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

 Semester Herbstsemester 2019 Dozierende T. Ilmanen, C. Busch Periodizität jährlich wiederkehrende Veranstaltung Lehrsprache Englisch

 Kurzbeschreibung Examples of partial differential equations. Linear partial differential equations. Separation of variables. Fourier series, Fourier transform, Laplace transform. Applications to solving commonly encountered linear partial differential equations (Laplace's Equation, Heat Equation, Wave Equation). Lernziel Classical tools to solve the most common linear partial differential equations. Inhalt 1) Examples of partial differential equations - Classification of PDEs - Superposition principle2) One-dimensional wave equation- D'Alembert's formula - Duhamel's principle3) Fourier series - Representation of piecewise continuous functions via Fourier series - Examples and applications4) Separation of variables - Solution of wave and heat equation - Homogeneous and inhomogeneous boundary conditions- Dirichlet and Neumann boundary conditions 5) Laplace equation- Solution of Laplace's equation on the rectangle, disk and annulus- Poisson formula - Mean value theorem and maximum principle6) Fourier transform - Derivation and definition - Inverse Fourier transformation and inversion formula- Interpretation and properties of the Fourier transform - Solution of the heat equation7) Laplace transform (if time allows)- Definition, motivation and properties- Inverse Laplace transform of rational functions - Application to ordinary differential equations Skript See the course web site (linked under Lernmaterialien) Literatur 1) S.J. Farlow, Partial Differential Equations for Scientists andEngineers, Dover Books on Mathematics, NY.2) N. Hungerbühler, Einführung in partielle Differentialgleichungenfür Ingenieure, Chemiker und Naturwissenschaftler, vdfHochschulverlag, 1997.Additional books:3) T. Westermann: Partielle Differentialgleichungen, Mathematik fürIngenieure mit Maple, Band 2, Springer-Lehrbuch, 1997 (chaptersXIII,XIV,XV,XII)4) E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons(chapters 1,2,11,12,6)For additional sources, see the course web site (linked under Lernmaterialien) Voraussetzungen / Besonderes Required background:1) Multivariate functions: partial derivatives, differentiability, Jacobian matrix, Jacobian determinant2) Multiple integrals: Riemann integrals in two or three variables, change of variables2) Sequences and series of numbers and of functions3) Basic knowledge of ordinary differential equations