Menù principale
B012967 - MATHEMATICAL METHODS FOR APPLICATIONS
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Further information
Type of Assessment
Course program
Academic Year 2021-22
Coorte 2021 - Second Cycle Degree in MATHEMATICS
Course year
First year - Second Semester
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/07 - MATHEMATICAL PHYSICS
Credits
9
Teaching Hours
72
Teaching Term
21/02/2022 ⇒ 17/06/2022
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Teaching Language
Italian
Course Content
The lectures concern the application of methods of harmonic analysis to boundary-value problems for linear partial differential equations.
Suggested readings (Search our library's catalogue)
Lecture notes.
Learning Objectives
Knowledge: method of separation of variables, methods of harmonic analysis, distributions, semigroup theory.Skills: to analyze from the point of view of harmonic analysis a boundary-value problem for a partial differential equation in order to provide an explicit solution in terms of a suitable basis.Skills acquired at the end of the course: to analyze an EDP-like problem both from a mathematical point of view and from a physical-applicative point of view.
Prerequisites
Linear algebra. Mathematical analysis. Differential equations. Physics (mechanics, electromagnetism). Equations of mathematical physics.Note: these are suggested prerequisites and not formal constraints.
Teaching Methods
Lectures. Some exercises and examples are left to students.
Further information
Teacher’s email: luigi.barletti@unifi.it
Moodle platform: https://e-l.unifi.it/
Type of Assessment
Oral examination consisting of a "practical" part (solving a simple problem with the techniques learned in the course) and a theoretical part (to enunciate and prove the mathematical results exposed in the course).
Course program
1. Fourier Series.
1.1 Convergence of Fourier series.
1.2 The problem of vibrating string with fixed extremities.
1.3 The problem of "rectangular drum".
1.4 Multiple Fourier series and periodic lattices
2. Sturm-Liouville problems
2.1 The circular drum problem.
2.2 A class of Sturm-Liouville problems.
2.3 Spherical harmonics.
2.4 Study of some systems in quantum mechanics.
3 Fourier Transforms.
3.1 Fourier transform of integrable functions.
3.2 Inversion theorems.
3.3 Fourier transform of L2 functions.
3.4 Solution of partial differential equations.
3.5 The sampling theorem.
4 Distributions
4.1 Distributions.
4.2 Derivatives of distributions.
4.3 Fourier transform of tempered distributions.
4.4 Periodic delta distribution.
4.5 Solution of the Poisson equation.
4.6 Solution of the Wave equation in R3 and R2.
4.7. Green function for the Helmoltz equation.
5 Semigroups
5.1 Semigroups of operators.
5.2 Group generated by a bounded operator.
5.3 Outline of the unbounded generator case.
5.4 Sources and perturbations.
5.5 Transport equation with collisions.
1.1 Convergence of Fourier series.
1.2 The problem of vibrating string with fixed extremities.
1.3 The problem of "rectangular drum".
1.4 Multiple Fourier series and periodic lattices
2. Sturm-Liouville problems
2.1 The circular drum problem.
2.2 A class of Sturm-Liouville problems.
2.3 Spherical harmonics.
2.4 Study of some systems in quantum mechanics.
3 Fourier Transforms.
3.1 Fourier transform of integrable functions.
3.2 Inversion theorems.
3.3 Fourier transform of L2 functions.
3.4 Solution of partial differential equations.
3.5 The sampling theorem.
4 Distributions
4.1 Distributions.
4.2 Derivatives of distributions.
4.3 Fourier transform of tempered distributions.
4.4 Periodic delta distribution.
4.5 Solution of the Poisson equation.
4.6 Solution of the Wave equation in R3 and R2.
4.7. Green function for the Helmoltz equation.
5 Semigroups
5.1 Semigroups of operators.
5.2 Group generated by a bounded operator.
5.3 Outline of the unbounded generator case.
5.4 Sources and perturbations.
5.5 Transport equation with collisions.