+C=0, B+√B2 Then, the equation becomes 0n=P(,7,中,中c,n) An alternative form can be obtained by setting X=S+n,Y=s-n pxx-pyy=F"(X,Y,,φx,φy) PARABOLIC case(B2-4AC=0) Here, we can only set a(or c) to zero(not both), other wise s and n are not independent. If we set a=0, then It can be verified, by direct evaluation, that in this case b=0, in which case we can pick n to be any function such that [J#0, and the equation becomes dm=F(s,m,,吹,) ELLIPTIC case(B2-4AC<0 This case is identical to the hyperbolic case but now s and n are complex conjugates(B4-4AC <0). Take X=S+n, y=i(s-n) and the equation F(X,Y,,中x,y Application +U. Vu=nvu+f
❩❭❬✄❪❴❫❪❴❵✮❛✲❜✽❝✠❞ ❬✙❪✗❫❪❴❵✮❛❡❝✜❢❤❣❥✐✍❦ ❩❭❬✙❧♠❫ ❧♠❵✮❛✄❜❅❝✜❞ ❬✙❧♥❫ ❧♠❵✮❛❡❝✜❢✧❣❨✐✍❦ ❪✗❫ ❣♣♦❞✥❝rq❞ ❜ ♦ts❩ ✉❩ ❢ ❪❵ ❦ ❧♥❫ ❣✈♦ ❞✇❝rq❞ ❜ ♦ts❩ ✉❩ ❢ ❧❵ ❦ ①③②✍④✣⑤✮⑥✹⑦✓②✴④✬④❍⑧⑩⑨✍❶✺⑦✿❷❹❸♠⑤❻❺✙④❀❼✗❸♥❽✘④❀❾ ❿☞➀✖➁ ❣❨➂✬➃➅➄ ❪ ❦ ❧ ❦ ❿ ❦ ❿☞➀ ❦ ❿✙➁✺➆✗➇ ➈⑤●❶♠➉➊⑦✓④✣➋✓⑤✍❶➌⑦✓❷➎➍♥④➐➏➑❸♥➋✿❽➒❼❍❶✺⑤❭❺☞④✬❸♠❺✹⑦✖❶✺❷❹⑤✴④❍➓➔❺➣→❉❾✦④✣⑦✦⑦✓❷➎⑤✴↔❡↕ ❣ ❪ ❝ ❧ ⑥❭➙ ❣ ❪ ♦ ❧✄➛ ❿☞➜✽➜ ♦ ❿✄➝➞➝ ❣✇➂➃ ➃ ➄↕ ❦ ➙ ❦ ❿ ❦ ❿☞➜ ❦ ❿✄➝✽➆ ➟➈➐➠➐➈➐➡❯➢❲➤✮➥✿➦ ❼✣❶♠❾✿④ ➄➧❞❜ ♦➨s❩❢✧❣❥✐ ➆ ➛ ➩❯④❍➋✿④♥⑥✱➫✽④❉❼❍❶✺⑤➭❸♠⑤✍➉➎→➯❾✦④✣⑦✘➲ ➄❸♥➋✒➳➆ ⑦✓❸➸➵✣④❍➋✿❸ ➄⑤✴❸✺⑦✘❺✙❸♠⑦✿②➆ ⑥➺❸✺⑦✓②✴④✣➋✓➫❖❷❹❾✿④ ❪ ❶♠⑤✍➓ ❧ ❶✺➋✓④❡⑤✴❸✺⑦ ❷❹⑤✍➓✹④✣➻☞④✣⑤✙➓✹④✣⑤⑩⑦ ➇ ➥➏➺➫✽④✬❾✦④✣⑦❖➲ ❣❨✐ ⑥➣⑦✿②✴④❍⑤ ❪❴❫ ❪❵ ❣ ♦ ✉ ❞ ❩➽➼ ➥⑦✽❼❍❶✺⑤❊❺☞④❲➍♠④✣➋✓❷➎➾✍④❍➓✲⑥♥❺➣→➚➓✴❷➎➋✓④❍❼✗⑦❅④✣➍➌❶♠➉➎⑨✍❶✺⑦✿❷❹❸♠⑤✮⑥♥⑦✿②✙❶➌⑦③❷➎⑤❡⑦✓②✴❷➪❾✽❼❍❶♠❾✿④✁➶ ❣❥✐ ⑥⑩❷❹⑤❊➫❖②✴❷➪❼✖②❉❼❍❶♠❾✿④❯➫❅④ ❼✣❶♠⑤➔➻✍❷❹❼✖➹ ❧ ⑦✿❸❡❺☞④❉➘➌➴✙➷➐➏➑⑨✴⑤✙❼❴⑦✿❷❹❸♠⑤●❾✿⑨✍❼✖②➔⑦✿②✍❶✺⑦✶➬➱➮✷➬☞✃❣✇✐ ⑥✴❶♠⑤✍➓❊⑦✓②✴④✬④❍⑧⑩⑨✍❶✺⑦✿❷❹❸♠⑤❻❺✙④❀❼✗❸♥❽✘④❀❾ ➛ ❿➁✗➁ ❣❨➂➃ ➄ ❪ ❦ ❧ ❦ ❿ ❦ ❿➀ ❦ ❿➁ ➆ ➼ ❐➤✱➤✮➥➟❅①➥✿➦ ❼✣❶♥❾✦④ ➄➧❞❜ ♦ts❩❢✩❒❮✐ ➆ ➛ ①③②✴❷➪❾❭❼✣❶♠❾✿④❰❷➪❾❻❷❹➓✴④✣⑤⑩⑦✿❷➪❼✣❶♠➉✁⑦✓❸✜⑦✓②✴④➯②➣→⑩➻☞④✣➋✓❺☞❸♠➉❹❷❹❼t❼✣❶♥❾✦④➨❺✴⑨✹⑦●⑤✴❸➌➫ ❪ ❶✺⑤✍➓ ❧ ❶✺➋✓④➨❼✣❸♠❽➚➻✴➉❹④✗Ï ❼✗❸♥⑤➌Ð❂⑨✴↔⑩❶➌⑦✿④❀❾ ➄➧❞❜ ♦➯s❩❢Ñ❒✥✐ ➆❴➇ ①Ò❶✺➹♥④Ó↕ ❣ ❪ ❝ ❧ ⑥➯➙ ❣☎Ô❴➄ ❪ ♦ ❧ ➆ ❶✺⑤✍➓●⑦✿②✴④✶④❍⑧⑩⑨✍❶➌⑦✓❷➎❸♥⑤ ❺☞④❍❼✗❸♥❽➚④❍❾ ➛ ❿➜③➜ ❝ ❿➝➞➝ ❣❨➂➃ ➄↕ ❦ ➙ ❦ ❿ ❦ ❿➜ ❦ ❿➝ ➆✗➇ Õ☞Ö➧Õ☞Ö➧Õ ×❉Ø✱Ø➞Ù➧ÚÜÛ♠Ý✹Þ❀Ú➧ß☞à✱á â☞ã⑩äæåèçté ê☞ë ê☞ì ❝ríïî❀ð ë ❣✇ñ☞ð❜ ë ❝✕ò ó✗ô ë ❷➪❾ ➼❍➼✣➼ õ
Heat ransfer N ni Conceni a iion d Coastal Engineering nOby bili y Disi aibujion d Statistical Mechanics This equation is known in Statistical Mechanics as the Fokker-Planck n Paice of in Ojiion d Financial Engineering This equation is known in Financial Engineering as the Black-schole N In some of the above cases the equation is slightly different (e.g. particalar non-constant coefficients), however the basic form remains invariant 2 Limiting Cases 21 elliptic卫 quations slide 3 Poisson Equatio Convection-Diffusion even when the boundary conditions or f are not smooth ng he docu in of dey endence of u(a, y)is Q2 his means that a small perturbation of f, or boundary conditions, anywhere in the domain unill alter the value of u(a, y) of elliptic equations will be studied extensively in this course. For these s we will be presenting solution techniques using Finite Differ ences, Finite Elements and Boundary Integral Methods
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