# 401-0363-10L Analysis III

Semester | Autumn Semester 2016 |

Lecturers | M. Soner |

Periodicity | yearly recurring course |

Language of instruction | English |

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. The first lecture is on Thursday, September 29 13-15 in HG F 7 and video transmitted into HG F 5. The exercises Sheet are here: Link The coordinator is Claudio Sibilia (see https://www.math.ethz.ch/the-department/people.html?u=sibiliac) The first exercise session is on Thursday, September 22 or resp. Friday, September 23. If you would like feedback on your work, please give it to your course assistent or leave it in the box of your course assistant in HG F 27. The due Date is one week later the assignment. Office hour (Praesenz): Thursday 16-17, NO E 39. |

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 Download the syllabus: https://polybox.ethz.ch/index.php/s/bu5KY8vWNMOnaAa |

Lecture notes | Alessandra Iozzi's Lecture notes: https://polybox.ethz.ch/index.php/s/RcsFm70tWCheSqH Errata: https://polybox.ethz.ch/index.php/s/VKh86gvQRTwIE0w |

Literature | E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 9. Auflage, 2011 C. R. Wylie & L. Barrett, Advanced Engineering Mathematics, McGraw-Hill, 6th ed. 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) |