Courses - Applied Analysis

Lecturer: Stef van Eijndhoven The aim of this course is to provide an overview of techniques used in the analysis of the basic differential equations originating from mathematical physics. The Laplacian operator plays the leading role in diffusion equations, Laplace/Helmholtz equations and wave equations. The applied techniques are based on a spectral analysis of the Laplacian. Results from this analysis are the classical transformations (Fourier and Hankel) and the classical orthogonal decompositions (Fourier, Legendre and Hermite). Relevant links are made with the fundamentals from distribution theory. Further, the Laplace transformation and complex Fourier transformation are discussed. From a wider perspective, these transformations are related to spectral analysis also. Finally, though they are not the main topic of the course, functional analytic concepts are introduced. The duration of the course is 14 weeks. Each week covers two hours of lectures and one hour of discussion of assignments. The course is based on lecture notes and is divided in the following chapters: Ordinary differential equations and Sturm-Liouville theory The Laplacian operator for one-dimensional Euclidean space The Laplacian operator for two-dimensional Euclidean space in terms of cartesian and polar coordinates The Laplacian operator for three-dimensional Euclidean space in terms of cartesian, cylindrical and spherical coordinates Introductory complex function theory Eulerian functions Bessel functions Legendre polynomials and spherical harmonics Fourier theory in relationship to distribution theory Laplace transformation Complex Fourier transformation, Wiener-Hopf The course is meant for all trainees of the T-orientation. Trainees entering the course are expected to have basic knowledge of vector analysis and of the fundamentals of differential equations and functional analysis. The lecture notes are self-contained.

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