Abstract

A notable class of problems often employed in undergraduate courses on differential equations is that of mixing problems: those involving a number of brine-filled tanks equipped with a number of brine-transporting pipes. Closed mixing problems, which feature neither filler nor drainer pipes, have been studied on a general level, where the flow networks are represented by digraphs [A. Slav\'ı k, Mixing problems with many tanks, American Mathematical Monthly, 120 (2013), 806-821]. In this paper, we extend the study to open mixing problems, which may feature filler and/or drainer pipes, representing the flow networks by a generalization of digraphs: quasi-digraphs. We formulate sufficient conditions under which such a mixing problem can be modeled as a system of linear ordinary differential equations whose coefficient matrix is the negative of the transpose of the Laplacian of the associated quasi-digraph. Subsequently, we formulate the analogues for mixing problems represented by weighted quasi-digraphs, and by cascade-type multilayer weighted quasi-digraphs. At the end of this paper, we propose suggestions for instructors on how our materials could be distilled to form a set of taught materials or a mini-project enriching an undergraduate differential equations course.

Keywords: Differential equation; mixing problem; digraph; quasi-digraph; multilayer.

DOI: 10.57016/TM-NXML6989

Pages:  79-103     

Volume  XXVII ,  Issue  2 ,  2024