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38 Matrix Theory-Applications and Theorems

Mathematics. Vol. 265. Heidelberg: Springer; 2012

**Chapter 3**

Provisional chapter

**Square Matrices Associated to Mixing Problems ODE**

DOI: 10.5772/intechopen.74437

In this chapter, mixing problems are considered since they always lead to linear ordinary differential equation (ODE) systems, and the corresponding associated matrices have different structures that deserve to be studied deeply. This structure depends on whether or not there is recirculation of fluids and if the system is open or closed, among other characteristics such as the number of tanks and their internal connections. Several statements about the matrix eigenvalues are analyzed for different structures, and also some questions and conjectures are posed. Finally, qualitative remarks about the differential equation system solutions and their stability or asymptotical stability are included. Keywords: eigenvalues, Gershgorin circle theorem, mixing problems, linear ODE

Mixing problems (MPs), also known as "compartment analysis" [1], in chemistry involve creating a mixture of two or more substances and then determining some quantity (usually concentration) of the resulting mixture. For instance, a typical mixing problem deals with the amount of salt in a mixing tank. Salt and water enter to the tank at a certain rate, they are mixed with what is already in the tank, and the mixture leaves at a certain rate. This process is modeled by an ordinary differential equation (ODE), as Groestch affirms: "The direct problem for one-compartment mixing models is treated in almost all elementary differential equations

Instead of only one tank, there is a group, as it was stated by Groestch: "The multicompartment model is more challenging and requires the use of techniques of linear algebra" [2].

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Square Matrices Associated to Mixing Problems ODE

**Systems**

Systems

Victor Martinez-Luaces

Victor Martinez-Luaces

Abstract

1. Introduction

texts" [2].

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74437

systems, associated matrices
