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Integration and differentiation to a variable fractional order. (English) Zbl 0820.26003

Summary: Integration and differentiation of functions to a variable order \((d/dx)^ n f(x)\) is studied in two ways: 1) using the Riemann-Liouville definition, 2) using Fourier transforms. Some properties and the inversion formula are obtained.

MSC:

26A33 Fractional derivatives and integrals
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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References:

[1] Erdelyi A., Higher transcendental functions (1953)
[2] Samko S.G., ”Nauka i Tehnica”, 1987. English Transl.: Fractional integrals and derivatives. Theory and Applications (1992)
[3] Miller K.S., An Introduction to Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002
[4] DOI: 10.1007/BFb0067095
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