Journal
INTERNATIONAL JOURNAL OF THEORETICAL PHYSICS
Volume 53, Issue 11, Pages 3698-3718Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10773-014-2123-8
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
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Funding
- PAPD (A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions)
- National Natural Science Foundation of China [10972053, 51203114]
- Special Program of China Postdoctoral Science Foundation [2013T60559]
- China Postdoctoral Science Foundation [2012M521122]
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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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