$\vec{w}h\alpha\mathfrak{t}\;\; i\mathbb{S}\ldots$


Maciek Korzek (TU Berlin)
2011/10/21, 13:00
Before the BMS Friday Colloquium by Prof. Charlie Elliot
Urania Berlin, at the BMS Loft (3rd floor)
About what?

One of the most fundamental differential operators appearing in partial differential equations (PDEs) is the Laplace operator. Its understanding is essential to be able to treat models describing real-world problems. One of the most basic PDEs relates the rate of change of some quantity to the Laplace operator applied to the same function: the diffusion equation $u_t = k \nabla^2 u$, also known as heat equation. In this talk several examples for diffusion will be explained. A connection between random walks and continuous diffusion will be established, a Gaussian filter will be linked to diffusion in image processing and the anisotropic diffusion equation $u_t = \nabla \cdot k(x) \nabla u$ will be used to improve an image by advocating diffusion in small slope regions only. In this way the edges in an image remain intact while noise or similar image failures are diffused out. Finally the heat equation will be derived in a bulk material, and also on a regular surface, resulting in a surface diffusion equation, $u_t = \nabla_{\!s} \cdot k(x) \nabla_{\!s} u$.