OVERVIEW

The module aims to present the most common tools for solving partial differential equations (PDEs) through the analysis of various applications. Emphasis is placed on second-order PDEs, on understanding specific techniques for the elliptic, parabolic, and hyperbolic cases, and on the introduction to discrete systems. Only for students of Civil Eng, curriculum "Strutture", the module provides also with knowledge of numerical methods for solving Engineering problems, implemented using Matlab.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to provide a study 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 analytical techniques for solving elliptic, parabolic and hyperbolic cases. The course also provides the tools to solve problems in various applications with numerical methods implemented through the use of Matlab.

AIMS AND LEARNING OUTCOMES

Active participation in lectures and individual study will enable students to:

- Classify the main types of partial differential equations.
- Calculate the analytical solutions of elliptic, parabolic, and hyperbolic partial differential equations.
- Apply techniques such as separation of variables, series expansion, and Fourier transforms, as well as special functions.

Only for students of Civil Eng, curriculum "Strutture":
- Select the most appropriate numerical methods for solving problems requiring numerical resolution.
- Understand and mitigate numerical instabilities or lack of convergence.
- Implement these methods using Matlab, the world's most widely used scientific computing software.
- Utilize Matlab functions beyond those covered in the course and debug code effectively.

PREREQUISITES

Basics of Calculus (suggested)

Basics of Linear Algebra (Matrices, Eigenvalues, Eigenvectors - suggested)
Basics of ODE and PDE theory (suggested)

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 (Points 1–10 are part of the core syllabus for all students, while points 11–15 are included in the extended syllabus for students taking the course for 8 credits.):

1. Analysis of phenomena and motivations leading to the study of partial differential equations. The elastic string and the transition from discrete to continuous systems.

2. Second-order differential equations. Classification and normal form. Elliptic, hyperbolic, and parabolic equations.

3. Elliptic equations. Properties of harmonic functions; Dirichlet and Neumann problems; Poisson’s formula for the circle.

4. General solution techniques: separation of variables; Fourier series and transform; Gibbs phenomenon; normal mode analysis; the Dirac delta "function"; bi- and three-dimensional cases.

5. Special functions: Bessel functions J, Y, I, K; Fourier-Bessel and Dini series; Fourier transforms in polar coordinates: the Hankel transform. Applications to problems in polar coordinates.

6. Parabolic differential equations; the diffusion and heat equations; descriptions in space and time domains; the heat kernel.

7. Hyperbolic equations: D'Alembert's equation, method of characteristics, the elastic membrane, mechanical-dynamic interpretation of normal modes; the Cauchy problem and the future domain of dependence.

8. Higher-order PDEs: the biharmonic equation; the related Cauchy problem.

9. Non-homogeneous equations: distributed and point sources; the Green's function and its interpretation as a system transfer function; description using the Dirac delta function.

10. Discrete systems and difference equations.

Only for students of Civil Eng, curriculum "Strutture":

1. Numerical methods for solving nonlinear equations and systems.

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

3. Numerical solution of systems of ordinary differential equations.

4. Numerical methods for constrained and unconstrained optimization.

5. The finite difference method for solving partial differential 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.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/10799/p/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists a written test for everybody plus an oral exam Only for students of Civil Eng, curriculum "Strutture", . The written test typically includes five problems aimed at assessing the student's understanding of the theoretical content of the course. The oral exam consists of questions related to the numerical part. The final grade is the average of the two parts, rounded up.

Students with a valid certification of physical or learning disabilities who wish to discuss possible accommodations regarding lectures and exams must speak with the instructor.

ASSESSMENT METHODS

The exam assesses the student's ability to formulate equations that model simple phenomena, to set up their solution, to analyze key qualitative aspects, and to identify the most appropriate numerical methods.

FURTHER INFORMATION

Contact the instructor for further information not included in the module syllabus