# Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation

@article{Podlubny2001GeometricAP, title={Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation}, author={Igor Podlubny}, journal={arXiv: Classical Analysis and ODEs}, year={2001} }

A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and dieren tiation (i.e., integration and dieren tiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and dieren tiation, the Caputo fractional dieren tiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the…

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## References

SHOWING 1-10 OF 29 REFERENCES

### On the paper by R. R. Nigmatullin “fractional integral and its physical interpretation”

- Mathematics
- 1994

The goal of establishing a clear relationship between fractial geometry and fractional calculus has been long sought by the scientific community. A number of intuitive or heuris tic suggestions in…

### On physical interpretations of fractional integration and differentiation

- Mathematics
- 1995

Is there a relation between fractional calculus and fractal geometry? Can a fractional order system be represented by a causal dynamical model? These are the questions recently debated in the…

### Fractional integral and its physical interpretation

- Mathematics
- 1992

A relationship is established between Cantor's fractal set (Cantor's bars) and a fractional integral. The fractal dimension of the Cantor set is equal to the fractional exponent of the integral. It…

### Fractional integral associated to generalized cookie-cutter set and its physical interpretation

- Mathematics
- 1997

This paper is based on Nigmatullin's study. When the `residual' memory set is a generalized cookie-cutter set on , using various hypotheses it is proved that the fractional exponent of a fractional…

### Linear Models of Dissipation whose Q is almost Frequency Independent-II

- Mathematics
- 1967

Summary Laboratory experiments and field observations indicate that the Q of many non-ferromagnetic inorganic solids is almost frequency independent in the range 10-2-107 cis, although no single…

### Einstein ’ s Static Universe : An Idea Whose Time Has Come Back ?

- Physics
- 2000

W hat is the shape of space? Is the universe finite or infinite? Did the world have a beginning, or has it always existed? These fundamental questions have intrigued and baffled humans since the most…

### 愛麗思漫遊奇境記 = Alice's adventures in Wonderland

- Art
- 1929

This thesis has two aims. The first one is to elucidate how Alice’s Adventures in Wonderland (1865) functions as a Bildungsroman, and the other one is to demonstrate how the novel also has a coming…

### Adventures in wonderland

- EducationNature Biotechnology
- 2001

A personal perspective on the many challenges faced by scientists when becoming bioentrepreneurs. These can be overcome with a little curiosity, creativity, and commitment.