Geometric interpretation of fractional-order derivative. (English) Zbl 1488.26026
Summary: A new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.
MSC:
| 26A33 | Fractional derivatives and integrals |
| 55R10 | Fiber bundles in algebraic topology |
| 58A20 | Jets in global analysis |
Keywords:
fractional derivative; geometric interpretation; Riemann-Liouville derivative; Caputo derivative; derivative of Hadamard type; geometry of jets; jet bundles; jets of functions; infinite-order jetsSoftware:
Sinc-PackReferences:
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