partial differential equations (mth6151)...• partial differential equations are fundamental to...
TRANSCRIPT
![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](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/1.jpg)
Partial differential equations
(MTH6151)Dr. Juan A. Valiente Kroon
![Page 2: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/2.jpg)
About me:
(Bsc Physics/Maths)
(PhD in General Relativity)
(1st postdoc)(2nd postdoc)
(Reader)(Advanced Research Fellow)
![Page 3: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/3.jpg)
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
![Page 4: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/4.jpg)
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!
![Page 5: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/5.jpg)
Introduction
![Page 6: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/6.jpg)
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).
![Page 7: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/7.jpg)
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,…
![Page 8: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/8.jpg)
![Page 9: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/9.jpg)
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)
![Page 10: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/10.jpg)
More contributors
David Hilbert (1862-1943)
Emmy Noether (1882-1935)
Cathleen Morawetz (1923-2017)
Yvonne Choquet-Bruhat (1923-)
![Page 11: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/11.jpg)
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
![Page 12: Partial differential equations (MTH6151)...• Partial differential equations are fundamental to describe the fundamental interactions in Nature and in the modelling of a wide range](https://reader036.vdocuments.mx/reader036/viewer/2022062417/612fd4211ecc51586943b3c5/html5/thumbnails/12.jpg)
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.