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A Tutorial Review on Fractal Spacetime and Fractional Calculus

Abstract

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz’s notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie’s mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

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Acknowledgments

The work is supported by PAPD (A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions), National Natural Science Foundation of China under Grant Nos.10972053 and 51203114, Special Program of China Postdoctoral Science Foundation Grant No. 2013T60559, China Postdoctoral Science Foundation Grant No.2012M521122.

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He, JH. A Tutorial Review on Fractal Spacetime and Fractional Calculus. Int J Theor Phys 53, 3698–3718 (2014). https://doi.org/10.1007/s10773-014-2123-8

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Keywords

  • Fractal spacetime
  • Fractional differential equation
  • Fractal derivative
  • Q-derivative
  • Cantor set
  • El Naschie mass-energy equation
  • E-infinity theory
  • Hilbert cube
  • Kaluza-Klein spacetime
  • Zero set
  • Empty set
  • Fractal stock movement