Fractional Calculus in Russia at the End of XIX Century

Maryna Dubatovskaya,Sergei Rogosin

doi:10.3390/math9151736

Abstract

In this survey paper, we analyze the development of Fractional Calculus in Russia at the end of the XIX century, in particular, the results by A. V. Letnikov, N. Ya. Sonine, and P. A. Nekrasov. Some of the discussed results are either unknown or inaccessible.

Highlights

It should be noted that Complex Analysis was traditionally highly developed discipline starting from Leonard Euler, who worked for a long period in Russia (1726–1741 and 1776–1783)
We describe the contribution of Alexey Vasil’evich Letnikov (1837–1888), Nikolai Yakovlevich Sonine (1849–1915), and Pavel Alekseevich Nekrasov (1853–1924) to Fractional Calculus and the role of these results in the modern Fractional Calculus and its Applications
Since it was shown that by integration by parts Formula (29) is reduced to the definition of the fractional derivative (28) without any auxiliary function, Letnikov concluded that the following alternative holds: either (1) there exists no auxiliary function of the above form, or (2) formula (29) cannot be taken as the general definition of fractional derivative of an arbitrary order

Summary

Introduction

A deep analysis of the results on this subject was given in the article [2] by Letnikov. He recognized the basic role of Liouville’s approach. After contribution by Letnikov, the serious works on Fractional Calculus in Russia in the second part of the XIX century were published by N. Traditionally highly developed discipline starting from Leonard Euler, who worked for a long period in Russia (1726–1741 and 1776–1783) This part of Mathematical Analysis was essentially developed in the XIX century by M. We describe the contribution of Alexey Vasil’evich Letnikov (1837–1888), Nikolai Yakovlevich Sonine (1849–1915), and Pavel Alekseevich Nekrasov (1853–1924) to Fractional Calculus and the role of these results in the modern Fractional Calculus and its Applications

Liouville’s Approach and Its Analysis by Letnikov

Letnikov or Grünwald-Letnikov Derivative

Solution to Certain Differential Equations

Sonine-Letnikov Discussion

Sonine’s Fractional Derivative and Integral

Sonine Kernel and Sonine Integral Equations

Higher Order Hypergeometric Functions

Nekrasov’s Contribution to Fractional Calculus

Conclusions