A family of nonlinear diffusion equations related to the q -error function

Abstract

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