Colloquium Talk - Sverre Holm (UiO): Abel’s 1823 Paper on Fractional Derivatives and Waves with Power-Law Attenuation
In 1823 Niels Henrik Abel published a paper with title "Oplösning af et par opgaver ved hjelp af bestemte integraler," (Solution of a couple of problems by means of definite integrals), Magazin for Naturviden-skaberne, Aargang I, Bind 2, Christiania, which did not appear in French translation until 1881. Here he presented a complete framework for fractional-order calculus with appropriate notation for non-integer-order integration and differentiation. This seems to have been unknown until discussed in Podlubny, Magin, Trymorush, "Niels Henrik Abel and the birth of fractional calculus," Fractional Calculus and Applied Analysis, pp. 1068-1075, 2017.
I will give an introduction to non-integer order calculus and Abel's derivation of it. This will be done by going through his 1823 paper showing how he assumes a generalization of both Cauchy's formula for repeated integration and the fundamental theorem of calculus. In addition I will show how fractional calculus can be used to derive fractional-order partial differential wave equations. They have solutions with power-law attenuation which matches measurements in the complex media of medical ultrasound and elastography as well as in sediment acoustics and seismics.