Since differentiation decreases the rate of decrease of the Fourier coefficients, we must assume that our function is sufficiently sooth (k 0 fast enough in order to meaningfully per form this operation. In particular, it can be shown that, if a function has p bounded derivatives,uk1k+1<∞ask→∞ 3.4 Poisson Equation with (0)=u(2 The number and type of boundary conditions required for this equation will be discussed further during the course; however it should be clear that, in one mension, a linear equation involving an n-th order spatial operator will require Note 4 Existence and Uniqueness of the Solution Physically, there can be no net generation since the flux in(u())equals the flux out(u(2). Also note that, from the equation u is free to"Aoat"and the condition Jo u dr=0 pins the solution. Physically, the level of u is determined by initial conditions for the heat equation of which -xz=f is the long-time limit sLiDE 1 Aazà✍á◆â⑨ã❫ä➒å✕áæ✻ä✟ç❳ä♣â✗è➄á➊é✕è➄á➊ê✕â✸å✖ä❫ã♣ç❳ä❫é❻ë♣ä✟ë✰è☞ì❘ä➒ç❳é✪è➌ä✻ê❍í❯å✖ä❫ã♣ç❳ä❫é❻ë♣ä❋ê❍í❅è☞ì❘ä❒î☛ê✪ï❝ç➣á➊ä✟ç➒ã❫ê☎ä➊ðñã✟á➊ä♣â✗è❀ë❫ò✄ó➅ä ôï⑥ë❫è➒é✕ë❫ë❫ïôä✧è☞ì❘é✪è❋ê✕ï❝ç✰í❫ï❝â❑ã♣è➄á➊ê✕â❵á✏ë✴ë➣ï✟ð↕ã♣á➊ä✟â✍è❀õ➦ö➞ëôê★ê✪è◆ì↕÷❍ø ù⑨ú❝ø↕û ü➋í✟é❻ë➣è✝ä✟â⑨ê✕ï✖ý❻ì☎þ á◆â➢ê✕ç❳å✁ä✟ç❨è➡ê ôä❫é✕â✍á◆â✯ý❳í❫ï❝õ◆õ➓ö✡ÿ✗ä♣ç☞í✟ê✕çô è☞ì⑥á✏ë✑ê✱ÿ✗ä♣ç❳é✕è➄á➊ê✕â✁￾✄✂✟â➋ÿ✗é✕ç➣è➄á➊ã✟ï❝õ✒é✕ç❫ò✰á◆è❅ã❫é✪â✆☎❫ä❥ë✓ì✗ê✕ó➎â è☞ì❘é✕è❀ò➷áí❥é➒í❫ï❝â❑ã♣è❀á➊ê✪â❃ì❘é✕ë✞✝✟☎❫ê✕ï❝â❑å✁ä❫å➥å✁ä✟ç➣á✡✠✕é✕è➄á✡✠❻ä✟ë❫ò✰ø ù⑨ú❝ø✒ø ☛❧ø ☞✍✌✏✎✒✑✔✓ é❻ë❥ø ☛☛ø✖û✕✓✖￾ ✗✙✘✛✚ ✜✣✢✥✤✧✦★✦✩✢✥✪✬✫✮✭✙✯✱✰✳✲✴✤✵✢✥✪ ✶ ✷✩✸✡✹✻✺✟✼✾✽ ✿ ù✁❀❁❀❃❂❅❄ ❆✮❇❉❈➊ü✴❊●❋■❍✻❏ ❑▼▲❖◆◗P ù❘❈➊ü✩❏❙❂ ù❘❈❚❋■❍✻❏❯❊ ù❱❀❲❈➊ü✩❏❙❂ ù✁❀❳❈❚❋❨❍✻❏❩❊ ❬ì❘ä➞â✗ïô☎❫ä✟ç➋é✕â⑨å❷è❀ö✱ÿ✗ä➞ê✓í❭☎❫ê✕ï❝â❑å✁é✕ç➣ö➝ã❫ê✕â⑨å✕á◆è➄á➊ê✕â✗ë✧ç✱ä❫❪♣ï❝á◆ç❳ä❫å✡í✟ê✪ç✴è☞ì⑥á✏ë➞ä●❪♣ï✗é✕è➄á➊ê✕â❵ó➎á◆õ◆õ❴☎❫ä å✕á✏ë♣ã♣ï⑥ë❫ë✟ä❫å❷í❫ï❝ç➣è☞ì❘ä♣ç➠å✪ï❝ç➣á◆â✯ýñè☞ì❘ä➢ã❫ê✕ï❝ç❫ë♣ä❯❵➥ì❘ê✕ó➅ä✍✠✕ä✟ç✤á◆è✧ë❳ì❘ê✪ï❝õså❛☎❫ä❵ã✟õ✒ä✱é✪ç✬è☞ì❘é✪è➊ò➍á◆â ê✪â❑ä å✕áôä♣â❘ë➣á➊ê✕â✗ò❇é✡õ➦á◆â❑ä❫é✕ç✻ä●❪♣ï✗é✕è➄á➊ê✕â➥á◆â❳✠✕ê✕õ❜✠☎á◆â✯ý✧é✕â❞❝❢❡➡è◆ì➞ê✕ç❳å✁ä✟ç✰ë➄ÿ✗é✕è➄á➊é✕õ⑨ê✱ÿ✗ä✟ç❳é✪è➌ê✪ç❯ó➎á◆õ◆õ✗ç✱ä❫❪♣ï❝á◆ç❳ä ❝❣☎❫ê✕ï❝â❑å✁é✕ç➣ö➥ã❫ê✪â❑å✕á◆è➄á➊ê✕â✗ë✍￾ ❤❨✐❲❥ ❦♠❧❫♥ ♦ ù❃♣q❆r❂✭ü✴❊ ❦♠❧❫♥ ♦ ❄❞♣✾❆✣❂➜ü s✉t ✈✆✇②①❩③r④ ⑤⑦⑥⑨⑧❶⑩✩①❩③❨❷✥❸✾③❺❹❻❷✥❼❾❽✒❷✏⑧✵❿✏➀②③❨❷✥③❨⑩❁⑩❺✇✴➁✒①✧➂✥③✣➃❘✇❻➄◗➀✏①✧⑧✵✇★❷ ➅P★➆❻➇◗▲➉➈❤❨➊➉➊➆q➋❳◆❫P✴➌✍➍❫➌➎➈❤❨✐➐➏➌ ✐✴➑r✐➌❯◆➓➒q➌✐➌❁➍❤◆◗▲➑✾✐ ➇◗▲✐➈✍➌➔◆❫P✴➌➎→❳➣❻↔↕▲✐ ❈➊ù❀ ❈➊ü✩❏✧❏ ➌❁➙✩➣❤❨➊ ➇⑦◆◗P✴➌ →❳➣❻↔ ➑➣❻◆ ❈➊ù❀ ❈➛❋■❍✻❏◗❏❩➜✞➝➊ ➇➑❃✐✴➑◆❫➌✱◆❫P❤◆➞➋q➟✛➍➑✾➠ ◆◗P❳➌⑦➌➞➙q➣❤◆◗▲➑✾✐ ù ▲➉➇✥➟✛➍❫➌✍➌✱◆ ➑➢➡→➑q❤◆❫➤ ❤❨✐❲❥ ◆◗P✴➌ ➈➑q✐❳❥▲❖◆◗▲➑✾✐❭➥ ❧❫♥ ♦ ù✙♣q❆❺❂✜ü✒➦▲✐➇❘◆❫P✴➌✉➇➑✾➊➣❻◆◗▲➑✾✐ ➜ ➅P✩➆❻➇◗▲➉➈❤❨➊➉➊➆q➋➧◆◗P✴➌ ➊➌❁➨✾➌➊❳➑➟ ù ▲➉➇ ❥➌❯◆❫➌✍➍➠▲✐➌❥ ➏➆❺▲✐▲❖◆◗▲❤❨➊ ➈➑✾✐❲❥▲➩◆❫▲➑q✐➇▼➟➑➍✱◆❫P✴➌❃P✴➌❤◆✉➌➞➙q➣❤◆◗▲➑✾✐✮➑➟❢❑▼P✴▲➉➈●P ✿ ù✁❀➞❀➎❂➫❄ ▲➉➇✱◆◗P✴➌ ➊❖➑q✐➒■➭✩◆◗▲➠➌ ➊ ▲➠▲❖◆ ➜ ✶ ✷✩✸✡✹✻✺✟✼■➯ ùr❂ ➳➲ ú❩➵❘➸ ➲ ùú❴➺➧➻ ú ❀ ❊➼❄❺❂ ➳➲ ú❩➵❘➸ ➲ ❄ú➽➺➧➻ ú ❀ ❈❚❄♦ ❂✭üq❏ ✿ ù❱❀➞❀❃❂ ➳➲ ú❩➵❘➸ ➲ ☛ ❧ ù ú❴➺➧➻ ú ❀ û ù ú ❂ ❄✪ú ☛ ❧ ❈☞ù♦ ❂✜ü✩❏ ➾✾➾