Posted: January 17, 2017, 3:13pm
One of the most interesting classes that I took last semester was the Fokas Method, a hybrid analytical-numerical approach for solving partial differential equations.
It was also taught by Professor Athanasios S. Fokas himself, one of the topmost applied mathematicians now in the world. Professor Fokas was a visiting professor from the University of Cambridge last semester. I am lucky to have had the opportunity to learn from him.
The Fourier transform which was discovered in the 18th century has been used to solve linear PDEs for a very long time. But it has its limitations. However, the Fokas method has been an unexpected new development and is claimed as the first major breakthrough since the Fourier transform.
Since the Fokas method is analytical, we can use simple numerical evaluations to solve complex problems. This approach takes place entirely in the spectral space, i.e., the complex lambda-plane. An integral equation called the global relation is derived using the boundary values. A solution is constructed based on this global relation.
The following steps are followed while solving a partial differential equation using the Fokas method:
- Given a domain, the global relation (GR) is derived using the boundary conditions.
- The solution is then expressed as an integral in the complex lambda-plane.
- For given boundary conditions, employ the global relation and from the integral representation obtained in step 2 eliminate the transforms of the unknown boundary values.
We solved the heat equation on the finite and half interval using this method along with other third order PDE’s that are not possible to solve using the classical approach.
Published on July 26th, 2017
Last updated on July 26th, 2017