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

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.


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
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