The parameter is the increase in the number of individuals per time and per individual. The simple model (2.1) also controls the pressure in the atmosphere (under many assumptions such air is an ideal gas in equilibrium). For a decaying population, uc can be a characteristic number of members in the population, e.g. the original populationI.
The next task is to insert the new dimensionless variables into the governing mathematical model. In the present problem, we are happy to know the exact solution for any value of the input data as long as asa is a constant. In the present problem, where we are lucky to have an exact solution u(t) =Ie−at, we can look at this to see explicitly datu∈(0, I] so thatuc=I gives ¯u∈(0 ,1] ).
This is a remarkable result in that all physical parameters (aandI) have been removed from the model. We can use this special case solution to control our reasoning about scaling in the more general setting. Therefore, to balance the magnitude of ¯uandd¯u/d¯twe, tc= 1/a must be chosen, resulting in the scaled ODE problem.
In particular, if the motion starts from rest, β= 0, and there are no physical parameters in the scale model.
Variable coefficients
Scaling a cooling problem with constant temperature in the surroundings
A natural choice, as argued in other exponential decay problems, is to choose tck= 1, leaving us with the scaled problem. 2.18).
Scaling a cooling problem with time-dependent surroundings
The α quantity measures the ratio of temperatures: amplitude of oscillations versus distance from initial temperature to the average temperature for large times. The β-number is the ratio of the two time scales: the frequency of the oscillations inTs and the inverse e-folding time of the heat transfer. For clear interpretation of β we can introduce the periodP= 2π/ω of the oscillations in Ts and the e-folding time e= 1/k.
The original problem contains five physical parameters: k, T0, Tm, a, andω, but only two dimensionless numbers appear in the scale model (2.23). In fact, this is an example where application of the Pi theorem (see Section 1.1.3) falls short. Scaling the differential equations, on the other hand, shows us that the two parameters Tand T0 only affect the nature of the problem through their difference.
The fileosc_cooling.py contains solvers for the dimensions problem and for the scaled problem. For the small β value, the oscillations in the ambient temperature are slow enough compared to the heating and cooling process to follow the. There are two time variations of interest in the present problem: heat is transferred to the environment at a rate k, and the environment has a temperature variation with a period that goes like 1/ω.
If we are fortunate enough to have an analytical solution at hand, we can inspect this solution and find that k affects the problem through the decay factor kt and ω affects the problem with fluctuations sin(ωt).) The parameter k related to the decay temperature indicates time scale c= 1/k, and ambient temperature fluctuations on the time scale tc= 1/ω. The transfer of temperature from T0 to the ambient level, Tm, takes place as e−kt, so that in this process tc= 1/ki is a characteristic time. Then the body temperature just responds to the fluctuations and sin(ωt) (and cos(ωt)) dominates.
For this purpose, we scale Ts by (eg) Tm, which means calculating Tm in the denominator. We conclude that this scaling is inferior because using the temperature scale T0−Tm allows reusing the software for the unscaled problem and only two dimensionless parameters appear in the scaled model. However, the initial condition becomes ¯T(0) =T0/Tm and the ratio T0/Tm is a third dimensionless parameter, so this scaling is also inferior to the one above with only two parameters.
Scaling a nonlinear ODE
Let us briefly mention another possible temperature scale: ¯T=T /Tm, motivated by the fact that ast→ ∞, T will oscillate around Tm, so this ¯T will oscillate around unity. A natural choice is tc= 1/ as in other exponential growth models as this results in the term on the right going over unity, just like the left. If the scale is correct, ¯u and its derivatives are of order unity, so the coefficients must also be of order unity.).
We also introduce a dimensionless parameter. by measuring the proportion of the initial population compared to the maximum, we obtain a dimensionless model. 2.29). The advantage of (2.29) is that the software for the unscaled model can be easily applied to the scaled model by choosing I=α,M = 1 and = 1.
SIR ODE system for spreading of diseases
Therefore, R0 reflects the ability of the disease to spread and is consequently an important dimensionless quantity known as the basic reproductive number2. This number reflects the number of infected people caused by one infectious individual during the duration of the disease.
SIRV model with finite immunity
The quantitype-1 can be interpreted as the average time it takes to successfully inoculate a susceptible substance. Writing γ=ν−1/μ−1 and δ=ν−1/p−1 gives the interpretation that γ is the ratio of the average time to recover and the average time to lose immunity, while δ is the ratio of the average. time to recover and the average time to successfully vaccinate a susceptible person. The value δ= 0.5 reflects that it takes two weeks to successfully vaccinate a susceptible person, but the effect of vaccination is still dramatic.
Michaelis-Menten kinetics for biochemical reactions
From the equations, we can see that k+ is measured in s−1mol−1, while k−inq is measured in s−1. It is convenient to get rid of the mole unit for the amount of substance. When working with dimensionless quantities, only the ratios of the rate constants and not their specific values are needed.
A common assumption is that the formation of [ES] is very fast and that it quickly reaches an equilibrium state, [ES]= 0. The choice of scales is actually a rather complicated matter that requires extensive analysis of the equations to determine characteristics of the solutions. Much literature is written about this, but here we must take a simplified and pragmatic approach.
We will consider K and when choosing scales such that these symbols appear naturally in the scaled equations. The variable [ES] starts and ends at zero, and its maximum value can be approximated from the equation for [ES] by setting [ES]= 0, which gives Note that software developed for solutions cannot be reused for solutions as the final system has a slightly different structure.
Such checks along with the conserved quantities above must be performed at each time step in a simulation. In the scale system we can assume small, which from (2.60) gives rise to the simplification E¯= 0, and thereby the relation ¯Q= ¯ES. This is a more precise analysis than the one leading to (2.56), since we now realize that the mathematical assumption for the simplification is → 0.
It is easy to make algebraic mistakes when deriving scaling equations, so it is always wise to perform consistency checks. We also observe that all the unknowns vary between 0 and approximately 1, indicating that the scaling is successful for the chosen dimensionless numbers. This is reasonable since with very little enzyme compared to the substrate (→0) very little will happen.
Vibration problems
- Undamped vibrations without forcing
- Undamped vibrations with constant forcing
- Undamped vibrations with time-dependent forcing
- Damped vibrations with forcing
- Oscillating electric circuits
We see that the oscillations around the equilibrium point in the gravity field are identical to the horizontal oscillations without gravity, except for an offset mg/(kI) in the displacement. Furthermore, there are two time scales, one for the free vibrations of the systems and one for the forced vibrations F(t). We continue the SymPy session from the previous section and perform much of the same steps.
Note that we use wforω= k/m in the computer code (to get a more direct visual counterpart to ω). This is also a relevant choice in the more general case I=0, V=0, unless I or V is so large that it dominates the amplitude caused by the forcing. Going back to (2.71), we can demand that all three terms in the differential equation have the same magnitude.
The formula for uc is a kind of measure of the ratio of force to spring (the dimensionless number A/(kuc) would be this ratio). Note that this is a problem that requires very high precision in numerical calculations. Finally, we can look at the case where ψω such that uc=A/(mψ2) is a relevant scale (ie, dropping ω2 relative to ψ2 in the denominator), but in this case we must use tc= 1/ψ since the force changes much faster than the free vibrations of the system.
The graphs in Figure 2.10 and 2.11 are produced from numerical simulations with 160 time steps per duty period.). We are interested in the case with small damping, b <2mω, where the solution is an exponentially damped sinusoidal function. We have split the differential equation of order introduced B=12b/m and let A1 represent A/m to simplify the expressions and help SymPy with fewer symbols in the equation.
Without these simplifications, SymPy stalls in the calculations due to too many symbols in the equation. The problem is actually a solid argument for scaling differential equations before asking SymPy to solve them, since scaling effectively reduces the number of parameters in the equations. Much of the discussion of scales in the previous sections is relevant even when damping is included.
The rest of the ODE is given in the previous section, and the specific formulas depend on the choices of canduc. It is easy to reuse a solver for a general vibration problem also in the dimensionless case.
