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

SemesterAutumn Semester 2019
LecturersT. Ilmanen, C. Busch
Periodicityyearly recurring course
Language of instructionEnglish

AbstractExamples 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).
ObjectiveClassical tools to solve the most common linear partial differential equations.
Content1) 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
- 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 principle

6) Fourier transform
- Derivation and definition
- Inverse Fourier transformation and inversion formula
- Interpretation and properties of the Fourier transform
- Solution of the heat equation

7) Laplace transform (if time allows)
- Definition, motivation and properties
- Inverse Laplace transform of rational functions
- Application to ordinary differential equations
Lecture notesSee the course web site (linked under Lernmaterialien)
Literature1) S.J. Farlow, Partial Differential Equations for Scientists and
Engineers, Dover Books on Mathematics, NY.

2) N. Hungerbühler, Einführung in partielle Differentialgleichungen
für Ingenieure, Chemiker und Naturwissenschaftler, vdf
Hochschulverlag, 1997.

Additional books:

3) T. Westermann: Partielle Differentialgleichungen, Mathematik für
Ingenieure mit Maple, Band 2, Springer-Lehrbuch, 1997 (chapters

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)
Prerequisites / NoticeRequired background:

1) Multivariate functions: partial derivatives, differentiability, Jacobian matrix, Jacobian determinant

2) Multiple integrals: Riemann integrals in two or three variables, change of variables

2) Sequences and series of numbers and of functions

3) Basic knowledge of ordinary differential equations