Stationarity-conservation laws for certain linear fractional differential equations

TL;DR

This paper develops stationarity-conservation laws for linear fractional differential equations using a novel Leibniz rule in Laplace convolution algebra, with applications to fractional diffusion and nonlocal charges.

Contribution

It introduces a new Leibniz rule in convolution algebra to explicitly construct stationary-conserved currents for fractional differential equations.

Findings

Explicit stationary-conserved currents for fractional diffusion equations

Generalization to mixed fractional-differential systems

Construction of stationary nonlocal charges

Abstract

The Leibniz rule for fractional Riemann-Liouville derivative is studied in algebra of functions defined by Laplace convolution. This algebra and the derived Leibniz rule are used in construction of explicit form of stationary-conserved currents for linear fractional differential equations. The examples of the fractional diffusion in 1+1 and the fractional diffusion in d+1 dimensions are discussed in detail. The results are generalized to the mixed fractional-differential and mixed sequential fractional-differential systems for which the stationarity-conservation laws are obtained. The derived currents are used in construction of stationary nonlocal charges.