## Francesca Da Lio: Catalogue data in Autumn Semester 2020 |

Name | Prof. Dr. Francesca Da Lio |

Address | Dep. Mathematik ETH Zürich, HG G 37.2 Rämistrasse 101 8092 Zürich SWITZERLAND |

Telephone | +41 44 632 86 96 |

Fax | +41 44 632 10 85 |

francesca.dalio@math.ethz.ch | |

URL | http://www.math.ethz.ch/~fdalio |

Department | Mathematics |

Relationship | Adjunct Professor |

Number | Title | ECTS | Hours | Lecturers | |
---|---|---|---|---|---|

401-0363-10L | Analysis III | 3 credits | 2V + 1U | F. Da Lio | |

Abstract | Introduction to partial differential equations. Differential equations which are important in applications are classified and solved. Elliptic, parabolic and hyperbolic differential equations are treated. The following mathematical tools are introduced: Laplace transforms, Fourier series, separation of variables, methods of characteristics. | ||||

Objective | Mathematical treatment of problems in science and engineering. To understand the properties of the different types of partial differential equations. | ||||

Content | Laplace Transforms: - Laplace Transform, Inverse Laplace Transform, Linearity, s-Shifting - Transforms of Derivatives and Integrals, ODEs - Unit Step Function, t-Shifting - Short Impulses, Dirac's Delta Function, Partial Fractions - Convolution, Integral Equations - Differentiation and Integration of Transforms Fourier Series, Integrals and Transforms: - Fourier Series - Functions of Any Period p=2L - Even and Odd Functions, Half-Range Expansions - Forced Oscillations - Approximation by Trigonometric Polynomials - Fourier Integral - Fourier Cosine and Sine Transform Partial Differential Equations: - Basic Concepts - Modeling: Vibrating String, Wave Equation - Solution by separation of variables; use of Fourier series - D'Alembert Solution of Wave Equation, Characteristics - Heat Equation: Solution by Fourier Series - Heat Equation: Solutions by Fourier Integrals and Transforms - Modeling Membrane: Two Dimensional Wave Equation - Laplacian in Polar Coordinates: Circular Membrane, Fourier-Bessel Series - Solution of PDEs by Laplace Transform | ||||

Lecture notes | Lecture notes by Prof. Dr. Alessandra Iozzi: https://polybox.ethz.ch/index.php/s/D3K0TayQXvfpCAA | ||||

Literature | E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 10. Auflage, 2011 C. R. Wylie & L. Barrett, Advanced Engineering Mathematics, McGraw-Hill, 6th ed. S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Books on Mathematics, NY. G. Felder, Partielle Differenzialgleichungen für Ingenieurinnen und Ingenieure, hypertextuelle Notizen zur Vorlesung Analysis III im WS 2002/2003. Y. Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005 For reference/complement of the Analysis I/II courses: Christian Blatter: Ingenieur-Analysis https://people.math.ethz.ch/~blatter/dlp.html | ||||

401-3420-70L | Topics in Harmonic Analysis Number of participants limited to 20 | 4 credits | 2S | F. Da Lio, L. Kobel-Keller | |

Abstract | The aim of this seminar about harmonic analysis is to study the most important and most classical topics in that field, e.g. maximal functions, Marcinkiewicz interpolation, Fourier theory, distribution theory, singular integrals and Calderon-Zygmund theory. After an introduction delivered by the two organisers, each week participants will give a seminar talk (usually in groups of two). | ||||

Objective | The students will learn on one hand the most important concept in harmonic analysis and on the other hand improve their presentations skills (by delivering a seminar talk). | ||||

Literature | The main references are: E. Stein: "Singular integrals and differentiability properties of functions " E. Stein, G. Weiss: "Introduction to Fourier analysis on Euclidean spaces" L. Grafakos: "Modern Fourier Analysis" & "Classical Fourier Analysis" | ||||

401-5350-00L | Analysis Seminar | 0 credits | 1K | M. Struwe, A. Carlotto, F. Da Lio, A. Figalli, N. Hungerbühler, M. Iacobelli, T. Ilmanen, L. Kobel-Keller, University lecturers | |

Abstract | Research colloquium | ||||

Objective | |||||

406-0353-AAL | Analysis IIIEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | 4 credits | 9R | F. Da Lio | |

Abstract | Introduction to partial differential equations. Differential equations which are important in applications are classified and solved. Elliptic, parabolic and hyperbolic differential equations are treated. The following mathematical tools are introduced: Laplace transforms, Fourier series, separation of variables, methods of characteristics. | ||||

Objective | Mathematical treatment of problems in science and engineering. To understand the properties of the different types of partlial differentail equations. | ||||

Content | Laplace Transforms: - Laplace Transform, Inverse Laplace Transform, Linearity, s-Shifting - Transforms of Derivatives and Integrals, ODEs - Unit Step Function, t-Shifting - Short Impulses, Dirac's Delta Function, Partial Fractions - Convolution, Integral Equations - Differentiation and Integration of Transforms Fourier Series, Integrals and Transforms: - Fourier Series - Functions of Any Period p=2L - Even and Odd Functions, Half-Range Expansions - Forced Oscillations - Approximation by Trigonometric Polynomials - Fourier Integral - Fourier Cosine and Sine Transform Partial Differential Equations: - Basic Concepts - Modeling: Vibrating String, Wave Equation - Solution by separation of variables; use of Fourier series - D'Alembert Solution of Wave Equation, Characteristics - Heat Equation: Solution by Fourier Series - Heat Equation: Solutions by Fourier Integrals and Transforms - Modeling Membrane: Two Dimensional Wave Equation - Laplacian in Polar Coordinates: Circular Membrane, Fourier-Bessel Series - Solution of PDEs by Laplace Transform | ||||

Literature | E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 10. Auflage, 2011 C. R. Wylie & L. Barrett, Advanced Engineering Mathematics, McGraw-Hill, 6th ed. Stanley J. Farlow, Partial Differential Equations for Scientists and Engineers, (Dover Books on Mathematics). G. Felder, Partielle Differenzialgleichungen für Ingenieurinnen und Ingenieure, hypertextuelle Notizen zur Vorlesung Analysis III im WS 2002/2003. Y. Pinchover, J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005 For reference/complement of the Analysis I/II courses: Christian Blatter: Ingenieur-Analysis (Download PDF) | ||||

Prerequisites / Notice | Up-to-date information about this course can be found at: http://www.math.ethz.ch/education/bachelor/lectures/hs2013/other/analysis3_itet | ||||

406-2284-AAL | Measure and IntegrationEnrolment ONLY for MSc students with a decree declaring this course unit as an additional admission requirement. Any other students (e.g. incoming exchange students, doctoral students) CANNOT enrol for this course unit. | 6 credits | 13R | F. Da Lio | |

Abstract | Introduction to the abstract measure theory and integration, including the following topics: Lebesgue measure and Lebesgue integral, Lp-spaces, convergence theorems, differentiation of measures, product measures (Fubini's theorem), abstract measures, Radon-Nikodym theorem, probabilistic language. | ||||

Objective | Basic acquaintance with the theory of measure and integration, in particular, Lebesgue's measure and integral. | ||||

Literature | 1. Lecture notes by Professor Michael Struwe (http://www.math.ethz.ch/~struwe/Skripten/AnalysisIII-SS2007-18-4-08.pdf) 2. L. Evans and R.F. Gariepy "Measure theory and fine properties of functions" 3. Walter Rudin "Real and complex analysis" 4. R. Bartle The elements of Integration and Lebesgue Measure 5. P. Cannarsa & T. D'Aprile: Lecture notes on Measure Theory and Functional Analysis. http://www.mat.uniroma2.it/~cannarsa/cam_0607.pdf |