Normalization Method as a Potent Tool for Grasping Linear and Nonlinear Systems in Physics and Soil Mechanics

Abstract

To address physical problems that require solving differential equations, both linear and nonlinear analytical methods are preferred when possible, but numerical methods are utilized when necessary. In this study, the normalization technique is established, which is a simple mathematical approach that requires only basic manipulation of the governing equations to obtain valuable information about the solution. The methodology of this technique involves adopting appropriate references to obtain the dimensionless form of the governing equation, after which the terms of the equation are balanced, obtaining the dimensionless monomials governing the solution. Thorough knowledge of the physical processes involved is necessary to find the best references. The main advantages of this technique are the simplicity of the methodology, the acquisition of valuable information about the solution without the need for complex mathematical calculations, and its applicability to nonlinear problems. However, it is important to consider the difficulty in selecting appropriate references in more complex scenarios. This study applies this normalization methodology to different scenarios, showing how choosing appropriate references lead to the independent dimensionless monomials. Once obtained, it was possible to identify different situations concerning the value of monomials. It will be when they are close to unity, and therefore normalized, when they fundamentally affect the solution of the problem. Finally, we present two cases, one linear and one complex, about the application of normalization to the challenging problem of soil consolidation in ground engineering, illustrating how the technique was used to obtain the solution and its many advantages.

Understanding Complex Traffic Dynamics with the Nondimensionalisation Technique

“…Thus, α permits traffic dissipation in the model to occur. Our task is to solve this model by extracting the principal monomials using the method of nondimensionalisation [23,32]. First, we choose the dependent variables that will be related to one another through the independent variable; then, we define the dimensionless variables that span the monomials.…”

“…In cases where the problem exhibits linearity or pseudo-linearity, we can assume that each term has an order of magnitude around 1 [32,34,35]. Then, the factors can be extracted from the equations to obtain the relevant coefficients of the model, which are…”

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].…”

“…The application of the nondimensionalisation technique to engineering problems formulated in ordinary differential equations is well known because it allows one to obtain dimensionless groups by grouping the variables, allowing us to know both the universal solution of the problem and the influence of each variable [31]. This procedure has recently been applied to different engineering problems, such as the diffusion of chlorides in concrete or soil consolidation, and the basis of the methodology has been explained in detail [29,31].…”