Application of the three-dimensional telegraph equation to cosmic-ray transport

TL;DR

This paper derives an analytical three-dimensional solution to the telegraph equation, highlighting its advantages over diffusion models in explaining cosmic-ray intensity profiles, and compares it with numerical simulations.

Contribution

It provides the first closed-form three-dimensional solution to the telegraph equation, extending previous one-dimensional treatments and demonstrating its applicability to cosmic-ray transport.

Findings

The telegraph model captures features of cosmic-ray intensity profiles better than diffusion.

Analytical solutions are obtained using similarity to the Klein-Gordon equation.

Comparison with simulations shows improved explanation of observational data.

Abstract

An analytical solution to the the three-dimensional telegraph equation is presented. This equation has recently received some attention but so far the treatment has been one-dimensional. By using the structural similarity to the Klein-Gordon equation, the telegraph equation can be solved in closed form. Illustrative examples are used to discuss the qualitative differences to the diffusion solution. The comparison with a numerical test-particle simulation reveals that some features of an intensity profile can be better explained using the telegraph approach.