Title of the course: Pure and Applied problems of PDEs
Instructor: Assoc Prof. Konstantinos Kalimeris, Assoc. Prof. Türker Özsarı
Institution: Academy of Athens, Bilkent Ü.
Dates: 29 August – 3 September 2022
Prerequisites: A familiarity with graduate level Functional analysis, Complex analysis and PDE theory would be very useful, but an introduction to necessary themes of the above subjects will be given in the first lectures.
Level: Advanced undergraduate and graduate
Abstract: In this course we will study a class of partial differential equations (PDEs) which appears in physical phenomena. These include heat, wave, Schrödinger and Laplace equations equipped with certain (initial and boundary) conditions.
The aim of this course is to provide explicit solutions for problems associated with the above-mentioned equations and analyze those solutions. More precisely, we will focus on the following subjects:
(A) Analytical derivation of solutions of specific initial and boundary value problems.
(B) Wellposedness analysis for given linear and nonlinear PDEs in function spaces.
(C) Analytical and numerical approaches to inverse problems originating from control theory and fluid mechanics.
To achieve the aims of the course, we will first give a brief and coherent overview of the classical methods for solving aforementioned equations. Next, we will discuss the limitations and disadvantages of these methods. In order to overcome these limitations, we will present a new approach, which provides a unified perspective in addressing different problems associated with this large family of PDEs. This method, called the Unified Transform Method (UTM), employs basic (elementary) principles of Functional analysis and Complex analysis, and follows a straightforward pathway for the derivation of the solution. UTM formulas will be used explicitly to discover regularity and control theoretic properties of solutions to these PDEs.
The course will be taught jointly by K. Kalimeris and T. Özsarı.
Language: EN (some part may be delivered in Turkish.)