Extending the D'Alembert Solution to Space-Time Modified Riemann-Liouville Fractional Wave Equations

TL;DR

This paper extends the classical D'Alembert solution to fractional wave equations involving space-time fractional derivatives, providing explicit solutions and analyzing Lorentz symmetry implications in fractional dynamics.

Contribution

It introduces a generalized fractional D'Alembertian and derives explicit solutions for fractional wave equations with different space and time derivative orders.

Findings

Explicit solutions for fractional wave equations were obtained.

The generalized fractional D'Alembertian was formulated.

Lorentz symmetry considerations were analyzed.

Abstract

In the realm of complexity, it is argued that adequate modeling of TeV-physics demands an approach based on fractal operators and fractional calculus (FC). Non-local theories and memory effects are connected to complexity and the FC. The non-differentiable nature of the microscopic dynamics may be connected with time scales. Based on the Modified Riemann-Liouville definition of fractional derivatives, we have worked out explicit solutions to a fractional wave equation with suitable initial conditions to carefully understand the time evolution of classical fields with a fractional dynamics. First, by considering space-time partial fractional derivatives of the same order in time and space, a generalized fractional D'Alembertian is introduced and by means of a transformation of variables to light-cone coordinates, an explicit analytical solution is obtained. To address the situation of different orders in the time and space derivatives, we adopt different approaches, as it will become clear throughout the paper. Aspects connected to Lorentz symmetry are analyzed in both approaches.