The course will introduce the theory of partial differential equations (PDEs) of mathematical physics. At the end of the course, the students should be able to solve relatively basic theoretical problems involving the fundamental PDEs of mathematical physics. It is also expected that by the end of the course the students will acquire sufficient autonomy in reading and understanding the textbooks.
Analysis 1, Linear Algebra
1. Cauchy problem for systems of PDEs, Cauchy-Kovalevskaya theorem.
2. Linear differential operators.
3. Fundamental linear partial differential equations: Laplace, heat and wave equation.
4. The Fourier transform, applications to partial differential equations. The Schroedinger equation.
5. Topics in nonlinear partial differential equations. Burgers equation, Korteweg-de Vries equation
The exam consists of a written and of an oral part. The written part will assess the student's proficiency in solving the exercises and applying the theory to a level comparable to the exercises worked out in class. The oral component will assess the knowledge of the theoretical aspects as well as the property of the expression of mathematical concepts.
1. Boris Dubrovin's course notes: https://people.sissa.it/~dubrovin/fm1_web.pdf
2. L. C. Evans, Partial Differential Equations, AMS, 2010.
3. A. Tikhonov, A. Samarskij, Equazioni della fisica matematica, Edizioni Mir, 1977.
4. R. Courant, D. Hilbert, Methods of Mathematical Physics, New York Intersci. Publ., 1989.
5. E. H. Lieb, M. Loss, Analysis, AMS, 2001.