Geometric and physical interpretation of fractional integration and fractional differentiation. (English) Zbl 1042.26003

Podlubny, Igor

Fract. Calc. Appl. Anal. 5, No. 4, 367-386 (2002).

The author offers geometric visualizations (based on a technique called “shadows on the walls”) and physical interpretations (based on relationships between “individual time” and “cosmic time”) of important types of fractional differentiation and integration. He considers in detail the Riemann-Liouville fractional integral and derivative, the Caputo fractional derivative,the potentials of Riesz and Feller, and indicates analogues to more general convolution integrals of Volterra type and to Stieltjes integrals. While the geometric interpretations give illuminating insights into the meaning and way of acting of the operators considered, the physical interpretations presented seem to the reviewer to be of a more speculative character.

Reviewer: Rudolf Gorenflo (Berlin)

MSC:

26A33	Fractional derivatives and integrals
44A35	Convolution as an integral transform
26A42	Integrals of Riemann, Stieltjes and Lebesgue type
83C99	General relativity
45D05	Volterra integral equations

Keywords:

fractional derivative; fractional integral; fractional calculus; geometric interpretation; physical interpretation