Abstract
In this paper we define Riesz type derivatives symmetric and anti-symmetric w.r.t. the reflection mapping in finite interval [a,b]. Functions determined in [a,b] are split into parts with well determined reflection symmetry properties in a hierarchy of intervals [a m ,b m ], m ∈ ℕ, concentrated around an arbitrary point. For these parts - called the [J]-projections of function, we prove the representation and integration formulas for the introduced fractional symmetric and anti-symmetric integrals and derivatives. It appears that they can be reduced to operators determined in arbitrarily short subintervals [a m ,b m ]. The future application in the reflection symmetric fractional variational calculus and the generalization of previous results on localization of Euler-Lagrange equations are discussed.
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Klimek, M., Lupa, M. (2013). Reflection Symmetry in Fractional Calculus – Properties and Applications. In: Mitkowski, W., Kacprzyk, J., Baranowski, J. (eds) Advances in the Theory and Applications of Non-integer Order Systems. Lecture Notes in Electrical Engineering, vol 257. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00933-9_18
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DOI: https://doi.org/10.1007/978-3-319-00933-9_18
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00932-2
Online ISBN: 978-3-319-00933-9
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