Hello and welcome to my ongoing video course about Partial Differential Equations, already consisting of 7 videos. The video series is called Partial Differential Equations (PDEs). The course it’s not finished yet. This course builds on the foundation laid in the Ordinary Differential Equations (ODEs) series, expanding into the more complex and multi-dimensional world of PDEs. Alongside the videos, you’ll find additional text explanations to help reinforce the material. To test your knowledge, use the quizzes, and refer to the PDF versions of the lessons whenever needed. If you have any questions, feel free to participate in the community discussion forum. Without further ado, let’s get started!
Part 1 - Introduction and Definition
Let’s start the video series with some basic notions we will use throughout the course. For example, we have to know what we mean by a partial differential equation and a classical solution of it.
Part 2 - Laplace’s Equation
One of the most important partial differential equations is Laplace’s equation where the solutions are called harmonic fucntions. We will try to find solutions that respect the radial symmetry because these can be used a base for the construcion of other solutions.
Part 3 - Fundamental Solution of Laplace’s Equation
The fundamental solution for Laplace’s equation is sometimes also called Newtonian kernel because of its application in physics. Indeed, in three dimensions, we have $\gamma(x) = \frac{1}{4 \pi} \frac{1}{| x |}$ for the fundamental solution. Obviously, there is a singularity at the origin, but it turns out that inside an integral it is no problem at all. Let’s show that for an arbitrary dimension $n$ as well.
Part 4 - Mean-Value Property of Harmonic Functions
The solutions of Laplace’s equation have some nice properties that can be surpring. For example, we can show that the values of the function at a point is already determined by the values of the function on a sphere around this point. This is what we mean by the mean-value property of a function. Let’s prove this by using Green’s identity and the fact that we can move a derivative inside the integral under our assumptions here.
Other videos related to this topic:
- Multidimensional Integration 9 | Integration with Polar Coordinates
- Multidimensional Integration 11 | Green’s Identities
- Measure Theory 10 | Lebesgue’s Dominated Convergence Theorem
- Multidimensional Integration 12 | Differentiation Under The Integral Sign
Part 5 - Maximum Principle for Harmonic Functions
A direct consequence of the mean-value property of harmonic functions is the so-called maximum principle. It roughly says that the maximum of any harmonic function $u : \Omega \rightarrow \mathbb{R}$ is always found on the boundary $$\partial \Omega$. We can even distiguish a strong and weak maximum principle depending if we have a connected set $\Omega** or not.
Other videos related to this topic:
Part 6 - Proof of Maximum Principle
Let’s prove the two statements from the last video. We will need some topology knowledge for connected sets.
Other videos related to this topic:
Part 7 - Uniqueness of the Boundary Value Problem for Poisson’s Equation
We can generalize our the important PDE given by Laplace’s equation to Poisson’s equation by changing the right-hand side to a continuous function $f$. So we search for functions $u$ where the Laplacian $\Delta u$ is given by $f$ on the whole open domain $\Omega$. Now, if we also claim that $u$ should be equal to a continuous function $g$ on the boundary $\partial \Omega$, then we speak of a boundary-value problem. In particular, in this case, we have so-called Dirichlet boundary conditions. It turns out that such a boundary-value problem on a bounded set $\Omega$ has at most one solution, so we definitely have uniqueness if a solution exists. As we will show in the video, this property immediately follows from the maximum principle for harmonic functions.
Connections to other courses
Summary of the course Partial Differential Equations
-
You can download the whole PDF here and the whole dark PDF.
-
You can download the whole printable PDF here.
- Ask your questions in the community forum about Partial Differential Equations.
Ad-free version available:
Click to watch the series on Vimeo.