A family of nonlinear diffusion equations related to the [formula omitted]-error function

Abstract

We investigate a bi-parametric family of nonlinear diffusion equations that includes, as special members, the power-law nonlinear diffusion equation, and an evolution equation akin to Ubriaco’s nonlinear diffusion equation. It is well-known that the power-law nonlinear diffusion equation is closely related to the Sq-thermostatistics. We show that the Ubriaco-like member of the above mentioned bi-parametric family is also related to the Sq-theory, because it admits exact time-dependent solutions that exhibit the q-error-function shape. These solutions constitute a generalization, within the Sq-thermostatistical framework, of a celebrated class of solutions of the linear diffusion equation, that have the standard error-function form. Besides exploring the main aspects of a family of non-standard diffusion processes, our present developments constitute a physical-mathematical application of the q-error-function. This function, in spite of its potential relevance for q-statistics and its applications, has not yet attracted the attention it deserves from the Sq-thermostatistics research community.