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Mathematical modelling - Final report

This is the final report written by **Group 60** (FIRE) for the mathematical modelling course (**DAT026**) at Chalmers University of Technology.

Group 60 consists of:

**Mazdak Farrokhzad**901011-0279

twingoow@gmail.com

Program: IT

Time spent: \(22\) hours, \(06\) minutes

**Niclas Alexandersson**920203-0111

nicale@student.chalmers.se

Program: IT

Time spent: \(31\) hours, \(22\) minutes

The reason why some of our time differs is due to the time spent in making illustrations, which Mazdak has not measured.

**By submitting this report we confirm that the entire report is our own work, that it has been written exclusively for this course, and that we both actively participated in solving all the exercises of the course.****We will not spread our work in this course to others who take or can be expected to take the course.**

\label{section:purpose-of-this-report} During the past two months, we, the authors of this report, have been taking a course in mathematical modelling and problem solving at Chalmers university of technology. The course focuses on ways in which different aspects of reality, the world in which we live with all its details, imperfections and uncertainties, can be abstracted away and represented in simpler, more pure if you will, mathematical terms. These abstractions, these simplified representations of reality, are called *mathematical models*. Using these models, we gain the ability to take real world problems, and convert them into a form which we can then solve with the help of mathematics. If our model is good enough, the insights we gain through solving these simplified mathematical problems will then also tell us how to solve the real world problems on which our models were based. When we speak of reality, this can of course also be simulations of our realtities in computers, or other possible realities hypothesised by multiverses - but this is out of the scope of this course.

Unlike many other mathematics courses, the primary focus of this course has, perhaps counterintuitively, not been on learning mathematics. Instead, it has focused on mainly using mathematics that we already knew, in the context of solving problems from the real world. Doing this requires finding a way to model the problem using mathematics, and then solving the problem in its mathematical form. This focus on modelling and problem solving sets the course apart from other courses, which are often more theoretical and perhaps algorithmic in nature, with problems which can often be solved using well defined series of steps, such as taking the derivative of a function, multiplying two matrices, or evaluating a logical expression, while this course often requires more creativity and heuristic approaches such as trial and error.

\label{section:process-of-modelling}

Describing the process of modelling and problem solving is tricky business - it is not an exact science, there is no algorithm for it. Instead, we have to rely on heuristic methods. It is difficult to separate the process of modelling from the process of problem solving since the model we opt to use depends almost entierly on the problem we are trying to solve, there is no “one size fits all”.

However, we can make a rough description of the process. First, we start with **what we know** and **what we need** (want to know) and from there, we can make **assumptions** and **restrictions** about both. Now, things get messy - at any future point, we might already know how to solve or model a problem of this kind and thus we can skip a bunch of steps. At this point, we must try to represent what we need and know in a model of some type that we have heard of or used before, or otherwise might find using the various sources at our disposal.

From now on, the process diverges depending on what type or types of model that we chose. For a differential equation for example, we will need to solve it, while for a discrete optimization problem such as finding the the Shortest Path (discussed in \ref{section:optimization-in-mathematical-terms}) it will require that we first constructing a graph and decide upon a suitable algorithm such as *Dijkstra’s algorithm*.

It is often a good idea to delay the introduction of any data and numerical values that we have until we want to verify and test things and have a model and suitable algorithms or formulae to work with. Using abstract variables can help us notice relationships and connections which we may otherwise overlook. If by this stage we are not satisfied with the results and can not verify them against our perceptions of reality, we must go back to the drawing board and refine either our model and perhaps choose a different type of model this time, or revise assumptions and restrictions. Perhaps more radically, we can ask if the problem itself is the right one to solve.

\label{section:model-types}

Every week we have been given a set of problems to model and solve - with each week’s problems in some manner related to each other. The common factor has usually been how to conveniently represent them in the mathematical world - that is to say, they can often be represented using similar types of models.

The model types we have dealt with are the following:

Functions and equations as models

Optimization models

Dynamic models

Probability models

Discrete models

These types should not be thought of as mutually exclusive - the opposite is often true. It is possible to find a model that involves all of the types mentioned above.

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