The description of the mass transfer mechanisms in various physical and engineering fields, e.g., Li-ion batteries, is of significant importance for optimizing their performance. The present work introduces a comparative study describing the different responses of a perfectly elastic material when different non-Fickian diffusion situations are considered. The uncoupled theory of elastic diffusion, in which the diffusion process is described by non-Fickian laws, such as Cattaneo, Jeffreys-type, and Burgers-type constitutive laws, is employed in this modeling. The diffusion of lithium ions inside the silicon anode is one of the physical situations in which diffusion-induced stresses may be significant. An impulsive initial value problem, consisting of an initial lithium ions amount that starts impulsively to diffuse over the entire space of a silicon material, is considered. Direct approach together with Laplace and exponential Fourier transforms techniques are employed to obtain the solution in the Laplace transformed domain. The inverse Laplace transform is computed numerically to obtain the solution in the physical domain. Comparisons among the material responses to different diffusion regimes are presented.