Methodology to Obtain Universal Solutions for Systems of Coupled Ordinary Differential Equations. Examples of a Continuous Flow Chemical Reactor and a Coupled Oscillator

Abstract

This paper presents a concise and orderly methodology to obtain universal solutions to different problems in science and engineering using the nondimensionalization of the governing equations of the physical–chemical problem posed. For its application, a deep knowledge of the problem is necessary since it will facilitate the adequate choice of the references necessary for its resolution. In addition, the application of the methodology to examples of coupled ordinary differential equations is shown, resulting in an interesting tool to teach postgraduate students in the branches of physics, mathematics, and engineering. The first example used for a system of coupled ordinary differential equations is a model of a continuous flow chemical reactor, where it is worth noting; on the one hand, the methodology used to choose the reference (characteristic) time and, on the other, the equivalence between the characteristic times obtained for each one of the species. The following universal curves are obtained, which are validated by comparing them with the results obtained by numerical simulation, where it stands out that the universal solution includes an unknown that must be previously obtained. The resolution of this unknown implies having a deep knowledge of the problem, a common characteristic when using the methodology proposed in this work for different engineering or physicochemical problems. Finally, the second example is a coupled oscillator, where it is worth noting that the appearance of characteristic periods that implicitly or explicitly affect the particles’ movement is striking.

Understanding Complex Traffic Dynamics with the Nondimensionalisation Technique

“…These groups can be further manipulated with basic mathematical operations to be presented in the most advantageous form. For instance, one optimization strategy involves ensuring that each unknown is associated with only one group [36]. We now combine these coefficients to obtain dimensionless monomials Π i,j for each of the (i) equations in our traffic model:…”

Mathematical Modeling and Analysis Using Nondimensionalization Technique of the Solidification of a Splat of Variable Section

“…In this section, a study of the behavior of variables will be carried out through their grouping into dimensionless monomials. For this, a mathematical procedure known as the nondimensionalization technique will be applied where the variables are grouped into monomials that have a physical meaning and that control the problem [44,51]. One of the major difficulties of this technique is the appropriate choice of references to obtain the dimensionless variables, which implies a deep knowledge of the physical phenomena involved.…”

“…Both the network simulation model and the nondimensionalization technique have demonstrated the effectiveness of their application in numerous engineering problems such as diffusion of chlorides, chemical reactors, soil consolidation, etc. [41][42][43][44]. Finally, there are several main contributions of this paper.…”

Analysis of the Burgers–Huxley Equation Using the Nondimensionalisation Technique: Universal Solution for Dirichlet and Symmetry Boundary Conditions

“…Furthermore, several articles have focused on defining the behaviour of monomials based on the values obtained, establishing criteria in which some monomials can have little influence on the problem compared with others that govern it. Thus, as a summary, the following steps must be applied [30,31]:…”

“…Thus, it is necessary, on the one hand, to obtain a universal solution that allows this equation to be easily solved, and on the other, to study the influence of the variables involved. In this way, the nondimensionalisation technique allows us to both obtain universal solutions and study the variables that have been grouped to form monomials [29][30][31].…”