Partial fractional derivatives of Riesz type and nonlinear fractional differential equations

Abstract

Generalizations of fractional derivatives of noninteger orders for N-dimensional Euclidean space are proposed. These fractional derivatives of the Riesz type can be considered as partial derivatives of noninteger orders. In contrast to the usual Riesz derivatives, the suggested derivatives give the usual partial derivatives for integer values of orders. For integer values of orders, the partial fractional derivatives of the Riesz type are equal to the standard partial derivatives of integer orders with respect to coordinate. Fractional generalizations of the nonlinear equations such as sine-Gordon, Boussinesq, Burgers, Korteweg–de Vries and Monge–Ampere equations for nonlocal continuum are considered.