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

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.

arXiv comment5 pages, 2 figures, accepted for publication in The Astrophysical Journal

References in corpus (1)

• The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

Cited by in corpus (4)

• Parameter estimation of superdiffusive motion of energetic particles upstream of heliospheric shocks
• Turbulent diffusion of streaming cosmic rays in compressible, partially ionised plasma
• New Solution of Diffusion-Advection Equation for Cosmic-Ray Transport Using Ultradistributions
• Numerical investigation of particle acceleration at interplanetary shocks: diffusive and superdiffusive scenarios