LEARNING OUTCOMES

Many applications require the solution of partial differential equations. The module is intended to provide the student with the ability to solve the most common elliptic, parabolic, and hyperbolic PDEs analytically, using various techniques, including the series and Fourier transform. The module aims to provide the student with the ability to correctly choose a numerical method to solve a problem, understand and fix instabilities, and use Matlab to compute the solution.

AIMS AND LEARNING OUTCOMES

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;

- choose the most appropriate numerical method to solve some problems that require numerical resolution;

- understand why numerical instabilities or lack of convergence may appear and how to avoid such difficulties;

- implement these methods using Matlab, the most widely used scientific computing software in the world;

- be able to use Matlab functions other than those seen in the course and debug the code.

TEACHING METHODS

The module is based on theoretical lectures, supported for the part of numerical methods by exercises with the use of Matlab.

Working students and students with DSA certification, disabilities or other special educational needs are advised to contact the lecturer at the beginning of the course to agree on teaching and exam methods which, in compliance with the teaching objectives, take into account  individual learning methods.

SYLLABUS/CONTENT

The main topics covered are listed below:

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.

8. Matlab: matrices and vectors, 1d and 2D graphics, control structures, functions, creation of an app.

9. Numerical methods for solving equations and nonlinear systems.

10. Polynomial interpolation, data fitting, least squares method.

11. Numerical resolution of systems of ordinary differential equations.

12. Numerical methods for constrained and unconstrained optimization.

13. Finite difference method for solving partial derivative equations.

RECOMMENDED READING/BIBLIOGRAPHY

The notes taken during the lessons and the material provided (notes of the theoretical part and tutorial of Matlab) are sufficient for the preparation of the exam. The books listed below are suggested as possible support texts and in-depth study.

• A.N.Tichonov, A.A.Samarskij: Equazioni della Fisica matematica, Problemi della fisica matematica, Mosca,1982;
• R. Courant, D. Hilbert, Methods of Mathematical Phisics vol I e II, Interscience, NY, 1973;
• R. Bracewell, The Fourier Transform and Its Applications, New York: McGraw-Hill, 1999;
• P. V. O’ Neil, Advanced engineering mathematica, Brooks Cole, 2003;
• H. Goldstein, Meccanica Classica, Zanichelli, Bologna, 1985;
• V. I. Smirnov. Corso di Matematica superiore, Vol. 3. MIR (1978).
• Quarteroni, F. Saleri, Introduzione al Calcolo Scientifico, Sprinter-Verlag 2006.
• Quarteroni, Modellistica Numerica per Problemi Differenziali, Springer-Verlag 2008.
• S. Chapra, R. Canale, Numerical methods for Engineers, McGraw-Hill, 2018.