Differential Equations for Mathematical Physics

1. Give the basic elements of the partial differential equations of mathematical physics.

2. Understanding of physical phenomena governed by partial differential equations; Construction of mathematical models: equations of the waves, heat, Laplace and Boltzmann equation.

3. Understanding of the different solution methods: why it has been proposed a solution method? What are some other alternative methods? Understanding how the analytical solutions obtained are interpreting the physical real situations (wellposedness of the models or paradoxes).

4. It will be privileged the undertsnding the physical part, the models and their analytical solution.

Equazioni differenziali della Fisica Matematica - 6 CFU - 47 hours

Lectures and exercises done by students at home and in class.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, inline with the programme planned and outlined in the syllabus.

Knowledge  of ordinary differential equations, notions of Mathematical Physics. Fourier series (some concepts will be recalled in class)

Partial differential equations
Scalar conservation laws and first order partial differential equations
Second order semilinear equations in two variables

Heat equation
Diffusion and probability
Vibrating string equation
Laplace and Poisson equations

An introduction to the kinetic theory of gases
Boltzmann equation
Reversibility paradox
Method of moments

[1] M.M. SMIRNOV, Second-Order partial differential equations, ed. Noordhoff.
[2] F.JOHN, Partial differential equations, Springer-Verlag.
[3] V.I. SMIRNOV, Corso di matematica superiore II, Editori Riuniti.
[4] N.S.KOSHLYAKOV, M.M.SMIRNOV, E.B.GLINER, Differential equations of mathematical physics, ed. North-Holland.

The final exam consists of an oral test during which the candidate demonstrates that he has assimilated the topics covered in the course (understanding, reasoning and the ability to build examples will be privileged).

Verification of learning can also be carried out electronically, should the conditions require it.