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Fractal Laplace transform: analyzing fractal curves

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Abstract

The concept of Laplace transform has been extended to fractal curves, enabling the solution of fractal differential equations with constant coefficients. This extension, known as the fractal Laplace transform, is particularly useful for handling inhomogeneous differential equations that involve delta Dirac functions and step functions within the realm of fractal functions. A comprehensive table of essential formulas for the fractal Laplace transform has been compiled to facilitate its application in various scenarios. By utilizing this transformative approach, researchers can now delve into the study of fractal functions and address complex problems involving non-traditional geometries. To illustrate the practicality of the fractal Laplace transform, several examples are provided, showcasing its effectiveness in solving fractal differential equations. This advancement represents a significant augmentation of the classical Laplace transform, tailored to suit the distinctive characteristics of fractal systems and functions.

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References

  1. Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. New York: WH freeman.

    Google Scholar 

  2. Falconer, K. 2004. Fractal Geometry: Mathematical Foundations and Applications. New York: Wiley.

    Google Scholar 

  3. Freiberg, U., and M. Zähle. 2002. Harmonic calculus on fractals-a measure geometric approach I. Potential Analysis 16 (3): 265–277.

    MathSciNet  Google Scholar 

  4. Barlow, M.T., and E.A. Perkins. 1988. Brownian motion on the Sierpinski gasket. Probability Theory and Related Fields 79 (4): 543–623.

    MathSciNet  Google Scholar 

  5. West, B., M. Bologna, and P. Grigolini. 2003. Physics of Fractal Operators. New York: Springer.

    Google Scholar 

  6. Samayoa Ochoa, D., L. Damián Adame, and A. Kryvko. 2022. Map of a bending problem for self-similar beams into the fractal continuum using the Euler-Bernoulli principle. Fractal and Fractional 6 (5): 230.

    Google Scholar 

  7. Lapidus, M.L., G. Radunović, and D. Žubrinić. 2017. Fractal Zeta Functions and Fractal Drums. New York: Springer.

    Google Scholar 

  8. Strichartz, R.S. 2018. Differential Equations on Fractals. Princeton: Princeton University Press.

    Google Scholar 

  9. Stillinger, F.H. 1977. Axiomatic basis for spaces with noninteger dimension. Journal of Mathematics and Physics 18 (6): 1224–1234.

    MathSciNet  Google Scholar 

  10. Tarasov, V.E. 2010. Fractional Dynamics. New York: Springer.

    Google Scholar 

  11. Kigami, J. 2001. Analysis on Fractals. Cambridge: Cambridge University Press.

    Google Scholar 

  12. Kesseböhmer, M., T. Samuel, and H. Weyer. 2016. A note on measure-geometric Laplacians. Monatshefte fur Mathematik 181 (3): 643–655.

    MathSciNet  Google Scholar 

  13. Giona, M. 1995. Fractal calculus on [0, 1]. Chaos, Solitons and Fractals 5 (6): 987–1000.

    ADS  MathSciNet  Google Scholar 

  14. Jiang, H., and W. Su. 1998. Some fundamental results of calculus on fractal sets. Communications in Nonlinear Science and Numerical Simulation 3 (1): 22–26.

    ADS  MathSciNet  Google Scholar 

  15. Bongiorno, D., and G. Corrao. 2015. An integral on a complete metric measure space. Real Analysis Exchange 40 (1): 157–178.

    MathSciNet  Google Scholar 

  16. Bongiorno, D. 2018. Derivatives not first return integrable on a fractal set. Ricerche di Matematica 67 (2): 597–604.

    MathSciNet  Google Scholar 

  17. Bongiorno, D., and G. Corrao. 2015. On the fundamental theorem of calculus for fractal sets. Fractals 23 (02): 1550008.

    ADS  MathSciNet  Google Scholar 

  18. Parvate, A., and A.D. Gangal. 2009. Calculus on fractal subsets of real line-I: Formulation. Fractals 17 (01): 53–81.

    MathSciNet  Google Scholar 

  19. Parvate, A., and A.D. Gangal. 2011. Calculus on fractal subsets of real line-II: Conjugacy with ordinary calculus. Fractals 19 (03): 271–290.

    MathSciNet  Google Scholar 

  20. Satin, S.E., A. Parvate, and A. Gangal. 2013. Fokker-Planck equation on fractal curves. Chaos, Solitons and Fractals 52: 30–35.

    ADS  MathSciNet  Google Scholar 

  21. Golmankhaneh, A.K. 2022. Fractal Calculus and Its Applications. Singapore: World Scientific.

    Google Scholar 

  22. Golmankhaneh, A.K., and A.S. Balankin. 2018. Sub-and super-diffusion on Cantor sets: Beyond the paradox. Physics Letters A 382 (14): 960–967.

    ADS  CAS  Google Scholar 

  23. Golmankhaneh, A.K., A. Fernandez, A.K. Golmankhaneh, and D. Baleanu. 2018. Diffusion on middle-\(\xi\) cantor sets. Entropy 20 (7): 504.

    ADS  MathSciNet  Google Scholar 

  24. Golmankhaneh, A.K., and D. Baleanu. 2016. Non-local integrals and derivatives on fractal sets with applications. Open Physics 14 (1): 542–548.

    Google Scholar 

  25. Banchuin, R. 2022. Noise analysis of electrical circuits on fractal set. COMPEL—The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 41 (5): 1464–1490.

    Google Scholar 

  26. Banchuin, R. 2022. Nonlocal fractal calculus based analyses of electrical circuits on fractal set. COMPEL—The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 41 (1): 528–549.

    Google Scholar 

  27. Golmankhaneh, A.K., and K. Welch. 2021. Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review. Modern Physics Letters A 36 (14): 2140002.

    ADS  MathSciNet  Google Scholar 

  28. Pietronero, L., and E. Tosatti, eds. 1986. Fractals in Physics. Amsterdam: Elsevier.

    Google Scholar 

  29. Deppman, A., E. Megias, R. Pasechnik, 2023. Fractal derivatives, fractional derivatives and \(q\)-deformed calculus. arXiv preprint arXiv:2305.04633.

  30. Shlesinger, M.F. 1988. Fractal time in condensed matter. Annual Review of Physical Chemistry 39 (1): 269–290.

    ADS  CAS  Google Scholar 

  31. Vrobel, S. 2011. Fractal Time. Singapore: World Scientific.

    Google Scholar 

  32. Welch, K. 2020. A Fractal Topology of Time: Deepening Into Timelessness. Austin: Fox Finding Press.

    Google Scholar 

  33. Nottale, L. 1993. Fractal Space-time and Microphysics: Towards a Theory of Scale Relativity. Singapore: World Scientific.

    Google Scholar 

  34. Gowrisankar, A., A.K. Golmankhaneh, and C. Serpa. 2021. Fractal calculus on fractal interpolation functions. Fractal and Fractional 5 (4): 157.

    Google Scholar 

  35. Golmankhaneh, A.K., and S.M. Nia. 2021. Laplace equations on the fractal cubes and Casimir effect. The European Physical Journal Special Topics 230 (21): 3895–3900.

    ADS  Google Scholar 

  36. Golmankhaneh, A.K., and C. Tunç. 2019. Sumudu transform in fractal calculus. Applied Mathematics and Computation 350: 386–401.

    MathSciNet  Google Scholar 

  37. Golmankhaneh, K.A., K.K. Ali, R. Yilmazer, and M.K.A. Kaabar. 2022. Local fractal Fourier transform and applications. Computational Methods for Differential Equations 10 (3): 595–607.

    MathSciNet  Google Scholar 

  38. Golmankhaneh, A.K., and A. Fernandez. 2018. Fractal calculus of functions on cantor tartan spaces. Fractal and Fractional 2 (4): 30.

    Google Scholar 

  39. Golmankhaneh, A.K., K. Welch, C. Serpa, and P.E. Jørgensen. 2023. Non-standard analysis for fractal calculus. The Journal of Analysis 31: 1895–1916.

    MathSciNet  Google Scholar 

  40. Golmankhaneh, A.K., and D. Baleanu. 2016. Fractal calculus involving gauge function. Communications in Nonlinear Science 37: 125–130.

    ADS  MathSciNet  Google Scholar 

  41. Golmankhaneh, A.K., C. Tunç, and H. Şevli. 2021. Hyers-Ulam stability on local fractal calculus and radioactive decay. The European Physical Journal Special Topics 230 (21): 3889–3894.

    ADS  Google Scholar 

  42. Golmankhaneh, A.K. 2021. Tsallis entropy on fractal sets. Journal of Taibah University for Science 15 (1): 543–549.

    Google Scholar 

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Acknowledgements

Cristina Serpa acknowledges partial funding by national funds through FCT-Foundation for Science and Technology, project reference: UIDB/04561/2020.

We express our gratitude to the reviewers for their valuable contributions in enhancing the quality of this article.

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Correspondence to Alireza Khalili Golmankhaneh.

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Communicated by S Ponnusamy.

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Khalili Golmankhaneh, A., Welch, K., Serpa, C. et al. Fractal Laplace transform: analyzing fractal curves. J Anal 32, 1111–1137 (2024). https://doi.org/10.1007/s41478-023-00677-1

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  • DOI: https://doi.org/10.1007/s41478-023-00677-1

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