|SETTORE SCIENTIFICO DISCIPLINARE||MAT/07|
The course aims to provide a presentation of the most common partial differential equations (PDE) and their solution techniques through an analysis of various applications. The emphasis is devoted to second order PDE and the understanding of the specific techniques for elliptic, parabolic and hyperbolic cases.
The unit deals with the most important partial differential equations through their most important mathematical physical in the pleasure of craft sector.
Active participation in lectures and individual study will enable the student to:
- be able to classify the main partial differential equations;
- calculate the analytical solution of partial differential equations of elliptic, parabolic and hyperbolic types;
- use the techniques of separation of variables, series and Fourier transform, special functions.
The module is based on theoretical lessons.
Introduction to partial differential equations (PDE). The elastic string and the transition from discrete systems to continuous systems .
2 . The differential equations of the second order. The classification and the normal form; elliptic , hyperbolic and parabolic PDE.
3 . Elliptic equations. The harmonic functions. The problems of Dirichelet and Neuman , The Poisson formula for the circle.
4 . The variable separation technique. The series and the Fourier transform . The Gibbs effect, the analysis of normal modes , the delta Dirac "function”.
5 . The Bessel functions, the problems in polar coordinates.
6 . The parabolic differential equations, diffusion and heat equations; descriptions in the domain of space and time.
7 . The hyperbolic equations: the equation of D' Alembert. The method of characteristics, the elastic membrane, the mechanical interpretation of the normal modes
8 . PDE of higher order: the biharmonic equation, its Cauchy problem. The vibration of bars and plates.
L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.
Examinations are oral. Some mandatory exercizes can be requested.
The examination mode consists of an oral test to ensure learning of the course content.