# Partial Differential Equations (AN6)

### General

School School of Science Department of Mathematics Graduate AN6 2 Partial Differential Equations Lectures (Weekly Teaching Hours: 3, Credits: 7.5) General Background Vector Analysis (undergraduate), Real Analysis, Functional Analysis Greek Yes (in English) See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes The course, apart from the instruction in the classical quartet of Partial Differential Equations (PDEs) (transport, Laplace, heat and wave) in several space variables, aims, first, to highlight the modern, analytic approach to the theory of PDEs and the reasons that suggest it, and, second, to provide an introduction to nonlinear PDEs, especially concerning first-order and hyperbolic equations. The skills and competences the students will acquire concern, on the one hand, the paradigmatic transition from the resolution of a problem to the theoretical analysis of its properties and the investigation of its structural foundation, and, on the other hand, the recognition of the essential difference between linear and nonlinear problems and the limitations of the method of approximation of a nonlinear problem by linear problems. Search for, analysis and synthesis of data and information Adapting to new situations Decision-making Working independently Working in an interdisciplinary environment Criticism and self-criticism Production of free, creative and inductive thinking

### Syllabus

Transport, Laplace, heat and wave equations for several space variables. Nonlinear first-order equations (method of characteristics, introduction to Hamilton-Jacobi equations and to conservation laws, weak solutions). The Cauchy-Kovalevskaya Theorem. Sobolev Spaces and weak derivatives. Theory of second-order linear equations. Semigroup Theory. Nonlinear hyperbolic and dispersion equations.

### Teaching and Learning Methods - Evaluation

Delivery

Face-to-face

Use of Information and Communications Technology -
Teaching Methods
Lectures 39
Self-study 78
Homework 70.5
Course total 187.5
Student Performance Evaluation

The evaluation is carried out as a combination of

• Written exam.
• Assigned homework.
• Presentation and oral examination.

### Attached Bibliography

• H.Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011
• L.C. Evans, Partial Differential Equations (2nd ed.). AMS, 2010
• G. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1976
• L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1-4, Springer, 1983-85
• J. Jost, Partial Differential Equations (2nd ed.), Springer, 2007
• T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS, AMS, 2006
• M. Taylor, Partial Differential Equations, Vol. I-III, Springer, 1996
• G.B.Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974.