2.3 Examples 2.3.1 Linear Advection Equation SLIDE 13 p(a, t)=po(a-at) +at 2.3.2 Burgers' Equation SLIDE 14 Recall f(u)=2u2, so a(u) at ar 0 Solution: u(, t)=uo(a-ut) The solution is constant along the characteristic lines defined by a -ut=To We note that the above solution is defined implicitly(e.g. the definition of the function requires the function itself and therefore it is often not very useful We can verify however by direct differentiation that it is in fact a solation of the partial diffe Consider the initial data <0 0<x<1 16☎✝✆✟✞ ✠☛✡✌☞✎✍✑✏✓✒✕✔✗✖ ✘✚✙✜✛✗✙✣✢ ✤✦✥✣✧✩★✫✪✭✬✯✮✱✰✁✲✳★✵✴✷✶✸✥✣✹✺✧✼✻✓✽✿✾✁✪❀✶❁✥✣✹✺✧ ❂✺❃✫❄❆❅✗❇❉❈✷❊ ❋✵●✷❍✜■❀❏▲❑✄●◆▼ ❖✺P✣◗✗❘❚❙❱❯❳❲❨❖✫❩✫P✣◗❭❬❫❪◆❙❱❯ ❴✦❵❜❛✷❝▲❛◆❞❡❏❚❢✸❝❚❑❤❣❱❏❚❑❤❞✐❍✜❑✄▼❜❢✸❣ ◗✯❲❥◗❩❧❦ ❪✫❙ ✘✚✙✜✛✗✙❤✘ ♠♥✾✗✬♣♦✺★✫✬rq◆st✻✉✽✿✾✩✪❀✶❁✥✣✹✺✧ ❂✺❃✫❄❆❅✗❇❉❈♣✈ ✇❢❁❞①❛②❍✜❍✗③ P⑤④✿❯⑥❲⑧⑦⑨ ④⑨②⑩ ❣❚● ❪✺P✣④❶❯⑥❲✼④ ❷ ④ ❷ ❙ ❦ ④ ❷ ④ ❷ ◗ ❲❥❸ ❋❀●✷❍✜■❀❏❚❑✜●✷▼❺❹ ④✩P✣◗✗❘❚❙❱❯❳❲❨④✺❩◆P✣◗❭❬❻④✺❙❱❯ ❼❵✭❢❽❣❚●✷❍✜■❀❏❚❑✜●✷▼☛❑✜❣✐❞❾●✷▼✳❣✕❏❿❛②▼✫❏➀❛②❍✜●✷▼✭➁➂❏❚❵✭❢➃❞❿❵✳❛②❝❿❛✷❞❡❏▲❢①❝▲❑✜❣❱❏❚❑❤❞✉❍✄❑✜▼✭❢✸❣✌➄✭❢❾➅❜▼✭❢❁➄☛➆✫➇ ◗❭❬❺④✳❙t❲✼◗✳❩✷➈ ➉➋➊♥➌✺➍♣➎➏➊♥➎⑤➐✭➑♣➎✌➎⑤➐✭➊➒➑◆➓❿➍♣➔♣➊♥→①➍♣➣↕↔❀➎➛➙✣➍♣➌❫➙❤→➝➜◆➊✣➞❳➌✺➊❿➜✯➙❆➟➀➠❶➣↕➙✣➡①➙❆➎✟➣↕➢❫➤✕➊❁➥➧➦◆➥➝➎⑤➐✭➊➒➜◆➊✣➞❳➌❜➙❆➎➛➙✣➍♣➌➨➍❱➩➫➎⑤➐✭➊ ➩❿↔❀➌✺➡①➎➛➙✣➍♣➌➯➭❚➊❿➲❾↔❀➙❆➭❚➊❾→❭➎⑤➐✭➊✓➩❿↔❀➌✺➡①➎✟➙✣➍②➌❉➙❆➎✟→①➊❾➣➩①➳❻➑②➌✺➜❻➎⑤➐✭➊①➭❚➊✣➩❾➍②➭❚➊☛➙❆➎➵➙❤→✯➍✕➩❿➎➏➊❾➌➯➌✺➍②➎➵➔r➊①➭❡➢➸↔✵→①➊✟➩❿↔❀➣✄➥ ➉➋➊✯➡❿➑♣➌❉➔r➊①➭❡➙➩❿➢☛➐❜➍♣➺❳➊①➔r➊①➭☛➓①➢➸➜♣➙❆➭❚➊❿➡①➎➵➜②➙➻✐➊①➭❚➊❾➌✳➎✟➙✣➑②➎✟➙✣➍②➌❉➎⑤➐✭➑②➎✉➙❆➎✓➙❤→➒➙❆➌✯➩❾➑◆➡❾➎➃➑➼→①➍②➣↕↔❀➎✟➙✣➍②➌➯➍❱➩ ➎⑤➐✭➊✌➠✭➑②➭❡➎✟➙✣➑②➣✩➜♣➙➻✉➊❾➭❚➊①➌❜➎➛➙✣➑♣➣✎➊❿➲❾↔❜➑②➎✟➙✣➍②➌✭➽⑥➙✣➥❆➊❁➥✄➾ ④✺➚➫❲✼④✺➪❩ ❬❺④❶➪❩ ④✺➚✷❙ ➶ ④✺➚➃❲ ④❩➪ ➹ ❦ ④❩➪ ❙ ④✺➘✎❲➴❬✌④❩➪ ④❭❬❺④❩➪ ④✺➘➛❙ ➶ ④✺➘✎❲ ❬✌④❩➪ ④ ➹ ❦ ④❩➪ ❙❀➷ ❂✺❃✫❄❆❅✗❇❉❈✷➬ ❴✦●✷▼✳❣❱❑❤➄❀❢①❝❧❏▲❵✭❢❽❑✄▼❜❑➮❏▲❑✜❛✷❍✗➄✭❛♣❏❿❛ ④✎P⑤◗✗❘▲❸◆❯⑥❲ ➱✃❐ ➹ ◗➋❒❨❸ ➹ ❬❻◗ ❸➝❮➯◗➋❮ ➹ ❸ ◗➋❰ ➹ ❂✺❃✫❄❆❅✗❇❉❈✷Ï Ð