- Classify ordinary and partial differential equations (ODE and PDEs)
- Solve general 2nd order ordinary differential equations
- Understand how common ODE arise from the separation of variables of physically important PDEs
- Know how and when to use the solution in series method for solving ODEs
- Understand the properties of difference equations and how they arise via numerical methods from ODEs
- Understand how to solve linear systems of ODEs
- Know how to use phase-portrait to study the behavior of non-linear systems; including understanding limit cycles
- Apply perturbation methods to the solution of ODE and algebraic equations
- Use the method of multiple scales when it is appropriate
A wide range of physics phenomena are described by differential equations (both partial and ordinary); these include but are by no means limited to sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, and quantum mechanics. It is clear that to design modern planes, car, bridges and other structures an understanding of how to solve these type of equations is essential. In the modern world cheap computation power is available and hence the solution of these equations is often done using (commercial) computer codes and methods like the Finite Volume of Finite Element method.|
In this course we take a different approach and look for analytical solutions that is solutions without the aid of a computer. Armed with these 'old' techniques, we ask the questions: Why are there different methods for solving ODEs? Which method is best for which job; How do we know our (commercial) numerical solution is correct?
We start by introduction the idea of a partial differential equations, that is, a differential equation that contains unknown multivariable functions and their partial derivatives; and the highly important special case the ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives. The course will show that solutions techniques for these equations build on each other and solving a more complex form is often facilitated by reducing to one of the simpler forms that your already know how to solve. For this reason the course will focus on ODEs and their solution techniques, which themselves appear in many applications; for example the flow of a liquid in an emptying barrel or the vibrations of a string.
The purpose of this course is: to become familiar with solution methods for differential equations, to know which tools to use when, and know how to check if the solution you obtain is correct