Analytical solutions of a fractional diffusion-advection equation for solar cosmic-ray transport

TL;DR

This paper derives an exact analytical solution to a fractional diffusion-advection equation relevant for modeling superdiffusive transport of energetic particles in the heliosphere, highlighting the importance of shock travel time in realistic models.

Contribution

It provides a new exact Fourier transform solution to the fractional diffusion-advection equation and compares it with existing solutions, improving modeling of particle transport in space physics.

Findings

Exact Fourier transform solution derived

Weak diffusion approximation simplified

Shock travel time is crucial for accurate modeling

Abstract

Motivated by recent applications of superdiffusive transport models to shock-accelerated particle distributions in the heliosphere, we solve analytically a one-dimensional fractional diffusion-advection equation for the particle density. We derive an exact Fourier transform solution, simplify it in a weak diffusion approximation, and compare the new solution with previously available analytical results and with a semi-numerical solution based on a Fourier series expansion. We apply the results to the problem of describing the transport of energetic particles, accelerated at a traveling heliospheric shock. Our analysis shows that significant errors may result from assuming an infinite initial distance between the shock and the observer. We argue that the shock travel time should be a parameter of a realistic superdiffusive transport model.