Partial differential equations

(MTH6151)Dr. Juan A. Valiente Kroon

About me:

(Bsc Physics/Maths)

(PhD in General Relativity)

(1st postdoc)(2nd postdoc)

(Reader)(Advanced Research Fellow)

Syllabus•Introduction to partial differential equations •First order partial differential equations •Introduction to second order linear partial differential equations

•The wave equation •Laplace equation •The heat equation

Course notes

• Available in QMplus

• There will be some changes in the material so check regularly for the latest version.

• If you spot any mistake please let me know!

Introduction

What is a partial differential equation?

• A partial differential equation (pde) is an equation for a function on several variables involving partial derivatives of the function.

• If the equation depends on only one variable the one speaks of an ordinary differential equation (ode).

What for?• Partial differential equations are fundamental to

describe the fundamental interactions in Nature and in the modelling of a wide range of systems:

• Nature: electromagnetism, sound, gravitation, fluids,vibrating strings,…

• Mathematics: shape of soap bubbles, complex variables, geometry,…

• Modelling: economics, finance, population dynamics,…

Contributors

Leonard Euler (1707-1783)

Carl Friedrich Gauss (1777-1855)

Joseph Fourier (1768-1830)

Pierre-Simon Laplace (1749-1827)

Bernhard Riemann (1826-1866)

Sofia Kowaleskaya (1850-1891)

More contributors

David Hilbert (1862-1943)

Emmy Noether (1882-1935)

Cathleen Morawetz (1923-2017)

Yvonne Choquet-Bruhat (1923-)

Nobel Prize (Economy)

• The Black-Scholes equation to set the prize of derivative financial products (around 1973)

• Fyscher Black and Myron Scholes were awarded the Nobel Prize of Economy for their model in 1997

Millenium problems ($1,000,000)

• Poincaré conjecture: classification of 3-dimensional surfaces —requires understanding a pde. Solved in 2003 by Grigory Perelman (1966-)

• Navier-Stokes equations: understand the equations of fluid dynamics.