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

Mathematical and Numerical Methods in Engineering

GOALS AND CONTENTS

The didactical goal is twofold. First we intend to present some classical differential models of Continuum Mechanics, developing and analyzing some finite difference schemes for their approximation. Second we want to introduce the variational formulation of some boundary value problems together with the finite element method for their numerical approximation. The course is characterized by a constant synergy between modeling, theoretical aspects and numerical simulation.

TOPICS First module of the course

Mathematical Methods

Differential calculus for functions of several real variables. Series of functions. Transport equation. Traffic flow models. Method of characteristics. Rankine-Hugoniot relation. Shock and rarefaction waves. Entropy condition. Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Cauchy problem in the half-space. Duhamel principle. Harmonic functions. Mean value properties. Maximum principles. Well-posed problems. Poisson’s formula for the disk. Newtonian potentials. String equations. Well-posed problems and separation of variables. D’Alembert formula. Kirchhoff formula and Huygens principle. Lebesgue integral. Projection theorem and Riesz representation theorem. Bilinear form, abstract variational problems and Lax-Milgram lemma. Schwartz distributions. Function spaces of finite energy (Sobolev spaces). Variational formulation of elliptic problems and applications to transport-reaction-diffusion equations.