We present an extension of the linear Luikovs system of equations of the coupled heat and mass transfer through porous media. This extension is based on generalizations of the Fourier and Fick laws connected to non-Markovian processes, i.e., anomalous diffusion, which has as particular case the framework of fractional diffusion equations. The generalization procedure led to a mathematical formalism involving integro-differential equations in which an integration kernel plays a key role. This kernel depends on the diffusive system and according to its mathematical expression analytical solutions can be obtained. As an illustration, we analyze, by using the Laplace transform and Fourier series, a one-dimensional system of finite size. (C) 2013 Elsevier Ltd. All rights reserved.