An Introduction to the Modelling of Real-World Problems by the Simplest Ordinary Differential Equations

Stanisław Migórski

1 Introduction and preliminary terminology

This note is a preliminary version of a self-contained introduction to the modeling of selected real-life phenomena by simple ordinary di.erential equations.We restrict ourselves to a discussion on first order ordinary di.erential equations, leaving the detailson models described by higher order equations and systems of first-order equations to the sequel of the present work.

The text was designed for computer science students who have a calculus back ground1 and have taken prior physics courses (cf. [4]). It is our belief that computer science students should know how to model a selected problem, particularly in light of rapidly changing technologies. We begin with preliminary notations and defnitions.

By a dierential equation we mean a mathematical equation involving an unknown function of one or several variables and its derivatives of various orders. The order of the di.erential equation is the order of the highest derivative of the unknown func- tion involved in the equation. For instance, the equation

$(1)$ $x^' = f(t,x)$

is called an ordinary differential equation (often abbreviated to ODE) of first order² in the normal form. It relates an independent variable t to an unknown function x and its first order derivative. The function f is given f: I × E → E, where I is an interval³ and E is a prescribed space and we are looking for a function x: I → E. Of course the equation (1) obliges the unknown function x to have some restrictions: it should be differentiable in a suitable sense and the compose function f(t, x(t)) should have a meaning. Classically, we admit the following

Definition 1

A function x: I → E is called a solution of the equation (1) on the interval I, if x C¹(I,E)⁴ and x’(t) = f(t, x(t)) for all t ∈ I.

Remark 2

We remark that if f: I × E → E is a continuous function, we may require only that a solution x: I → E is only differentiable. This follows from the observation that in this case the composition of continuous functions f(t, ·) and x(·) is continuous and it is equal to x’(·). Hence x’ is continuous on I and thus x is a solution.

If the right hand side of the equation (1) can be expressed as a product g(t)h(x) where g depends only on t and h depends only on x, then the differential equation (1) is called separable. If the right hand side f is independent of t, then the resulting differential equation is called autonomous. Every autonomous differential equation is separable.

When the function f in (1) is (affine) linear with respect to the second variable, the first order equation of the form x’ = α(t)x + β(t) with prescribed functions and is called a nonhomogeneous linear differential equation. If β= 0, then the equation x=α(t)x with a prescribed function, is called a homogeneous linear differential equation.

Definition 3

Let t₀ I and x₀ ∈ E. A problem in which we are looking for the unknown function x: I → E of a differential equation (1) where the value of the unknown function at some point is known x(t₀) = x₀⁵ is called an initial value problem (in short IVP) or a Cauchy⁶ problem for (1).

Analogously as above, we have

Definition 4

Let t₀ ∈ I and x₀ ∈ E. We say that a function x: I → E is a solution of the IVP

$\begin{cases}x'=f(t,x)\\x(t_{0})=x_{0}\end{cases}$

if x is a solution to equation (1) on the interval I and it satisfies x(t₀) = x₀. Sometimes if no initial condition is given, we call the family of all solutions to the differential equation (1) the general solution.

2 Modelling

“How do we translate a physical phenomenon into a set of equations which describes it?” – this is certainly one of the most di.cult problems that scientists deal with intheir everyday research. This problem is a di.cult one since it is usually impossible to describe a phenomenon totally, so one often tries to reformulate a real-world problem as a mathematical one making certain simplyfying assumptions. As a result one usually describes the system approximately and adequately.

The problem of generating “good” equations is not an easy task. The set of equations one deals with is called a model for the system. In general, once we have built a set of equations, we compare the data generated by the equations with real data collected from the system (by measurement). When the two sets of data “agree” (or are “sufficiently” close), we gain con.dence that the set of equations will lead to a good description of the real-world system. If a prediction from the equations leads to some conclusions which are by no means close to real-world future behavior, we should modify and “correct” the underlying equations. When creating a model, it is necessary to formulate the problem under consideration into questions that can be answered mathematically.

The following are basic steps in building a model.

(1) Clearly state the assumptions on which the model will be based. These assumptions should describe the relationships between the quantities to be studied.

(2) Completely describe the parameters and variables to be used in the model.

(3) Use the assumptions (from step (1)) to derive mathematical equations relating the parameters and variables (from step (2)).

A large number of laws of physics, chemistry, economics, medicine, etc. can be for mulated as di.erential equations. They serve as models that describe the behavior of complex systems. The mathematical theory of differential equations first developed together with the sciences where the equations had originated and where the results found application. Whenever a mathematical model involves the rate of change of one variable with respect to another, a di.erential equation is apt to appear. As we will see in the following sections, diverse problems, sometimes originating in quite distinct scienti.c .elds, may give rise to identical di.erential equations. Whenever this happens, the mathematical theory behind the equations can be viewed as a unifying principle behind diverse phenomena.

In the following sections we provide examples of mathematical models of several rela- tively simple phenomena which are described by ordinary differential equations of the form (1). We mainly concentrate on modeling issues. The modern theory of ordinary differential equations together with known techniques, methods and applications can be found in references [1] and [2].

In the following sections we provide examples of mathematical models of several relatively simple phenomena which are described by ordinary differential equations of the form (1). We mainly concentrate on modeling issues. The modern theory of ordinary differential equations together with known techniques, methods and applications can be found in references [1] and [2].

Newton’s law of cooling

It is known from experimental observations that the surface temperature of an object changes at a rate proportional to its relative temperature. The latter is the dierence between its temperature and the temperature of the surrounding environment. This is what is known as Newton’s law of cooling. Find the dependence of the temperature of the object on time.

We denote by T(t) the temperature of the object at time instant t and by M the constant temperature⁷ of the environment. The Newton law of cooling gives a differential equation of the first order

$(13)$ $\frac{dT}{dt}=-k(T-M)$

where k > 0. The proportionality constant in (13) is negative since the temperatures decrease in time and the derivative $\frac{dT}{dt}$ must be negative for T > M. The equation (13) is a separable ODE. We assume that the initial temperature of the object is prescribed

$(14)$ $T(t_{0})=T_{0}$

where T₀ stands for the initial temperature of the object. Thus, the dependence of the temperature of the object on time is described by the initial value problem (13) and (14). Its solution has a form⁸

$(15)$ $T(t)=M+(T_{0}-M)e^{-kt}$

Example 18 (Time of death)

It was noon on a cold May day in Krakow: 160C. Detective John Kowalski arrived at the crime scene in a hotel room to find the sergeant leaning over the body. The sergeant said there were several suspects. If they knew the exact time of death, then they could narrow the list. Kowalski took out a thermometer and measured the temperature of the corpse: 34.50C. He then left for lunch. Upon returning at 1:00pm, he found the body temperature to be 33.70C. Find the time the murder occured.

Solution.

We proceed in two steps. Let T₀ = 34.5, M = 16, t = 60 and T(60) = 33.7. First, we find the value of the coefficient k as follows

$k=\frac{1}{t}ln\frac{T_{0}-M} T(t)-M}=\frac{ln8.5-ln17.7}{60}\approx7.3677\cdot10^{-4}$

In other words, we have determined k from the information on two temperatures measured by Kowalski, at the point T₀ = 34.5 and the point T(60) = 33.7.

Knowing the value of k, we are able to find a time instant td at which the crime happened. To this end we take into account other two points T(0) = 37⁹ and T(t_d) = 34.5.

$t_{d}=\frac{1}{t}ln\frac{T(0)-M}{T(t_{0})-M}=\frac{ln(37-16)-ln(34.5-16)}{k}=\frac{ln21-ln18.5}{k}\approx$
$\approx172.036$

Therefore the murder occured about 172 minutes before noon, that is, around 9:08am. We conclude that in order to find the time of death it is necessary to measure the body temperature twice¹⁰.

References

[1] Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003).

[2] Z. Denkowski, S. Migórski and N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston, Dordrecht, London, New York (2003).

¹ Some universities, the Jagiellonian among them, make linear algebra a prerequisite for differential equations. Many schools, especially engineering, only require calculus.

² The most general (implicit) form of an ordinary differential equation of the first-order is as follows: F(t, x, x’) = 0 with a prescribed function F.

³ The interval I may have a form [a, b], [a, b), (a, b], (a, b),-., b] and [a,+.).

⁴ A function x: I → E is C1, if x is dierentiable on I and its derivative x’ is a continuous function.

⁵ This condition is called an initial condition.

⁶ Augustin Louis Cauchy (1789–1857), a French mathematician.

⁷ The constant M is called the medium temperature since it is the temperature of the medium an object is immersed in.

⁸ This can be easily checked by the reader.

⁹ Assuming the dead person was not sick and had a temperature of 37⁰C. The interested reader may estimate the time of death, if instead of 37⁰C we suppose 36.6⁰C.

¹⁰ Remember this method unless you are sergeant Colombo.