### Integralgleichungen / Integral Equations (Summer Semester 2012)

- Lecturer: PD Dr. Frank Hettlich
- Classes: Lecture (0156900), Problem class (0157000)
- Weekly hours: 4+2
- Audience: Mathematics (6.-10. semester)

This is a class for students of any mathematics programm. Topics include integral equations of the second kind and their solution theory, the so-called Riesz-Fredholm theory. Furthermore a topic will be the Fourier transform and convolution equations.

A prerequesite to understand the material of this class are the basic lectures of the Bachelor in Mathematics. Necessary results from functional analysis will be discussed in the lecture. In fact, this class can be seen as an expansion and application of the concepts and methods of functional analysis.

Optionally, the lecture will be given in English

Schedule | |||
---|---|---|---|

Lecture: | Monday 9:45-11:15 | Z1 | Begin: 16.4.2012, End: 18.7.2012 |

Thursday 9:45-11:15 | Z1 | ||

Problem class: | Monday 15:45-17:15 | Z1 | Begin: 16.4.2012, End: 16.7.2012 |

# On the content of the lecture

Besides the differential equations also integral equations are important mathematical concepts in physical, technical or medical applications. The formulation of boundary value problems by integral equations are often of theoretical interest (existence and uniqueness of solutions) as well as a basis of efficient numerical solution schemes. The lecture and the exercise session will give an introduction to a functional analytic approach to linear integral equations. We will discuss fundamental typs like Volterra equations, Fredholm equations and convolution equations.

# problem sheets

Additionally there will be problem sheets and an exercise session, where we work on the problems. Afterwards also suggestions of solutions to the problems will be offered.

Problem sheet | Themen |

Problem sheet 1 | ODE <-> IEQ |

Problem sheet 2 | completion, resolvent |

Problem sheet 3 | compact sets, compact operators |

Problem sheet 4 | compact operators, Riesz number |

Problem sheet 5 | dual systems |

Problem sheet 6 | Fredholm's alternative |

Problem sheet 7 | Hilbert spaces, proejection theorem |

Problem sheet 8 | adjoint operators, spectrum |

Problem sheet 9 | eigenvalues, eigenfunctions |

Problem sheet 10 | the neumann problem |

Problem sheet 11 | Fourier transformation |

Problem sheet 12 | convolution operators |

There are solutions to the problems available in German. Please consider the German version of this page

# Skriptum (pdf-Dateien)

Presumably, a lecture note will be available here by the end of the semester.

# References

H. Engel | Integralgleichungen | Springer, 1997 |

H. Hochstadt | Integral Equations | Wiley, 1973 |

W. Hackbusch | Integralgleichungen | Teubner, 1989 |

K. Joergens, | Lineare Integraloperatoren | Teubner, 1971 |

R. Kress | Linear Integral equations | Springer, 1989 |

W. McLean | Strongly Elliptic Systems and Boundary Integral Equations | Cambridge University Press, 1999 |