Welcome to the website for MA34110: Partial Differential Equations. As we progress through the module, this is where you'll find all sorts of goodies to hopefully guide you on your path towards becoming a PDEs expert, such as practice questions, revision videos and short summary notes. As with anything in the course, I'm always interested in getting feedback from you, so feel free to grab me after a lecture with any suggestions or comments, or get in touch with me via some other method.
Hopefully finding content this website is fairly intuitive. The course fairly naturally splits up into a few major topic areas (there's an expanded and to some extent interactive list on the course outline page):
Each major topic now has its own entry in the navigation bar, which contains relevant content (such as handouts, videos, interactive applets etc.) and links to further resources that may be interesting or useful. Keep checking back as the course progresses as more content is added.
Partial differential equations (PDEs) are an incredibly important topic across the board in mathematics. As well as leading to lots of very interesting abstract questions in pure mathematics, they underpin almost all applied mathematics. PDEs arise all over the place in mathematical models of all sorts of phenomena, be they physical, astronomical, quantum, financial or mechanical.
So, put simply, if you want to precisely understand how real-life systems behave and evolve, be they on tiny subatomic quantum scales or on the scales of galaxies, you're going to need to look at some PDEs. If you want to understand why the signal on your phone is rubbish if you're holding it in a certain way, then you'll need Maxwell's equations. If you want to design a supersonic car, you'll need the Navier-Stokes equations for the aerodynamics. The list goes on; PDEs are everywhere in models of real-world phenomena.
MA34110 will approach PDEs from a fairly applied perspective; we'll mainly be concerned with methods used to solve important types of partial differential equations. Much of this is theoretically underpinned by the kind of techniques you'll meet in MA30210 - Norms and Differential Equations. We'll occasionally touch on concepts like "well-posedness" for example, that are precisely quantified using norms, and will solve PDEs using Fourier Transform techniques which have a rich theoretical underpinning for which the material in the Norms course would be good preparation.
If you've taken (or are taking) any of the fluids modules (e.g. Hydrodynamics I or II) then you've probably met the Euler equations and will later meet the Navier Stokes equations; these are all partial differential equations (and if you can rigorously prove existence and smoothness of their solutions, you win $1m [please remember to share the prize!]).
Croeso i wefan MT34110: Hafaliadau Differol Rhannol. Wrth inni barhau trwy'r modiwl, dyma le byddwch chi'n darganfod pob math o ddanteithion i'ch arwain ar eich llwybr i fod yn arbenigwr mewn hafaliadau differol rhannol, fel cwestiynau ymarfer, fideos adolygu a chrynodebau byr. Fel gydag unrhyw beth yn y cwrs, rwy'n diddori yn eich adborth, felly mae croeso mawr ichi fy nal ar ôl darlith gydag unrhyw awgrym neu sylw, neu cysylltwch â mi mewn ffordd arall.
Rwy'n gobeithio byddwch yn darganfod gwybodaeth ar y wefan yma yn hawdd. Mae'r cwrs yn rhannu'n naturiol mewn i ychydig o brif destunau (mae yna restr hirach a rhannol ryngweithiol ar dudalen Amlinelliad y Cwrs):
Mae gan bob prif destun ei eitem ei hun ar y bar llywio, sy'n cynnwys gwybodaeth berthnasol (fel taflenni, fideos, rhaglennig rhyngweithiol ayyb.) a dolenni i adnoddau ychwanegol a all fod yn ddiddorol neu'n ddefnyddiol. Cadwch i wirio'r tudalennau yma wrth i fwy o wybodaeth cael ei ychwanegu wrth inni barhau trwy'r cwrs.
Mae Hafaliadau Differol Rhannol (HDRh) yn destun hynod o bwysig o fathemateg. Ynghyd ag arwain at lawer o gwestiynau haniaethol ym mathemateg bur, maent yn greiddiol i ran helaeth o fathemateg gymhwysol. Mae hafaliadau differol rhannol yn ymddangos pobman mewn modelau mathemategol o bob math o ffenomenau, boed yn ffisegol, seryddol, cwantwm, ariannol neu'n fecanyddol.
Felly, yn syml, os ydych am wir ddeall sut mae systemau go iawn yn ymddwyn ac yn datblygu, boed nhw ar raddfeydd is-atomaidd cwantwm bach neu ar raddfeydd galaethau, bydd yn rhaid ichi weithio â hafaliadau differol rhannol. Os ydych am ddeall pam fod eich signal ffôn yn ddiwerth os rydych yn ei ddal mewn ffordd arbennig, yna bydd angen hafaliadau Maxwell arnoch. Os ydych am ddylunio car uwchsonig, yna bydd angen hafaliadau Navier-Stokes arnoch ar gyfer yr aerodynameg. Mae'r rhestr yn mynd ymlaen; mae hafaliadau differol rhannol ymhobman mewn modelau o ffenomenau byd go iawn.
Bydd MT34110 yn mynd i'r afael â hafaliadau differol rhannol o safbwynt eithaf cymhwysol; byddwn yn ymwneud yn bennaf â dulliau a defnyddir i ddatrys mathau pwysig o hafaliadau differol rhannol. Mae llawer o hyn wedi'i danategu'n ddamcaniaethol gan y math o dechnegau byddwch yn dysgu yn MT30210 - Normau a Hafaliadau Differol . Byddwn yn cyffwrdd â chysyniadau fel "iawn-osodedig" er enghraifft, sy'n cael eu meintoli'n gywir gan normau, a byddwn yn datrys hafaliadau differol rhannol gan ddefnyddio technegau Trawsffurfiadau Fourier sydd â sail ddamcaniaethol gyfoethog y byddai'r deunydd yn y cwrs Normau yn baratoad da ar eu cyfer.
Os rydych wedi cymryd (neu os ydych yn cymryd) unrhyw un o'r modiwlau llifyddion (e.e., Hydrodynameg I neu II) yna rydych mwy na thebyg wedi cwrdd â hafaliadau Euler a byddwch yn cwrdd yn hwyrach â hafaliadau Navier-Stokes; mae'r rhain yn hafaliadau differol rhannol (ac os allwch chi brofi'n drwyadl bodolaeth datrysiadau llyfn , byddwch yn ennill $1m [cofiwch rannu'r gwobr!].