Mathematical Modelling

Mathematical modelling is based on the concept of a mathematical model. A model is a simplification of a complicated phenomenon with the help of mathematical terms and symbols [4]. The construction of a model may require knowledge of other scientific fields besides mathematics, as well as the ability to make sophisticated guesses when it comes to collecting information, testing, and so forth.

The interplay between mathematics and its applications has been going on for a long time. In the early days of our civilization, mathematical skills and practices grew out of the need to carry out everyday duties. A society gradually developed that was based on agriculture. The annual rhythms of the flooding, draining, and growing seasons required flood control, irrigation systems, and a calendar system. Also, storage facilities had to be built and bioprocesses like beer, wine and cheese making to be controlled. The early mathematical skills like arithmetic, elementary algebra and geometry were developed to serve these simple but crucial tasks.

In this chapter, we attempt to shed light on modelling different kinds of systems with the help of some examples. To start off, we should explain what we mean by the system itself, and what mathematical modelling of such a system denotes. A system stands for a group of different objects, or elements, that are interacting with each other and their associated characteristics (numerical values). Systems are often dynamic, which means that the state of the system changes with time. Hereby, the state of the system accounts for the numerical values of its components. Usually, the most interesting aspect about a system is its equilibrium state, i.e. a state in which the component values remain constant with respect to time. For example, a hospital can be considered a system with the patients, doctors and nurses its components. The patients might have their reasons of seeking medical care as a characteristic, and respectively the staff’s characteristic could be their field of expertise. In addition, the hospital has different equipment (resources) such as an x-ray machine, a laboratory, beds etc. A lack of these resources might cause bottlenecks in the flow of patients. Such a system is an example of a traditional discrete system, in which the arrival or departure of a patient quickly changes the hospital’s state. Typically, these kinds of systems are also stochastic systems, because the patient’s arrival and staying time are often random variables. For such system, we are usually interested in the adequateness of the resources or in the duration of a patient’s stay until he has passed through the treatment system. Figure 3.1 shows a process flowchart for the functionality of a hospital’s dispensary outpatient clinic [4].

The examples on “network design” (p. 15), “river and flood models” (p. 20) and “urban water systems” (p. 21) lead us to consider networks. A useful way to describe a network is to define for each pair of nodes a function whose value is 1 if there is a direct connection between these nodes in the network, and 0 otherwise. More generally, x = 1 can be used to indicate that a certain event occurs and x = 0 that it does not. Indeed, binary (i.e., 0-1-valued) variables appear in many models, and so do also other integer-valued variables. In this chapter we shall take a look at such models.

Soft computing methods of modelling usually include fuzzy logics, neural computation, genetical algorithms and probabilistic deduction, with the addition of data mining and chaos theory in some cases. Unlike the traditional “hardcore methods” of modelling, these new methods allow for the gained results to be incomplete or inexact. Methodologically, the different approaches of these soft methods are quite heterogeneous. Still, all of them have a few things in common, namely that they have all been developed during the last 30–50 years (Bayes formula in 1763 and Lukasiewicz logic in 1920 being the exceptions), and that they would probably have not achieved their current standards without the exceptional growth in computational capacities of computers.

Let us consider a system and let us assume that we have some theoretical knowledge (or at least reasonable assumptions) about its behaviour. Such knowledge typically includes being aware of the underlying phenomena and recognizing the entities that describe these phenomena and their interactions. Yet, we do not necessary need to know formulae for explicitly describing all interactions. In such a situation, dimensional analysis can help us in rendering our knowledge more explicit.

Helium filled carnival balloons take off easily and can fly long distances. How high can they rise? Can they be of concern to airplanes? Now you might say that this latter question sounds a bit artificial and naive. However, helium balloons are regularly sent to the upper atmosphere for observing weather data. When designing and controlling such measurements, one has to know how high the balloons will rise and how much time they need for the journey.

In practical applications the “complete” model, i.e., a model that contains all features that the experts in the application domain consider important, is often quite complicated and difficult to analyse mathematically. A straightforward numerical realization is often costly and may give very little qualitative understanding of the situation. It is therefore important to study if the model can be systematically simplified in order to enhance a qualitative analysis/understanding.

Let us examine the behaviour of sound in a gas or in a liquid medium. From a physical point of view, the sound we hear is created by the pressure change in the medium surrounding us that is sensed by our ears. The equations describing the behaviour of a liquid or a gas are based on well-known equations of fluid mechanics. Therefore in acoustics, they are often referred to as fluids. In the following sections we present a simple wave equation, which is the simplest of (linear) equations used to model acoustical phenomena. Even though the wave equation is quite a simplified model, it has proven to be extremely useful for describing the behaviour of sound in the most common fluid we face every day, namely air.

Inverse problems can best be characterized through their counterparts, namely direct or forward problems. A classical forward problem is to find a unique effect of a given cause using an appropriate physical or mathematical model. Forward problems are usually well-posed, i.e., they have a unique solution which is insensitive to small changes of the initial values. Inverse problems are the opposite to forward problems, meaning that one is given the effect and the task is to recover the cause. Inverse problems do not necessarily have unique and stable solutions, i.e., they are often ill-posed in the sense of Hadamard [7].

Examine a spring-mass oscillator with a gradually changing spring constant, such that the spring is half as rigid at the other end. Describe the activity of such a “slacked” oscillator.