Lecture 13. Fourier Transform, part II

5 Abril 2019, 12:00 • Hugo Tavares

The Schwarz space: embeddings and dense sets.

The Fourier Transform and its inverse from the Schwarz space into itself, 
Extension to L^2: Plancherel's and Parseval's formulas.

Lecture 12: Distributions part II; Fourier Transform part I

2 Abril 2019, 10:30 • Hugo Tavares

Higher order derivative of a distribution. COmputation of the derivative of the fundamental solution of the Laplace Equation in the sense of distributions. 
Definition of Fourier Transform and Inverse Fourier Transform in L^1. 
The Schwarz space, definition of convergence.

Lesson 11: Distributions, part I

29 Março 2019, 12:00 • Hugo Tavares

The theory of distributions.

- Definition of the test function space, and notion of convergence

Lecture 10: The Wave Equation, part II

26 Março 2019, 10:30 • Hugo Tavares

The wave equation in dimension 1: a stability result, cone of dependence, cone of influence.

The Nonhomogeneous equation: Duhamel's principle.The wave equation in dimension N>1: uniqueness results in a bounded domain and in the whole R^N by energy methods.Solution to the exercise 6 from sheet no.2 and exercise 2 from  sheet no. 3.

Lecture 9: The Heat Equation, part III. The Wave Equation, part I

22 Março 2019, 12:00 • Hugo Tavares

The homogeneous problem in the whole space: some remarks about admissible initial data.The nonhomogeneous problem: Duhamel's principle.

The Wave Equation: - deduction- initial and boundary conditions.- The D'Alembert formula and stability result (dimension 1)