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 |
Keywords:
Riemann-Liouville fractional derivative; Marchaud fractional derivative; variable order; Fourier transforms; inversion formulaReferences:
| [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 · doi:10.1007/BFb0067095 |
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