential equations. North Holland Math. Studies, Vol. 204, Elsevier, Amsterdam, 2006
Kilbas, A. A., Titioura, A. A.: Nonlinear differential equations with Marchaud–Hadamard-type fractional derivative in the weighted sapce of summable functions. Math. Model. Anal., 12, 343–356 (2007)
Kiryakova, V.: A brief story about the operators of the generalized fractional calculus. Fract. Calc. Appl. Anal., 11, 875–885 (2008)
Kolwankar, K. M., Gangal, A. D.: Fractional differentiability of nowhere differentiable functions and dimensions. Chaos Solitons Fractals, 6, 505–513 (1996)
Kolwankar, K. M., Gangal, A. D.: Local fractional Fokker–Planck Equation. Phys. Rev. Lett., 80, 214–217 (1998)
Liang, Y. S.: The relationship between the Box dimension of the Besicovitch functions and the orders of their fractional calculus. Appl. Math. Comput., 200, 197–207 (2008)
Liang, Y. S.: On the fractional calculus of Besicovitch function. Chaos Solitons Fractals, 42, 2741–2747 (2009)
Liang, Y. S.: Box dimensions of Riemann–Liouville fractional integrals of continuous functions of bounded variation. Nonlinear Anal., 72, 4304–4306 (2010)
Liang, Y. S.: Fractal dimension of Riemann–Liouville fractional integral of 1-dimensional continuous functions. Fract. Calc. Appl. Anal., 21, 1651–1658 (2018)
Liang, Y. S.: Estimation of fractal dimensions of Weyl fractional integral of certain continuous functions. Fractals, 28, 2050030 (2020)
Liang, Y. S.: Progress on estimation of fractal dimensions of fractional calculus of continuous functions. Fractals, 27, 1950084 (2019)
Liang, Y. S., Liu, N.: Fractal dimensions of Weyl–Marchaud fractional derivative of certain one-dimensional functions. Fractals, 27, 1950114 (2019)
Liang, Y. S., Su, W. Y.: Riemann–Liouville fractional calculus of 1-dimensional continuous functions. Sci. China Math., 46, 423–438 (2016)
Liang, Y. S., Su, W. Y.: Fractal dimensions of fractional integral of continuous functions. Acta Math. Sin. (Engl. Ser.), 32, 1494–1508 (2016)
Liang, Y. S., Su, W. Y.: Von Koch curves and their fractional calculus. Acta Math. Sin. (Chin. Ser.), 54, 227–240 (2011)
Liang, Y. S., Su, W. Y.: Fractal dimension of certain continuous functions of unbounded variation. Fractals, 25, 1750009 (2017)
Liang, Y. S., Wang, H. X.: Upper Box dimension of Riemann–Liouville fractional integral of fractal functions. Fractals, 29, 2150015 (2021)
Liang, Y. S., Zhang, Q.: 1-dimensional continuous functions with uncountable unbounded variation points. Chinese Journal of Comtemporary Mathematics, 39, 129–136 (2018)
Liang, Y. S., Zhang, Q., Yao, K.: Fractal dimension of fractional calculus of certain interpolation functions. Chinese Journal of Comtemporary Mathematics, 38, 93–100 (2017)
Machado, J. T., Mainardi, F., Kiryakova, V.: Fractional Calculus: Quo Vadimus? (Where are we going?) Fract. Calc. Appl. Anal., 18, 495–526 (2015)
Mandelbrot, B. B.: The Fractal Geometry of Nature, Freeman, San Francisco, 1982
Miller, K. S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1976
Mu, L., Yao, K., Qiu, H., et al.: The Hausdorff dimension of Weyl–Marchaud fractional derivative of a type of fractal functions. Chin. Ann. Math. Ser. B, 38, 257–264 (2017)
Mu, L., Yao, K., Wang, J.: Box dimension of Weyl fractional integral of continuous functions with bounded variation. Anal. Theory Appl., 32, 174–180 (2016)
Navascués, M. A.: Fractal polynomial interpolation. J. Math. Anal. Appl., 24, 401–418 (2005)
Navascués, M. A.: Fractal approximation. Complex Anal. Oper. Theory, 4, 953–974 (2010)
Nigmatullin, R. R., Baleanu, D.: New relationships connecting a class of fractal objects and fractional integrals in space. Fract. Calc. Appl. Anal., 16, 1–26 (2013)
Nigmatullin, R. R., Baleanu, D.: Relationships between 1D and space fractals and fractional integrals and their applications in physics. Handbook of Fractional Calculus with Applications, 4, 183–219 (2019)
Oldham, K. B., Spanier, J.: The Fractional Calculus, Academic Press, New York, 1974
Patzschke, N., Zähle, M.: Fractional differentiation in the self-affine case III–the density of the cantor set. Proc. Amer. Math. Soc., 117, 137–144 (1993)
Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, 1999
Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal., 5, 367–386 (2002)
Pooseh, S., Almeida, R., Torres, F. M.: Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim., 33, 301–319 (2012)
Ross, B.: Fractional calculus and its applications. Proceedings of the International Conference Held at the University of New Haven, June 1974, Lecture Notes in Math., Vol. 457, Springer, Berlin, 1975
Ruan, H. J., Su, W. Y., Yao, K.: Box dimension and fractional integral of linear fractal interpolation functions. J. Approx. Theory, 161, 187–197 (2009)
Samko, S. G., Kilbas, A. A., Marichev, O. I.: Integrals and Derivatives of Fractional Order and Some of Their Applications, Naukai Tekhnika, Minsk, 1987
Shen, W. X.: Hausdorff dimension of the graphs of the classical Weierstrass functions. Math. Z., 289, 223–266 (2018)
Stein, E. M.: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970
Su, W. Y.: Construction of fractal calculus. Sci. China Math., 45, 1587–1598 (2015)
Sun, Q. J., Su, W. Y.: Fractional integral and fractal function. Numer. Math. J. Chin. Univ. (Engl. Ser.), 11, 70–75 (2002)
Tatom, F. B.: The relationship between fractional calculus and fractals. Fractals, 3, 217–229 (1995)
Teodoro, G. S., Machado, J. A., Oliveira, E. C.: A review of definitions of fractional derivatives and other operators. J. Comput. Phys., 388, 195–208 (2019)
Tian, L.: The estimates of Hölder index and the Box dimension for the Hadamard fractional integral. Fractals, 29, 2150072 (2021)
Tian, L.: Hölder continuity and Box dimension for the Weyl fractional integral. Fractals, 28, 2050032 (2020)
Verma, S., Viswanathan, P.: A note on Katugampola fractional calculus and fractal dimensions. Appl. Math. Comput., 339, 220–230 (2018)
Verma, S., Viswanathan, P.: Bivariate functions of bounded variation: Fractal dimension and fractional integral. Indagat. Math. New. Ser., 31, 294–309 (2020)
Wang, C. Y.: R-L Algorithm: An approximation algorithm for fractal signals based on fractional calculus. Fractals, 29, 2150243 (2021)
Wang, J., Yao, K.: Construction and analysis of a special one-dimensional continuous functions. Fractals, 25, 1750020 (2017)
Wang, J., Yao, K., Liang, Y. S.: On the connection between the order of Riemann-Liouvile fractional falculus and Hausdorff dimension of a fractal function. Anal. Theory Appl., 32, 283–290 (2016)
Wen, Z. Y.: Mathematical Foundations of Fractal Geometry (in Chinese), Science Technology Education Publication House, Shanghai, 2000
Wu, B.: The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field. Fract. Calc. Appl. Anal., 23, 875–885 (2020)
Wu, J. R.: The effects of the Riemann–Liouville fractional integral on the Box dimension of fractal graphs of Hölder continuous functions. Fractals, 28, 2050052 (2020)
Wu, J. R.: On a linearity between fractal dimensions and order of fractional calculus in Hölder Space. Appl. Math. Comput., 385, 125433 (2020)
Wu, X. E., Du, J. H.: Box dimension of Hadamard integral of continuous functions of bounded and unbounded variation. Fractals, 25, 1750035 (2017)
Wu, Y. P., Zhang, X.: The Hadamard fractional calculus of a fractal function. Fractals, 26, 1850025 (2018)
Xie, T. F., Zhou, S. P.: On a class of fractal functions with graph Box dimension 2. Chaos Solitons Fractals, 22, 135–139 (2004)
Xie, T. F., Zhou, S. P.: On a class of singular continuous functions with graph Hausdorff dimension 2. Chaos Solitons Fractals, 32, 1625–1630 (2007)
Xu, Q.: Fractional integrals and derivatives to a class of functions. Journal of Xuzhou Normal University (Natural Science Edition), 24, 19–23 (2006)
Yao, K., Liang, Y. S., Fang, J. X.: The fractal dimensions of graphs of the Weyl-Marchaud fractional derivative of the Weierstrass-type function. Chaos Solitons Fractals, 35, 106–115 (2008)
Yao, K., Liang, Y. S., Su, W. Y., et al.: Fractal dimension of fractional derivative of self-affine functions. Acta Math. Sin. (Engl. Ser.), 56, 693–698 (2013)
Yao, K., Su, W. Y., Zhou, S. P.: On the fractional calculus of a type of Weierstrass function. Chin. Ann. Math. Ser. B, 25(A), 711–716 (2004)
Zähle, M.: Fractional differentiation in the self-affine case V-the local degree of differentiability. Math. Nachr., 185, 297–306 (1997)
Zähle, M., Ziezold, H.: Fractional derivatives of Weierstrass-type functions. J. Comput. Appl. Math., 76, 265–275 (1996)
Zhang, Q.: Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus. Acta Math. Sin. (Engl. Ser.), 30, 517–524 (2014)
Zhang, Q., Liang, Y. S.: The Weyl–Marchaud fractional derivative of a type of self-affine functions. Appl. Math. Comput., 218, 8695–8701 (2012)
Zhang, X., Peng, W. L.: Connection between the order of Katugampola fractional integral and fractal dimensions of Weierstrass function. College Math. J., 35, 25–31 (2019)
Zheng, W. X., Wang, S. W.: Real Function and Functional Analysis (in Chinese), High Education Publication House, Beijing, 1980
Acknowledgements
We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest The authors declare no conflict of interest.
Additional information
Supported by National Natural Science Foundation of China (Grant No. 12071218) and the Fundamental Research Funds for the Central Universities (Grant No. 30917011340)
Rights and permissions
About this article
Cite this article
Liang, Y.S., Su, W.Y. A Geometric Based Connection between Fractional Calculus and Fractal Functions. Acta. Math. Sin.-English Ser. 40, 537–567 (2024). https://doi.org/10.1007/s10114-023-1663-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-023-1663-3