|SCIENTIFIC DISCIPLINARY SECTOR||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.
1. Introduction to partial differential equations (PDE). The elastic string and the transition from discrete systems to continuous systems. Second order partial differential equations. Classification and normal form. Elliptic, hyperbolic and parabolic PDE.
2. Elliptic equations. The harmonic functions. Dirichlet and Neumann boundary conditions, the Poisson formula for the circle.
3. Separation of variables technique. Series and Fourier transform. The Gibbs effect, the analysis of normal modes, the delta Dirac "function”. Bessel functions and problems in polar coordinates.
4. Parabolic differential equations, diffusion and heat equations; descriptions in space and time domain.
5. Hyperbolic equations: the equation of D'Alembert. The method of characteristics, the elastic membrane, the mechanical interpretation of the normal modes.
6. Some concept on PDE of higher order: the biharmonic equation and its Cauchy problem. The vibration of bars and plates.
7. Non homogeneous PDE and Green functions.
All class schedules are posted on the EasyAcademy portal.
The examination mode consists of an oral test to ensure learning of the course content.
The oral exam focuses on the learning of one or two subjects from those discussed in class.