Title of the course: Solution strategies of Partial Differential Equations
Instructor: Dr. Konstantinos Kalimeris, Asst. Prof. Leonidas Mindrinos
Institution: Academy of Athens, Agricultural University of Athens
Dates: 31 August – 6 September 2026
Prerequisites: A familiarity with basic level numerical analysis, complex variables and PDE theory would be very useful, but the course will be as self-contained as possible.
Level: Advanced undergraduate and graduate
Abstract: In this course we will study a class of partial differential equations (PDEs) which appear in a plethora of physical phenomena such as the heat, the Schrödinger and the Laplace equations. Such equations for simple geometries and simple boundary conditions are traditionally solved via separation of variables or transform methods. However, these methods have limited applicability for more complicated initial-boundary value problems (IBVPs), and even in cases where they can be applied, they have several disadvantages.
Numerical approaches to such problems will be presented at the beginning of the course. For elliptic problems, such as Laplace equation, the numerical solution typically relies on the finite difference method (FDM) to discretize the spatial domain using a grid. This discretization results in a large but sparce system of algebraic equations. Then, iterative schemes like the Jacobi method can be applied. For parabolic problems, like the heat equation, numerical methods combine FDM for the spatial variable with time-stepping schemes. After discretizing space, the PDE reduces to a system of ordinary differential equations in time, which can be solved using the Euler method (forward or backward). Advantages and disadvantages will be discussed.
The course will also contain a brief introduction to complex variables, which will be employed to the presentation of a new approach, serving as the natural extension of the Fourier transform to IBVPs: the so-called Unified Transform Method. The students will be given an easy to apply guide for studying problems with this method, as well as an overview of the several analytical and numerical advantages that this unified approach possesses.
Based on this method, during the last lectures of the course computational advances applied to inverse problems associated to these IBVPs will be presented.
Language: EN