Searching fundamental information in ordinary differential equations. Nondimensionalization technique

Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.


Introduction
The search of the dimensionless groups of a physical or engineering problem is a basic subject which has interest to most of the researchers, since any system of equations, containing mathematical formulation of physical laws can be represented as a relation between dimensional quantities [1,2]. To attain this aim, two techniques are currently used: dimensional analysis and nondimensionalization. The first one is a classical approach and, in spite of its detractors [3,4], there are many publications and books about this topic [5][6][7][8][9][10]. According to Gibbings [11], the delight of dimensional analysis is the combination of its great utility with a demanding intellectual rigor. However, it often has imperfections as a result of a careless treatment, and it is difficult to offer a defense against the assertion about that this technique is only effective because the correct answer is previously known. Dimensional analysis is a formidable tool in the most difficult problems, but their results are sometimes incomplete, so it requires explanations in order to get a complete understanding of the solutions. Nevertheless, it is rare to a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 find applications of dimensional analysis to linear and nonlinear problems, ruled by ordinary differential equations and coupled systems of these equations. Problems such as the motion of a simple pendulum, the flow of a heavy fluid through a spillway, fluid motion in pipes, the motion of a body in a fluid and other problems, which are studied as typical examples in books about dimensional analysis [11,12,13], in order to demonstrate that this technique is worth applying. Even though they are more difficult to analyze, applications to coupled partial differential equations, in their discriminate version, are frequently published in the scientific literature [14][15][16].
The technique of nondimensionalization dates back to a long time. In 1935, Ruark [17] wrote: 'Inspectional analysis (the name he gives to nondimensionalization) consists in transforming the equations of the problem, differential or otherwise, so that all the variables are dimensionless. Simple inspection then shows how these dimensionless variables are related. . . A formal solution of the equations may then be obtained by writing each dependent dimensionless variable as a power series in which the arguments are the independent dimensionless variables.' For some authors, in fact, nondimensionalization and dimensional analysis provide the same information [18], while for others the technique of nondimensionalization may reveal more accuracy than dimensional analysis and, in that sense, it is more powerful [19]. But, as occurs with dimensional analysis, nondimensionalization is applied to complex problems ruled by coupled partial differential equations [4,20,21], but it is scarcely found in problems ruled by ordinary differential equations. As we will see later, the reason of that lies in that there is more than one reference to make dimensionless the variables, and this technique is rarely applied to this kind of equation.
In this work, we introduce the nondimensionalization, in a formal protocol, whereby dimensionless variables are defined being their range of variation extends within the interval [0-1], covering it, either completely or near completely. The resulting equations are not only dimensionless, but also what it is called normalized, in the sense their solutions are universal. Suitable references, related between them, must be chosen to satisfy this assumption.
In this way, once the dimensionless governing equations are established, each of their addends is the product of two factors, as it is appreciated in the applications: one formed by the dimensionless variables and/or their changes (derivatives), and the other formed by a grouping of parameters of the problem. Assuming that the mean value of the first factor of all the addends is of the order of magnitude unity-an admissible hypothesis unless the nonlinearity of the problem is too acute, as we will check in every example-, the other factors must also be of the same order of magnitude. Therefore, the ratios of the last factors (the dimensionless numbers) have an order of magnitude unity, and may be physically interpreted as balances of the quantities counteracting in the process, within the temporal or geometrical domain of the problem.
The essential for an optimal nondimensionalization, what might be called normalized nondimensionalization, resides in a correct choice of the magnitudes of reference that convert the dependent and independent variables of the problem to their dimensionless form. These references can be explicitly contained in the statement of the problem or, on the contrary, be implicitly; In this case, they can be call hidden variables. In any case, it is necessary that they relate to each other. For example, if the time reference is the time interval between t = 0 y t = t 0 , then the reference for the dependent variable associated should be the difference between of the values of this one for those times, and not any other value, although having the same physical dimension.
If the selection of references is such that it confines the values of the variables in the interval [0-1], as has already been said above, the resulting equation, dimensionless and normalized, will allow determination of the dimensionless groups of the problem. These groups will be of order of magnitude unity and with clear physical meaning, in terms of the balance of the variables that interact in the problem.
The functional dependence between these dimensionless groups leads to the solution of the problem and the expression of the unknowns in terms of the other parameters.
The above protocol provides solutions that, in general, overcome the results of dimensional analysis, which never refers to any order of magnitude of the unknowns sought. Gibbings [11] is one of the few recent authors that: i) clearly ascribes a physical significance to some of the classical dimensionless numbers (Reynolds, Euler, Grashof. . .), while recognizing that this is not always possible, and ii) affirms that these numbers, neither give a direct numerical measure of the unknowns nor an order of magnitude of the ratios that they represent. The reason, clearly, is that the variables in the relevant list, when it is applied the dimensional analysis, are generally chosen without any kind of spatial and/or temporal connection between them.
Three applications, that include nonlinear problems, are presented to illustrate the proposed protocol for nondimensionalization: a pendulum over an accelerated platform, a pendulum over a sliding support and simple interaction between two species. In all of them, the solutions provided by nondimensionalization have more accuracy (it is exact in the second application) than those given by dimensional analysis. Discussions about the choice of suitable references (whether or not established in the statement of the problem) and their relations, the existence of hidden references, references for asymptotic problems and other aspects related to the process of nondimensionalization, are discussed in each application. Finally, the solutions are checked by numerically solving each problem using the network method [22,23].

Application 1. Pendulum over an accelerated platform
A platform with acceleration a o , transports a physical pendulum of length l o and mass m o , as shown in Fig 1. The angle θ o , measured from the vertical, will be the displacement of the pendulum and, releasing it with zero velocity, give rise to oscillations of period T o . The aim is to study this period.
The platform-pendulum system has an equilibrium position defined by the angle θ e , around which the oscillations take place. This angle is related to a o and g by the expression For example, for l o = 4 and a o = g = 1, we obtain θ e = π/4 (0.785) rad. Fig 2 shows the solution θ(t) for the initial conditions θ o = -1 y dθ/dt (t = 0) = 0. Note that θ o is quite close to the average value of the maximum and minimum angular displacements, which are always negative.
Firstly, we will apply the nondimensionalization process to this problem. Without losing generality, we fit the study, initially, to cases with small values of θ e and θ o , and large values of θ o compared with θ e ; i.e., θ o -θ e %θ o .
Under this assumption, which implies cos(θ o )%1 and sen(θ o )%θ o , the governing equation is Choosing the references θ o and t o (the time necessary to reach the lowest point of the way from the initial position), the dimensionless variables are written as θ' = θ/θ o and t' = t/t o . Substituting them in the Eq (1) gives the dimensionless form of the governing equation, which is This equation provides three coefficients: and two dimensionless groups (by dividing the two first coefficients by the last one): Then, the order of magnitude of t o is given by the expression The application of dimensional analysis begins with the list of relevant variables of the problem. These are: l o , g, a o , θ o and the unknown T o (m o is not relevant, since is a common proportionality factor of the two forces that counteract in the problem, weight and inertia). The choice of t o , rather than T o , is more appropriate since it is an unknown that is directly related to θ o . However, such a choice is never made by the dimensional analysis. The list provides three dimensionless groups and a less precise solution for the unknown The solution given by nondimensionalization for cases with large values of θ o can be obtained using the series expansion of sin and cos functions: Retaining the two first terms of the series (using more terms, it leads to the same results), the governing equations can be written in the form and its corresponding dimensionless form The coefficients of this equation are and the independent dimensionless groups formed by these coefficients (dividing by the last and reorganizing them) So that, the solution is the same as that provided by dimensional analysis, substituting t o for the period T o .
To check the reliability of the results given by the nondimensionalization, the cases of Table 1 were run numerically (simulation is carried out by the network method). To simplify the study, dθ/dt (t = 0) = 0.
We first check cases 1 to 3, related to proportional changes in the parameters l o , g y a o and a large value for θ o . The results of the simulation (θ(t), dθ/dt and FFT spectra) are shown in   To summarize the information provided, we can observe that Fig 3, showing the results of the numerical simulation for cases 1-3, matches the expected results of the nondimensionalization, since the dimensionless numbers are of the order of magnitude unity, which is what it is intended to demonstrate.
Likewise, Figs 4 to 7 represent the remaining cases of Table 1, in which changes are introduced into parameters of the equation of the model that vary proportionally dimensionless numbers, verifying the consistency of the results observed in these Figs.

Application 2. Pendulum over a sliding support
From a support of mass m 1 that slides freely over a surface, hangs a physical pendulum of mass m 2 and length l o , Fig 8. When the small ball is moved to a disequilibrium location defined by θ o , the ball support system oscillates freely.
To simplify the governing equations we assumed two hypotheses: θ o is small and the initial velocity of the ball is zero. Under these assumptions, the mathematical model is defined by the with g being the gravitational acceleration. Choosing as reference quantities θ o , x o (half of the displacement of the support in its oscillatory movement, a hidden unknown quantity) and t o (half of the time required by the ball to swing from one end to other), the dimensionless variables are defined in the form:θ' = θ/θ o , x' = x/x o and t' = t/t o . Substituting these variables in Eqs https://doi.org/10.1371/journal.pone.0185477.g005 (14) and (15) enables the dimensionless governing equations to be derived From the first Eq (16), two coefficients emerge  giving rise to one dimensional group The order of magnitude is then given by From the second Eq (17), three coefficients emerge which provide two dimensionless groups Nondimensionalization technique in ordinary differential equations with C, an unknown function of (x 0 /l 0 θ o ). Using the result (20), t o is also given by From the relevant list of variables, m 1 , m 2 , l o , g, θ o and the unknowns, x o and t o , dimensional analysis provides three dimensionless groups that can be expressed as From these groups, the solution for x o and t o is Again, for t o , the solutions of dimensional analysis and nondimensionalization are the same, while for x o the solution of dimensional analysis is less precise.
To check the results of the nondimensionalization we will simulate the cases of Table 2. Cases 1 to 3, whose numerical solution is shown in Fig 9, Table 2 and Figs 10 and 11, by introducing changes in the parameters that modify the dimensionless numbers in proportion. As it can be seen, in Table 2. Cases for the second application.

Application 3. Simple interaction between two species
One of the simplest models that simulates the interaction between two species is governed by the equations Nondimensionalization technique in ordinary differential equations These are The first leads to the coefficients a o x i Dy o a o x i y i which determine two dimensionless groups Nondimensionalization technique in ordinary differential equations while the second leads to the coefficients which, in turn, provides the groups From (31) and (32), the order of magnitude of t o is given by The ratio of this time allows us to write As we will see later, F 3 (Δy 0 /y i ) = Δy 0 /y i , so that the last equation allows us to relate the unknown Δy o with the parameters given in the statement of the problem and to re-write the order of magnitude of t o as a function of known parameters: The application of classical dimensional analysis to this problem is immediate. Relevant list, formed by the magnitudes x i , y i , a o and b o , lead to the dimensionless groups (π-theorem) π 1 = x i /y i and π 2 = a o /b o . The solution for the dimensionless form of the unknowns Δy o /y i and t o , a o , x i (or t o , b o , y i ) are then a less precise solution of that provided by nondimensionalization. Now some cases are simulated to check the above results, Table 3. Case 1 is simulated in Fig 13. Since the curves are asymptotic, we choose t o as the time required by x variable to reduce its value to 0.2x i . From the numerical solution, t o = 1.099 and Δy o = 1. Using the expressions (33) and (34), the values (of order of magnitude unity as Nondimensionalization technique in ordinary differential equations expected) of the functions ϕ 1 , ϕ 2 and ϕ 3 , for argument unity, are obtained: Taking case 2, whose simulation shown in Fig 14, we obtain Δy o = 6, so that C 3 (4) = 4. Generalizing this result, i.e., making C 3 (c) = c, the expression of Δy o , in terms of the parameters of the statement (35), is demonstrated.
To finish, we describe some cases in which all the parameters presented vary. Cases 3 to 6 retain the same value for the group b o x i /a o y i so that C 1 (and C 2 ) do not change. Since b o x i / a o y y = 1 for all cases, C 1 (1) = C 2 (1) = 1,099according to the results of case 1. In this way, from (35) t o = 1.099a o y i = 1.099b o x i , an expression that is coherent with the simulation, Fig 15. As for case 7, Fig 16, b o x i /a o y y = 6, and t o = 0.24, from simulation. It is deduced that C 2 (6) = 0.96, a value of the order of magnitude unity as expected.
As we can see from these results, with normalized nondimensionalization technique, we extract dimensionless numbers of the order of magnitude unity that leads to solutions which fits perfectly with the obtained results of the numerical simulation, where we can check from

Final comments and conclusions
Although many authors consider that dimensional analysis and nondimensionalization are two similar approaches in the search for the dimensionless groups that rule a given problem, both formulated by ordinary or partial differential equations, coupled or not, the application of the second approach may lead to a more precise solution under certain hypotheses. The choice of suitable references, for which a profound knowledge of the physical phenomenon involved is required, that allow dimensionless variables to be defined by a range of values within the interval [0-1], provides the same order of magnitude to the coefficients of the new 'normalized' equations. The complete set of independent ratios between these coefficients has two inherent properties: they can be physically interpreted in terms of a balance of the quantities that counteract in the problem, and they are of the order of magnitude unity.
The explanation of the advantage of the nondimensionalization may seem over-subtle but it is not. Classical dimensional analysis starts with the definition of the relevant list of variables-  parameters and quantities-that influence the problem, including the unknowns being sought. However, no requirements are imposed on these variables except a physical reasoning, frequently insufficient, that justifies their inclusion in that list.
This, together with a definition of the dimensional basis (for example, length, mass and time in mechanics), allows us to obtain the dimensionless groups, to which, on the one hand, it is difficult to attribute a balance and, on the other hand, it is impossible to attribute an order of magnitude. By contrast, the choice of references to make the variables dimensionless in the proposed nondimensionalization processes is not arbitrary, but is subject to a kind of discrimination that normalizes the dimensionless equations that these variables define. The difference between dimensional analysis and nondimensionalization, as proposed in this work, means that many of the direct groups that emerge in dimensional analysis, such as mass ratios or geometrical factor, do not appear with nondimensionalization which leads to a more precise solution, as the above applications illustrate.
Concerning what is referred to the future prospects of the use of this technique, we can say that during tens of years, the process of nondimensionalization has been applied (generally to models defined by coupled partial differential equations) in a manner that we might call classical, and the scientific community has agreed with the existence of dimensionless numbers with orders of magnitude far greater than the unity. Can they have any physical meaning, in terms of balance, these numbers? Of course not, but there are texts in which recognizes them this meaning. They are poorly learned numbers and now this is difficult to renew. However, the processes made by ordinary differential equations are not usually subject of controversy in the scientific literature. It should be noted that the problems dealt with in this work refer to physical processes governed by natural laws. is an issue to which we will dedicate efforts in the future. We can say, without fear of error that the field is open, especially in coupled problems, and we hope to be able to spread our humble contributions to the scientific community.