# Partial differential equations

# Partial differential equations

Spring 2014

–

Room T1, Facultat de Matemàtiques, UB

The course will start with a modern review of the key topics learnt in a first PDE course. It will continue with the essential and more advanced techniques based on Functional Analysis and the Calculus of Variations, including an introduction to the finite element method to compute solutions. Theoretical results will be explained in relation with concrete models from Physics, Finance, Image Processing, and Biology.

Spring 2014

–

Room T1, Facultat de Matemàtiques, UB

The course will start with a modern review of the key topics learnt in a first PDE course. It will continue with the essential and more advanced techniques based on Functional Analysis and the Calculus of Variations, including an introduction to the finite element method to compute solutions. Theoretical results will be explained in relation with concrete models from Physics, Finance, Image Processing, and Biology.

Xavier Cabré ICREA (UPC)

Maria del Mar Gonzalez (UPC)

##### Contents

**One-dimensional transport, waves, and diffusion:**

Mathematical modelling, linear first order transport, d’Alembert’s formula, reflection of waves, energy, separation of variables, Fourier series, Sturm-Liouville theory. Applications: Schrödinger equation and Black-Scholes equation in Finance**The diffusion equation in Rn:**

Maximum principle, divergence theorem and energy method, fundamental solution in Rn, the Dirac delta, convolutions, random walks, probabilities and the diffusion equation. Applications: image processing**The Laplace and Poisson equations:**

Convex and holomorphic functions, mean value property, maximum principle, Newtonian potential, Green functions, relation between random walks, the (discrete) Laplacian and exit probabilities, finite differences method. Applications: ideal fluids**The Calculus of Variations and Hilbert space techniques:**

Dirichlet minimization principle and the energy method, nonlocal Dirichlet principle, orthogonal projections in Hilbert spaces, introduction to Sobolev spaces, the finite element method. Applications: minimal surfaces and image processing**Nonlinear first order equations:**

Quasilinear first order equations, Burgers equation, method of the characteristics, introduction to Hamilton-Jacobi equations. Applications: traffic dynamics, optimal control problems

##### Bibliography

**Basic:**

- Strauss, Walter.
*Partial Differential Equations: an Introduction*. Second edition. Wiley, 2008. - Pinchover, Yehuda; Rubinstein, Jacob.
*An introduction to partial differential equations.*Cambridge University Press, 2005.

**More advanced:**

- Salsa, Sandro.
*Partial Differential Equations in Action, From Modelling to Theory.*Springer, 2008. - Evans, Lawrence C.
*Partial differential equations.*Second edition, American Mathematical Society, 2010. - Brezis, Haim.
*Functional analysis, Sobolev spaces and partial differential equations*. Universitext, Springer, 2011.