Prof. Dr. Francesca Da Lio

ANALYSIS III, D-MAVT, D-MATL

Autumn Semester 2020

Lecturer: Prof. Francesca Da Lio

Coordinator: Stefano D'Alesio

Class Content

Lecture Topic Bibliograpghy
21.09.2020 Presentation of the course. Definition Laplace Transform and of inverse Laplace Transform. Examples. Linearity. Sufficient condition for the existence of the LT (exponential growth condition) Sections 2.1,2.2,.2.3 in Iozzi's Notes, Class Notes/Further material (optional)/Table of Laplace Transform (useful)/
28.09.2020 Definition of the inverse Laplace Transform. Linearity of the inverse Laplace Tranform How to use the inverse Laplace Transform. The LT of derivatives. S-shifting property. Examples. Sections 2.3.1,2.3.2,2.3.2 in Iozzi's Notes, Class Notes, Further reading: Kreyszig 6.2
5.10.2020 The Heaviside function and examples of discontinuous signals. Impulses. Dirac's delta function (definition and properties). Sifting property, Laplace transform of Dirac's delta function. Sections 2.4, 2.6,2.7 in Iozzi's Notes, Class Notes
12.10.2020 Convolution. Definition and Properties. Proof of the fact that the LT of the convolution of two functions is the product of the respective LT. Derivative and integral of the LT. Periodic functions. Examples. Section 2.5. 2.8 and 3.1 Iozzi's Notes until page 23, Class Notes
19.10.2020 Sum of two periodic functions. Some examples. Trigonometric polynomials, trigonometric series, orthogonality relations. Fourier Series of the square wave function. Square wave function and its approximations. Discrete spectrum of a periodic function. Section 3.1 Iozzi's Notes, Class Notes
26.10.2020 Representation result for periodic functions. Computation of a Leibniz series through the Fourier series of the square wave. Odd and even functions. Half-range expansions. An example: the triangular wave function. Sections 3.1 & 3.2 in Iozzi's Notes.Class Notes.
2.11.2020 Complex Fourier series. Example. Relation between complex and real coefficients. Fourier integral representation of a given absolutely integrable function. Dirichlet's discontinuous factor and the sine integral. Section 3.5 in Iozzi's Notes.Class Notes.
9.11.2020 Complex Fourier integrals. Fourier Transform. Inverse Fourier Transform. Some properties of the Fourier Transform. Inversion Formula. Some examples. Section 3.6 in Iozzi's Notes. Class Notes.
16.11.2020 Fourier transform of the Gaussian. FT of the convolution. Introduction to PDEs : examples. Classification of PDEs (order, linear, nonlinear,homogeneous).Elliptic,parabolic,hyperbolic second order linear PDEs. Superposition Principle. Introduction of the method of separation of the variables applied to the wave equation. Remark 4.5 , Sections 4.2 & 4.3 in Iozzi's Notes. Class Notes.
23.11.2020 Method of separation of the variables to solve the wave equation (continuation). D'Alembert solution. Section 4.3 & 4.2 in Iozzi's Notes. Examples of solutions to wave equation with zero initial velocity and speed of propagation c=1: Example1,Example2. Look at (for curiosity) Online PDE solvers by Luis Silvestre, Class Notes.
30.11.2020 Interpretation of the characteristics in the case of the wave equation. Characteristic triangle, domain of dependence, region of influence. Example of solution of wave equation by graphical method. Heat equation via Fourier series. Sections 4.4 & 4.5 in Iozzi's Notes. Class Notes.
7.12.2019 Solution of the Laplace equation in a rectangle. Example. Heat equation: modeling very long bars. Solution by Fourier integrals and transforms. Sections 4.4 & 4.5 in Iozzi's Notes, Class Notes
14.12.2020 Dirichlet problem for Laplace equation on a disk. Poisson formula.Mean value formula for harmonic functions. Maximum Principle. Examples of solution to the Laplace equation in a disk. See Iozzi's Notes Sections 4.8 & 4.9. Class Notes
*** Frohe Festtage und Viel Erfolg! ****

Recommended bibliography (Undergraduate-Master level):