Reframing existence and uniqueness theorems for the non-homogeneous wave equation

Bachelor thesis (2022)

Authors

C.R. van Ooijen Electrical Engineering, Mathematics and Computer Science

Contributors

Carolina Urzúa-Torres Numerical Analysis - EEMCS (supervisor 1)
M.C. Veraar Analysis - EEMCS (supervisor 2)
Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright: © 2022 Cas van Ooijen
Numerical Analysis Differential Equations Analysis Wave Equation
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Published Date

30-06-2022

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

This thesis is about the wave equation. The wave equation describes waves that propagate through a certain medium. The solution of the wave equation is a mathematical description of what that wave looks like. There are many fields of study where the wave equation is used to model processes that behave like travelling waves. For instance, a vibrating string or membrane can be modelled by the wave equation very well. Furthermore, the wave equation can be used to model light or sound waves and how they reflect and refract when travelling
through different materials.
Because the wave equation is such an important tool to model these phenomena, there are many people working on solving the wave equation in different contexts, i.e. finding a solution. However, it is difficult (often impossible) to find an exact solution. Therefore, people usually calculate a ’solution’ that is approximately correct.
An important question to ask is: ’Does the wave equation always have a solution?And is there only one solution?’. Physically, the answer is clear. If a string is vibrating, it clearly cannot vibrate in two ways at the same time and it will always vibrate in some particular way. Even though the answer is clear physically, answering this question mathematically is more difficult. It is good to remember that the wave equation is just a mathematical model of propagating waves, and it could very well be that the wave equation has more than one solution or no solution at all in some particular context.
The answer of this question is important for the people calculating approximate solutions to the wave equation. If there does not exist a unique solution, the process of calculating the approximate solution may break down.
Answering that question is the subject of this thesis. We will first introduce the necessary mathematical tools, after which we will use existing research to find out under which conditions the wave equation has one, and only one solution. Finally, we will extend our results to more general equations than just the wave equation.