Analysing cosmic ray density distribution using variable separable method in diverse spatial domains

This study employs the variable separable method to investigate the intricate dynamics of cosmic ray density distribution across diverse spatial domains. The exploration spans cylindrical, spherical, and Cartesian coordinates, delving into the effects of diffusion and velocity on cosmic ray propagation. The focal point is the evolution of cosmic ray density over time, scrutinizing the influence of parameters like diffusion coefficient, velocity, and eigenvalue. The simulations unveil compelling insights into the spatial and temporal behaviour of cosmic rays, yielding patterns that transcend coordinate systems. In the cylindrical region, the initial density distribution undergoes radial decay as cosmic rays disperse with time. In spherical coordinates, the simulation elucidates radial and angular patterns, revealing anisotropic behaviours that depend on the eigenvalue and velocity. Cartesian coordinates unfold a similar narrative, with radial decay along each axis and anisotropic tendencies along different directions. The outcomes of these simulations contribute substantively to our understanding of cosmic ray behaviour. The diffusion-driven homogenization effect becomes apparent as density gradients diminish, underscoring the integral role of diffusion in cosmic ray dynamics. By studying cosmic ray propagation in various spatial settings, this work not only elucidates fundamental principles but also lays a foundation for astrophysical applications. In essence, this research underscores the potency of the variable separable method in unravelling intricate phenomena, enabling a comprehensive exploration of cosmic ray behaviour in different environments. The insights gained from these simulations pave the way for deeper investigations into high-energy astrophysical processes and contribute to the broader understanding of cosmic ray interactions.