# 401-0373-00L Mathematics III: Partial Differential Equations

Semester | Autumn Semester 2018 |

Lecturers | C. Busch |

Periodicity | yearly recurring course |

Language of instruction | German |

Abstract | Examples of partial differential equations. Linear partial differential equations. Introduction to Separation of Variables method. Fourier Series, Fourier Transform, Laplace Transform and applications to the resolution to some partial differential equations (Laplace Equation, Heat Equation, Wave Equation). |

Learning objective | The main objective is that the students get a basic knowledge of the classical tools to solve explicitly linear partial differential equations. |

Content | 1) Examples of partial differential equations - Classification of PDEs - Superposition principle 2) One-dimensional wave equation - D'Alembert's formula - Duhamel's principle 3) Fourier series - Representation of piecewise continuous functions via Fourier series - Examples and applications 4) Separation of variables - Resolution of wave and heat equation - Homogeneous and inhomogeneous boundary conditions, Dirichlet and Neumann boundary conditions 5) Laplace equation - Resolution of the Laplace equation on rectangle, disk and annulus - Poisson formula - Mean value theorem and maximum principle 6) Fourier transform - Derivation and Definition - Inverse Fourier transformation and inversion formula - Interpretation and properties of the Fourier transform - Resolution of the heat equation |

Lecture notes | There are available some Lecture Notes by F. Da Lio in English and also in German. These can be found following the links provided under the tab 'Lernmaterialien'. |

Literature | 1) N. Hungerbühler, Einführung in partielle Differentialgleichungen für Ingenieure, Chemiker und Naturwissenschaftler, vdf Hochschulverlag, 1997. 2) S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Books on Mathematics, NY. 3) E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons (only Chapters 1,2,6,11) 4) T. Westermann: Partielle Differentialgleichungen, Mathematik für Ingenieure mit Maple, Springer-Lehrbuch 1997. |

Prerequisites / Notice | It is required a minimal background of: 1) multivariables functions (Riemann integrals in two or three variables, change of variables in the integrals through the Jacobian, partial derivatives, differentiability, Jacobian) 2) numerical and functional sequences and series, basic knowledge of ordinary differential equations. |