The course aims at introducing the fundamental theory of partial
differential equations (PDEs) of mathematical physics, discussing their
classification and methods of solution. The student should be able to
apply the acquired knowledge to the solution of exercises and relatively
basic theoretical problems involving the classical equations of
mathematical physics. It is also expected that at the end of the course
the student will have acquired a sufficient autonomy in finding, reading
and understanding textbooks.

Analysis 1, 2. Linear Algebra.

1. Fundamental linear equations; wave, Helmholts, heat, Laplace and
Poisson equations. Classification of partial differential equations of
second order.

2. Boundary value problems for second order PDEs. Cauchy problem and
Cauchy–Kowalevskaya theorem. Boundary value problems for elliptic
equations. Well posed problems.

3. Banach and Hilbert spaces. L2 and C spaces. Self–adjoint operators.

4. Separation of variables for Laplace, Poisson and wave equations.

5. Spectrum of Sturm–Liouville operators. Special cases and classical
orthogonal polynomials. Spherical functions.

6. Topics in nonlinear differential equations. Burgers equation and ColeHopf transformation. Korteweg-de Vries equation.

7. (Time permitting). Solution of the Korteveg-de Vries equation: forward
and inverse scattering. Riemann– Hilbert problems

Of the following texts we will follow primarily the first, with support
material from the other two and possibly other material to be distributed
during the course.

1. Boris Dubrovin’s course notes: http://people.sissa.it/∼dubrovin/fm1
web.pdf
2. A.Tikhonov, A.Samarskij, Equazioni della fisica matematica, Edizioni
Mir, 1977.
3. R.Courant, D.Hilbert, Methods of Mathematical physics, New York
Intersci. Publ. 1989.

Lectures

The exam consists in a written and oral part. The written component will
assess the student’s proficiency in solving exercises and applying the
theory, at a level comparable to the exercises worked out in class. The
oral component will assess the knowledge of theoretical aspects as well
as the property of expression of mathematical concepts.