• 096229 - MATHEMATICAL AND NUMERICAL METHODS IN ENGINEERING [1]

We present some classical differential models of Mathematical Physics, developing and analyzing analytical methods for the computation of the solutions.

The lectures and the laboratories will provide students with:

i) knowledge and understanding of (DD1)
* the main properties of the solutions of some classes of linear partial differential equations;
* the basic tools used for solving linear partial differential equations;
* modeling of some physical problems

ii) the ability to apply the previous knowledge to simple examples on the calculator (DD2). In particular the student is required to
* solve linear partial differential equations
* discuss the properties of mathematical models

The instructor expects a broad comprehension of the subjects which sould not be limited to the statement of theoretical results.
Instead, the acquired knowledge should enable students to express
critical judgment and make informed choices on analytical methods for partial differential equations (DD3). The students are expected to express their answers in a mathematically rigorous and clear way.

Review: Differential calculus for functions of several real variables. Series of functions.

First-order conservation laws: Transport equation. Traffic flow
models. Method of characteristics. Rankine-Hugoniot relation. Shock
and rarefaction waves. Entropy condition.

Diffusion: Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Cauchy problem in a strip and in the half-space with several boundary conditions (isolation, transmission, mixed). Duhamel principle.

Laplace-Poisson equation: Harmonic functions. Mean value properties. Maximum principles. Well-posed problems. Poisson’s formula for the disk. Newtonian potentials.

Wave equation: String equations. Well-posed problems and separation of variables. D’Alembert formula.

Basics of Functional Analysis: Lebesgue integral. Projection theorem and Riesz representation theorem. Schwartz distributions. Sobolev spaces.

We recommend that students who attend this course have knowledge of linear algebra and calculus, in particular:

* differential calculus for functions of several real variables;
* surfaces and integral surfaces, the divergence theorem;
* series of functions, Fourier series;
* linear second order ordinary differential equations;

There are five examination dates (two in January-February, two in June-July, one in September).
The exam is written and consists in two questions on the theory (6 points each) and two exercises (10 points each). A positive evaluation requires at least 6 points on the theory, 10 points on the exercises and a total of 18 points.