Sumários

Lecture 11: Distributions, part I

27 Março 2018, 14:00 • Hugo Tavares

The theory of distributions.

- Definition of the test function space, and notion of convergence
- Definition of distribution and convergence in the sense of distributions. Examples.
- Derivative of a distribution. Examples.

Lecture 10: The Wave Equation, part II

23 Março 2018, 11:00 • 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.

Exercise class 5

22 Março 2018, 11:00 • Hugo Tavares

The Laplace Equation.
Solution of Exercises  3, 7 and 8 of Exercise Sheet 3.

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

20 Março 2018, 14:00 • 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 (dimension 1).

Lecture 8: The Heat Equation, part II

16 Março 2018, 11:00 • Hugo Tavares

Proof of the uniqueness result for the initial/boundary value problem using energy methods.

The fundamental solution of the heat equation. Solution to the initial value problem in the whole space.