Continuum Mech. Thermodyn. (2014) 26:447–463 DOI 10.1007/s00161-013-0313-x ORIGINAL ARTICLE Cs. Mészáros · I. Kirschner · Á. Bálint Relevance of the time–quasi-polynomials in the classic linear thermodynamic theory of coupled transport processes Received: 11 March 2013 / Accepted: 10 July 2013 / Published online: 25 July 2013 © Springer-Verlag Berlin Heidelberg 2013 Abstract A general description of the basic system of ordinary differential equations of coupled transport processes is given within framework of a linear approximation and treated by tools of matrix analysis and group representation theory. It is shown that the technique of hyperdyads directly generalizes the method of simple dyadic decomposition of operators used earlier in the classical linear irreversible thermodynamics and leads to possible new applications of the concept of quasi-polynomials at descriptions of coupled transport processes. Keywords Coupled transport processes · Non-equilibrium thermodynamics · Matrix analysis · Group representation theory 1 Introduction Since the non-equilibrium thermodynamics plays a decisive role [1, 2] in accurate description of the genuine (usually: coupled) transport processes taking place in macroscopic dissipative continua, the refined mathemat- ical modeling of such phenomena represents a permanent objective of the contemporary mathematical physics too. Namely, many transport processes also show features of the anomalous diffusion [3] of non-local charac- ter and with memory effects on macroscopic level (corresponding usually to the percolative fractal character on the mesoscopic level), and require, e.g., use of the Riemann–Liouville operators, leading to a widespread application of the fractional partial derivatives with respect to time and spatial coordinates, as well [4, 5]. The relevant mathematical features have also been analyzed in detail in studies related to general problems of fractional diffusion, e.g., [6–9] as well as in [10, 11] where the crucial importance of applying of the theory of Lie groups in this research field has been pointed out. The classic domains of the extended irreversible thermodynamics (EIT) have also many open fundamental questions even in some earlier developed areas of them. It is enough to mention the archetypal problem of Communicated by Andreas Öchsner. Cs. Mészáros Department of Physics and Process Control, Faculty of Mechanical Engineering, Szent István University, Páter K.u.1. 2103 Gödöll˝ o, Hungary I. Kirschner Institute of Physics, Eötvös University, 1117 Budapest, Hungary Á. Bálint (B ) Department of Chemistry and Biochemistry, Faculty of Agricultural and Environmental Sciences, Szent István University, Gödöll˝ o, Hungary E-mail: balint.agnes@mkk.szie.hu