Ordinary differential equations and initial value problems

Mathematical models of physical systems require the appropriate mathematical framework for describing specific processes and interactions. For example, if two variables,x and y, are directly proportional to each other we write,

y ∝x. (2.3)

This describes the relationship whereby an increase/decrease in the variablexresults in an increase/decrease in the variable y and vice versa. There is, however, a lack of information regarding the magnitude of the increase iny for a given increase in

x. We therefore transform (2.3) into the following mathematical equation,

y=kx, (2.4)

where k is a constant, known as the coefficient of proportionality, and hence the ratio ofx and y is equal to this constant value,

k= y

x, (2.5)

for all non-zero values ofxandy. Figure 2.5 illustrates how the relationship between

x and y is influenced by k. When k = 1, y = x and thus y bisects the positive quadrant of the axes through the origin. When the value ofkis increased i.e. k= 5, the magnitude ofy is increased fivefold for everyx value thus producing a steeper line. When the value ofk is decreased i.e. k= 0.2, the magnitude of y is decreased

Figure 2.5: Comparisons of varied k values in the linear function y = kx. The gradient of the line increases for larger values ofk and decreases for smaller values ofk.

fivefold for everyxvalue thus producing a line with decreased steepness. This direct proportionality is exhibited by the relationship between the diameter of a circle and its circumference, that is,

C=πd, (2.6)

whereCandddenote the circumference and diameter of a circle respectively and the coefficient of proportionality is equal to the irrational constant π. Proportionality gives rise to simple linear relationships such as (2.6) however, the same principles can also be applied to more intricate and complex relationships. Mathematical models are often formulated in light of information regarding the rate of change in the relevant variables over time. In order to describe the rate of change over time mathematically for a given variabley, we take the derivative with respect to time,

t,

dy

dt, (2.7)

wheredy anddtdenote the change in yand the change in trespectively. Using this framework, we are able to construct mathematical models of systems in which the rate of change in the output variable of interest is proportional to the variable itself. For example, consider a population growth model whereby the rate of change of the

population at any given time is directly proportional to that population,

dP

dt ∝P, (2.8)

whereP andtdenote population and time respectively. The variableP on the right hand side is positive, indicating population increase or growth; negative terms are used to model the behaviour of negative growth or decay in such systems. This relationship is transformed into a mathematical equation in the same way as (2.3), that is,

dP

dt =kP, (2.9)

where the constantk is a newly derived coefficient of proportionality. Both P and its derivative, or differential, appear in (2.9) and hence we refer to this equation as an ordinary differential equation (ODE). In order to determine the function that describes the growth of this population over time, we must solve this ODE and therefore obtain a function for P only. Solving ODEs can involve many different calculus-based methods and techniques depending on the nature of the given equa- tion. This particular case is sufficiently straightforward to employ the method of separation of variables as follows:

dP dt =kP, =⇒ 1 P dP =k dt, =⇒ Z 1 P dP = Z k dt, =⇒ lnP =kt+c, =⇒ P = exp(kt+c), =⇒ P = exp(kt) exp(c), ∴ P =Aexp(kt),

wherecis the constant of integration andA= exp(c) is also constant. Assuming that we have sufficient knowledge of the system, we can make an appropriate estimate for the value of the constant parameterk. Taking k= 0.5 we have,

At this stage, (2.10) is a general solution to the ODE by virtue of the fact that, although we have determined the function that describes the evolution of the popu- lation over time, the function can exhibit an infinite number of solutions dependent on the value of the constant A. In order to determine the exact solution to the ODE, we require additional information regarding the size of the population at a given time point. This is known as an initial condition since the information is given for the case when time is zero and consequently, together with the original ODE (2.9), describes an initial value problem (IVP). In this example, consider the initial condition that the population is equal to ten when time is zero, P = 10 at t= 0, then,

10 =Aexp(0),

=⇒ A= 10,

∴ P = 10 exp(0.5t). (2.11)

The function (2.11) is the exact solution to the IVP, exhibiting a single trajectory describing the growth of the population (Figure 2.6). The function can be used

Figure 2.6: Exact solution to the IVP described by (2.9) and the initial condition

P = 10 att= 0 (P(0) = 10). The dashed arrows depict the graphical prediction of the population size att= 4.

to predict the size of the population at future time points by simply evaluating the function at the time point of interest. For example, if we want to know the projected population at four time points after population growth was initiated, we determine

P att= 4,

P = 10 exp(0.5×4),

=⇒ P = 10 exp(2),

∴ P = 73.9. (1 d.p.)

Similarly, if we want the time taken for the population to reach a particular size then we can substitute the desired population size into (2.11) and solve fortinstead. Note that the choice of the constant parameterk in (2.10) was made assuming sufficient knowledge of the system. However, in practice, it is common that the relevant model parameters are not well established. This has implications regarding the interpretation and plausibility of solutions to IVPs that will be covered in further detail in the remainder of this chapter.