OVERVIEW

These classes provide an overview of the most common partial differential equations
(PDEs) and related solution techniques, with particular focus on second order equations. The
role of the teaching unit within the curriculum is to provide tools for the analysis of
mathematical models in various applications.

AIMS AND CONTENT

LEARNING OUTCOMES

Modeling and Simulation Fundamentals. Theory and Practice of Continuous Simulation and related Methodologies. Theory and Practice of Discrete Simulation and related Methodologies. Hybrid Simulation.

AIMS AND LEARNING OUTCOMES

Active participation in lectures and individual study will enable the student to:
- (D1 - Knowledge and understanding) Classify the main partial differential equations
(content) presented during the course (condition), distinguishing between elliptic, parabolic,
and hyperbolic cases (criterion);
- (D2 - Applying knowledge and understanding) Calculate the analytical solution of elliptic,
parabolic, and hyperbolic partial differential equations (content) in exercises assigned during
the exam (condition), using the techniques learned (criterion);
- (D3 - Making judgements) Select and apply the most appropriate technique among
separation of variables, Fourier series, and Fourier transform (content) to specific problems
proposed during the course (condition), justifying the chosen method (criterion).

PREREQUISITES

Basic knowledge of real and complex numbers, circular and hyperbolic trigonometry, derivatives and integrals, ordinary differential equations.

TEACHING METHODS

Lectures. Attendance is not compulsory but strongly recommended. Students with learning
disabilities or special needs are invited to contact the teacher at the beginning of the course to
agree on personalized learning methods.

SYLLABUS/CONTENT

1. Elements of 3D vector calculus.
2. Convolutions and Dirac delta.
3. Fourier analysis (discrete and continuous).
4. Inhomogeneous PDE and Green functions.
5. Laplace equation: unicity theorems. Separation of variables. Examples.
6. Fourier equation: unicity theorems. Separation of variables. Examples.
7. D'Alembert equation. Method of characteristics. Examples.
8. Bi-Laplace equation: Cauchy problem. Examples.
9. Helmholtz theorem.

RECOMMENDED READING/BIBLIOGRAPHY

• 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).

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

https://corsi.unige.it/10728/p/studenti-orario

Class schedule

The timetable for this course is available here: EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a written test, possibly complemented by an oral test at the teacher's choice. Minimal score is 18/30.

ASSESSMENT METHODS

Learning assessment is carried out through a written and/or oral exam, during which the
ability to classify equations, solve exercises, and apply the learned techniques will be
evaluated. Assessment criteria include correctness of solutions, clarity of exposition, andappropriate use of terminology.

FURTHER INFORMATION

Please contact the teacher for further information not included in the teaching unit description.