Title of the course: Transform Methods for Differential Equations – A unified approach
Instructor: Dr. Konstantinos Kalimeris
Institution: University of Cambridge
Dates: 3-9 September 2018
Prerequisites: Basic knowledge of Complex Variables (An introduction to basic Complex Analysis themes will be given in the first lectures).
Level: Advanced undergraduate
Abstract: In this course we will study a class of differential equations which appear in a plethora of physical phenomena and include the heat, the wave, the Laplace, and the Helmholtz equations. Such equations for simple geometries and simple boundary conditions are traditionally solved via separation of variables or transform methods. However, these methods are limited, and even in the particular cases that they are applicable, they have several disadvantages.
After reviewing in a coherent way some of the classical transform methods, we will present a new approach, which has been acclaimed as the first major breakthrough in the solution of linear PDEs, since the discovery of the Fourier transform in the 18th century. In this course, this Unified Transform Method will be presented, making usage of basic mathematical tools and methods of complex analysis. The students will be given an overview of the several analytical and numerical advantages that this unified approach possesses. At the last lectures of the course the students will have the opportunity to apply this knowledge in a series of problems related to differential equations.
The course will also include an introduction of complex variables.