Partial differential equations (MTH6151)...• Partial differential equations are fundamental to...

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Partial differential equations (MTH6151) Dr. Juan A. Valiente Kroon

Transcript of Partial differential equations (MTH6151)...• Partial differential equations are fundamental to...

Page 1: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range

Partial differential equations

(MTH6151)Dr. Juan A. Valiente Kroon

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About me:

(Bsc Physics/Maths)

(PhD in General Relativity)

(1st postdoc)(2nd postdoc)

(Reader)(Advanced Research Fellow)

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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

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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!

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Introduction

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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).

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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,…

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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)

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More contributors

David Hilbert (1862-1943)

Emmy Noether (1882-1935)

Cathleen Morawetz (1923-2017)

Yvonne Choquet-Bruhat (1923-)

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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

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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.