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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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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.

Author information

Author notes

  1. K. Welch, C. Serpa, R. Rodríguez-López: These authors contributed equally to this work.

Authors and Affiliations

  1. Department of Physics, Urmia Branch, Islamic Azad University, Urmia, West Azerbaijan, 63896, Iran

    Alireza Khalili Golmankhaneh

  2. California Institute of Integral Studies, San Francisco, CA, USA

    Kerri Welch

  3. ISEL-Instituto Superior de Engenharia de Lisboa, Lisbon, Portugal

    Cristina Serpa

  4. CMAFcIO-Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Lisbon, Portugal

    Cristina Serpa

  5. Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Rosana Rodríguez-López

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  1. Alireza Khalili Golmankhaneh
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  2. Kerri Welch
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  3. Cristina Serpa
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  4. Rosana Rodríguez-López
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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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