Partial Differential Equations An overview Lecture 1
✂✁☎✄✝✆✟✞✠✁☎✡☞☛✌✞✎✍✑✏✒✄✓✏✕✔✖✆✟✞✠✁✗✡✙✘✛✚✢✜✣✁✤✆✟✞✦✥✧✔✩★✫✪ ✬✔✮✭✰✯✢✏✱✄✝✯✲✞✦✏✴✳ ✵✶✏✒✷✸✆✟✜✹✄✺✏✼✻
1 Model Equation U·u=rV >0, f, gi Despite its apparent simplicity this equation appears in a wide range of dis m heat 7 to financial er we will make extensive use of this equation, and several of the limiting cases contained therein, to illustrate the numerical techniques that will be presented In some cases U, k, and f will be functions of the solution u, in which case the equation is said to be nonlinear Note 1 Derivation of the Convection-Diffusion Equat ion for Heat Transfer We sketch below the derivation of the Convection-Diffusion equation for the particular problem of Heat Transfer in a moving fuid. Consider a velocity field U=(U(a, y), V(, y)) which is(for simplicity) time independent and incom au av v A streamline is a curve which is obtained by integrating the vector field and corresponds to the trajectory of a fuid parcel
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❉✲❊✮❋❍●❏■✏❑✒▲✏▼✔◆P❖◗●❘❋✴❙❚▼✔❯✖❱❲▼★❙✔●❘◆❳❱✘❋✱❑☞❨✱▼✛❨✱❩✲▼✭▼✔❬❪❭◗●❘❨✱❑✒❊✏◆P▼☎❫✠❖◗❋✱▼✍❴✱❴✱❑☞◆◗■❵❨✱❩◗▼★❛◗●✏❯✒●✏◆◗❙❚▼★❊❘❜❝▼✔◆✲▼☎❋✙■✏❞ ●✏❴ ❡✮❢❡✮❣✭❤❤ ❤ ❤ ✐✍❥❚❦❁❧♥♠✔♦✱♣✾q❁❦❁r s t✵✉ ✈✍✇❚①✍② ③⑤④✙⑥⑧⑦❄⑨✴⑩✖❶✍⑦✦⑥❸❷❹⑦❺④✙⑥ ⑥✪③✪④✴❶❹❻⑧⑩☞⑦✦③✪③✪⑦✦❼ ❽❶☎⑥⑧⑨✤❾✁④✴③⑧❿✦⑦❺➀ ➁ ➂➄➃ ✈✍✇➅①❹② ③✪④✙⑥✪⑦➆⑨✴⑩✷➇❚⑨➅➀➉➈✍➊✘⑦✦⑥⑧③❽ ❿ ❷✍⑦✱④✙⑥➌➋➅⑦❺❶✍⑦❺③✪④✙⑥ ❽⑨➅❶ ❽❶❹❻ ❽❼❹⑦❝❾✁④✴③✪❿✦⑦✦➀ ❱✘❩✲▼✔❋✱▼ ❢ s➎➍✠➏➅➐ ❑✒❴❲❨✙❩✲▼✬❑☞◆❪❨✙▼☎❋✙◆◗●❘❯✣▼✔◆✲▼☎❋✙■✏❞✳❖➑▼☎❋➒❭◗◆✲❑➓❨➒▲✮❊✏❯✒❭✲➔→▼✏➣ ➐ ❑✒❴❲❨✙❩✲▼↔❨✱▼✔➔→❖✎▼✔❋✙●❘❨✱❭✲❋✙▼✏➣ ➍ ❑↕❴❝❨✙❩✲▼♥➙➛▼✔◆◗❴✱❑➓❨✦❞✳●❘◆✎➙ ➏ ❑↕❴❄❨✱❩◗▼❍❴✱❖✎▼✍❙❚❑☞➜✎❙➒❩◗▼✔●✁❨✍➝❄➞✤❩✲▼➒❨✙▼☎❋✙➔ t✛✉ ❙✔●❘◆❵❛➑▼➟▼☎❫✠❖◗❋✱▼✍❴✱❴✱▼✔➙→❑✒◆✳❨✱▼✔❋✱➔✳❴ ❊❘❜➠❨✙❩✲▼♥❩◗▼✔●✁❨➆➡◗❭➛❫❵➢ s➥➤✾➦☎➧◗➨✱➦✔➩❘➫ ➣✠❛✠❞✭❙☎❊✏◆◗❴✱❑✒➙✲▼☎❋✙❑☞◆✲■✵●✏◆❵❑✒◆➛➜◗◆✲❑☞❨✱▼✍❴❺❑✒➔✳●❘❯➑❖◗●❘❋✴❙❚▼✔❯◗❊❘❜➄❴✱❑☞➭✔▼ ❡✮➯➒❡✏➲ ➝ ➞✤❩✲▼↔◆✲▼☎❨➒❋✙●❘❨✱▼↔❊❘❜✷❨✱❩✲▼✬❩◗▼✔●✁❨✤❨✙❋✙●✏◆◗❴✦❜➳▼✔❋✱❋✙▼✔➙❏❑☞◆❪❨✙❊→❨✱❩✲▼✬❖◗●✏❋✙❙☎▼☎❯✪➣➛❖✎▼✔❋✘❭✲◆✲❑☞❨➟●❘❋✙▼✔●◗➣➛❱✘❑☞❯✒❯➠❨✱❩✠❭◗❴✘❛➑▼ t✉ s ➵✬➸ ❡✮➯✛❡✏➲➻➺➅➼➦➧ ➼ ➯ ❡✏➯✭❡✏➲ ➁ ➼➦➩ ➼ ➲ ❡✮➯✛❡✮➲✮➽ s ➵➟➾➥➚ ➢➶➪ ➞✤❩✲▼↔❩✲▼✍●✁❨✘➡◗❭➛❫➹❑✒❴❲➜✎◆◗●❘❯✒❯☞❞❏❋✱▼✔❯✒●❘❨✱▼✔➙❏❨✱❊→❨✙❩✲▼↔❨✱▼✔➔→❖✎▼✔❋✙●❘❨✱❭✲❋✙▼❍❨✱❩✲❋✙❊✏❭✲■✮❩➹❉◗❊✏❭✲❋✙❑✒▼☎❋✍➘ ❴❲❯↕●❹❱ ➢ s ➵➟➴✲➾➐ ❱✘❩✲▼✔❋✱▼ ➴ ❑✒❴✤❨✙❩✲▼ ➤ ❙☎❊✏◆◗❴❺❨✙●✏◆❪❨ ➫ ❨✱❩✲▼✔❋✱➔✳●✏❯➄❙❚❊✏◆✎➙➛❭◗❙➅❨✙❑☞▲✠❑☞❨✦❞❏❊✏❜❸❨✱❩◗▼↔➡◗❭✲❑↕➙✣➝ ➞✤❩✲▼✛➙✲▼☎❋✙❑☞▲✁●✁❨✙❑☞▲✮▼↔❨✱▼✔❋✱➔ ❡❪❢❡✏❣ ❑✒❴♥❙☎●❘❯✒❯✒▼✔➙➷●✳➔✳●✁❨✙▼☎❋✙❑✒●✏❯❸➙➛▼✔❋✱❑✒▲✁●✁❨✱❑✒▲✏▼✛❛✎▼✍❙☎●❘❭✎❴❺▼✵❑☞❨➟❑↕❴➟●✮❴✱❴✱❊➛❙❚❑↕●✁❨✱▼✍➙ ❱✘❑☞❨✱❩➷●★➜✲❫➛▼✔➙✚➡◗❭✲❑↕➙➹❖◗●❘❋✴❙❚▼✔❯✎❨✙❩◗●✁❨➒➔→❊✁▲✮▼✔❴✤❱✘❑☞❨✱❩➹❨✱❩✲▼↔➡✎❊✁❱✬➝ ➬❄❫➛❖✲❋✙▼✔❴✙❴✱❑☞◆✲■✚❨✱❩◗▼✳➔→●❘❨✱▼✔❋✱❑↕●❘❯❄➙➛▼✔❋✱❑✒▲✁●✁❨✱❑✒▲✏▼ ➤⑧➮●❘■✮❋✙●✏◆✲■✏❑↕●❘◆✞◆✲❊❘❨✙❑☞❊✮◆➫ ❑☞◆➱❨✙▼☎❋✙➔→❴↔❊✏❜✃➜◗❫✠▼✍➙❳❨✱❑✒➔✭▼ ●❘◆✎➙✚❴✱❖◗●✏❙☎▼✬➙➛▼☎❋✙❑☞▲✁●❘❨✱❑✒▲✏▼✔❴ ➤➬❝❭✲❯✒▼☎❋✙❑↕●❘◆❏◆◗❊❘❨✱❑✒❊✏◆➫ ❱✃▼↔❊✏❛✲❨✙●❘❑✒◆ ❡❪❢❡✮❣ ❤ ❤ ❤ ❤ ✐✔❥❚❦❁❧↔♠✔♦✱♣➳q❁❦❁r s ➍➟➏ ❡ ❡✮❣ ➐✛➤⑧❐♠☎♦✙♣➳q❁❦❁r ➤ ❣ ➫➅➨ ❣ ➫ ❒
pc at dtpareel_at dyparcel_aT aT aT +U·VT t hich then yields the Wndlfip f the cnvectin-diffusi n equatin dfte-divid ing y pc dnd de wning fi dl trr ie at tion on DE'r(OptiontI Ret ding We eview the classi Wcati n f Wst dnd sec nd Gde lined Pd tid! diffe entia! dent vd This classiC t dete, ine the chd dcte dependent dll cn ddditi ndl e dding) Within this ngte the independent vd idles dnd o(a, y),the First Order pDE’s Fi st Gde pa tid diffe entigl equati ns d'e always hyperbolic type.A gene dl lined wst de equati n can ye w itten ds Aφx+B0y=F(x,y,) hee a and b day ye functins a nd g, yut nf Prdr +oydy,then Ado+y(Bda -Ady)= fda AIng the lines(chd dcte istics)such thdt Bdx-Ady=0 Cha cte istics d e B r-Ay=v, G- any a, dnd the gene dl sAutin yec es F da +g() A F da +g(Bx- Ay
❮ ❰➟ÏÑÐÒ ÒÓ➟Ô➑ÕÔ➑ÖÑ×Ö✎Ø☎Ù✙Ú➳Û❁Ü❁Ý ×✮Þ ß à❚á â ã ä Ô➑ÕÔ➑åæ×å❘Ø☎Ù✙Ú➳Û❁Ü❁Ý ×✮Þ ß à➅á â ç ä Ô➑ÕÔ Þ è☎éé ê ❮ ❰➟Ïìë Ô➑ÕÔ Þ äìí Ô➑ÕÔ➑Ö➎äïî Ô➑ÕÔ➑å ð ❮ ❰➟Ïìë Ô➑ÕÔ Þ äìñóò✍ôÕ ð õ✘ö✲÷↕ø✴ö✭ù✱ö✲ú✔û✳ü❪÷✒ú☎ý↕þ✲ÿ➌ù✙ö✲ú✁◗û✄✂❘ý✆☎✞✝✠✟☛✡☞✝✌☎➑ù✱ö✲ú♥ø✍✝✏û✏✎✮ú✔ø➅ù✙÷✑✝✮û✓✒❁þ➛÷✕✔✗✖◗ÿ✱÷✑✝✮û★ú✙✘✚✖✛✂✁ù✙÷✑✝✮û✜✂✌☎✼ù✱ú✢✟➆þ➛÷✣✎❪÷↕þ✏✒ ÷✒û✆✤✜✥❪ü ❰❪Ï ✂❘û✎þ✚þ➛ú✦◗û✲÷✒û✆✤★✧ ❮ ✩ ❰➟Ï✫✪ ✬ ❮ ✬✮✭ ❰♥Ï✫✯ ✰✲✱✫✳✵✴✷✶ ✸✺✹✼✻✓✽✙✽✙✾✞✿❁❀❂✻✫✳☛✾❃✱✓❄❅✱✆❆✁❇❉❈✷❊●❋✽✲❍❏■✺❑▲✳▼✾◆✱✏❄❖✻✓✹◗P❘✴❙✻❯❚✫✾❱❄❳❲✄❨ ❩ú❬✟✙ú✦✎✠÷✒ú☎õÑù✱ö◗ú✭ø☎ý✣✂✮ÿ✱ÿ✱÷✕➑ø✦✂✁ù✙÷✑✝✮û❘✝✌☎❭✛✟✴ÿ❺ù❪✂✏û◗þPÿ✱ú✔ø✍✝✮û◗þ❘✝✠✟✙þ➛ú✢✟♥ý✒÷✒û✲ú✢✂❙✟❴❫✛✂❙✟❺ù✙÷✣✂✏ý✖þ➛÷✑✔➠ú✦✟✙ú☎û❪ù✱÷❵✂❘ý ú✢✘✚✖✛✂❘ù✱÷✣✝✏û◗ÿ↔÷✒û ù✦õ◗✝➹÷✒û◗þ➛ú✢❫✎ú✔û◗þ➛ú✔û✮ù❛✎❜✂✌✟✙÷✣✂❙✥✲ý☞ú✍ÿ ✯❞❝ö◗÷✒ÿ★ø❚ý❵✂✏ÿ✙ÿ❺÷✑✎ø✦✂❘ù✱÷✣✝✏û➱÷✒ÿ❡✖◗ÿ❺ú✦☎✞✖✲ý ✪ û✆✝✏ù❡✝✏û✲ý✒ü ù☛✝✚÷✒þ✲ú☎û❪ù✱÷✑☎➳ü➷ÿ▼✝✮ý✑✖➛ù✙÷✑✝✮û❘✡→ú❚ù✱ö✛✝✠þ◗ÿ❴✂❙❫✆❫✲ý✒÷✒ø✢✂✌✥✲ý✒ú✵ù❏✝❢✂❣❫✛✂❙✟❺ù✙÷✒ø✦✖✲ý❵✂✌✟➟ú✙✘✚✖✛✂✁ù✙÷✑✝✮û ù✦ü✚❫➑ú ✪ ✥✆✖✲ù✺✂❘ý↕ÿ☛✝ ù☛✝ þ➛ú☎ù✱ú✦✟❏✡→÷☞û◗ú✛ù✙ö✲ú✭ø✴ö✄✂✌✟❤✂✏ø➅ù✙ú✦✟❴✝✌☎❝ù✙ö✲ú✳ÿ▼✝✮ý✑✖➛ù✙÷✑✝✮û◗ÿ ✯❪✐✝❙✟♥ù✦õ❁✝➹÷✒û◗þ➛ú✦❫➑ú☎û✎þ➛ú☎û❪ù❉✎❜✂✌✟✙÷✣✂❙✥✲ý☞ú✍ÿ❥✂✏ý☞ý ú✢✘✚✖✛✂❘ù✱÷✣✝✏û◗ÿ✛ø✦✂❘û❦✥➑ú❏ø☎ý✣✂✮ÿ✱ÿ✱÷✕✎ú✔þ❦✂✏ÿ✛ÿ❺ö✆✝✁õ✘û❦✥➑ú☎ý✣✝✁õ ✯ ❩ú❏û✆✝❘ù✙ú✳ù✱ö✛✂❘ù❧☎✞✝✠✟✵ú✢✘✚✖✛✂❘ù✱÷✣✝✏û◗ÿ✛õ✘÷➓ù✙ö ✡✜✝❙✟✙ú↔ù✱ö✛✂✏û➷ù✦õ◗✝❵÷☞û✎þ➛ú✦❫➑ú☎û◗þ✲ú☎û❪ù❴✎❜✂✌✟✙÷❵✂✌✥✲ý✒ú✔ÿ ✪ ù✙ö✲ú★ø☎ý✣✂✮ÿ✱ÿ✱÷✕➑ø✦✂✁ù✙÷✑✝✮û ÷✒ÿ❥☎❱✂✌✟❴✡❛✝✠✟✱ú★ø✍✝❙✡✜❫✲ý✒ú✍♠❘✂✏ÿ ÿ☛✝❙✡→ú↔ú✢✘✚✖✛✂✁ù✙÷✑✝✮û◗ÿ♥✡❣✂❹ü❣✥✎ú✍ø✍✝✠✡✭ú❉✝❙☎❳✡→÷✑♠✠ú✍þ✚ù✦ü✚❫➑ú❞♦✾ÿ✱ú☎ú❞♣✐❳q ☎✞✝❙✟❥✂✮þ✲þ➛÷☞ù✱÷✣✝✏û✛✂✏ý✫✟✱ú✙✂✏þ➛÷✒û✆✤✏r ❩÷☞ù✱ö✲÷✒û ù✙ö✲÷✒ÿ➹û✆✝✏ù✱ú ✪ ♦Ö❳s❺å r ✪ þ➛ú✔û✆✝❘ù✙ú✔ÿ❏ù✱ö✲ú➱÷☞û✎þ➛ú✦❫➑ú☎û◗þ✲ú☎û❪ù❘✎❂✂❙✟✱÷❵✂✌✥◗ý☞ú✍ÿ★✂❘û✎þ✉t✈♦Ö❳s❺å r ✪ ù✱ö✲ú þ➛ú✢❫✎ú✔û◗þ➛ú☎û❪ù▲✎❜✂✌✟✙÷❵✂✌✥✲ý✒ú ✪ ✝✠✟✤ÿ☛✝✏ý✣✖➛ù✙÷✑✝✮û❢✝❙☎✖ù✙ö✲ú❡✇②①✺③ ✯ ④✁⑤✞⑥❜⑦✢⑧❣⑨❣⑥❜⑩❳❶✚⑥★❷❉❸✷❹❧❺✣⑦ ✐ ÷✑✟✴ÿ❺ù★✝❙✟✴þ➛ú✦✟★❫✛✂✌✟✱ù✱÷❵✂❘ý❍þ➛÷✕✔➠ú✦✟✙ú☎û❪ù✙÷✣✂✏ý➟ú✢✘✚✖✛✂❘ù✱÷✣✝✏û◗ÿ❢✂✌✟✙ú❻✂❘ý✒õ♥✂❹ü✠ÿ✜✝❙☎❧❼✗❽✫❾ ❶✚⑥❂❿②➀✄➁✼⑤❱➂ ù✦ü✏❫➑ú ✯➄➃ ✤✏ú✔û✲ú✦✟❤✂❘ý➠ý✒÷☞û◗ú✢✂✌✟♥✄✟✙ÿ❺ù✁✝❙✟✴þ➛ú✦✟✤ú✢✘✚✖✛✂❘ù✱÷✣✝✏û ø✦✂✏û✷✥➑ú✬õ✁✟✙÷➓ù✱ù✱ú☎û●✂✏ÿ ➅t❯➆ ä ➇t❯➈ ❮➊➉♦Ö➋s❺å✗s t❯r õ✘ö✲ú✢✟✱ú ➅ ✂❘û◗þ ➇ ✡❣✂❹ü➌✥✎ú❛☎✞✖✲û✎ø➅ù✱÷✣✝✏û✎ÿ❉✝✌☎ Ö ✂❘û◗þ å ✪ ✥✆✖➛ù✬û✆✝✏ù❡✝✌☎♥t ✯✜➍☎✃õ❲ú→õ✁✟✱÷☞ù✱ú ×t ❮ t➆ ×Ö ä t➈ × å ✪ ù✱ö◗ú☎û ➅ ×t ä t❯➈✆♦➇ ×Ö★➎ ➅ × å r ❮➊➉×Ö➐➏ ➃ ý✣✝✏û✛✤★ù✙ö✲ú✬ý✒÷☞û✲ú✍ÿ❉♦✾ø✴ö✛✂❙✟❏✂✮ø➅ù✙ú✦✟✙÷✒ÿ❺ù✱÷↕ø☎ÿ❤r➆ÿ☛✖◗ø✴ö➹ù✱ö✄✂✁ù ➇ ×Ö★➎ ➅ × å ❮❅➑ ✪ ➅ ×t ×Ö ❮❅➉ ♦➓➒✺①✺③❁r ✯ ➔ö✛✂❙✟❏✂✮ø➅ù✱ú✢✟✱÷↕ÿ❺ù✱÷↕ø☎ÿ◗✂✌✟✙ú ➇ Ö❞➎ ➅ å ❮➊→ ✪ ☎✞✝❙✟▲✂❘û✠ü → ✪ ✂❘û◗þ✚ù✙ö✲ú❉✤✏ú✔û✲ú✦✟❤✂❘ý✣ÿ☛✝✏ý✣✖➛ù✱÷✣✝✏û✷✥✎ú✍ø✍✝✠✡✭ú✍ÿ t ❮↔➣ ➅ ↕ ➉ ×Ö ä❦➙ ♦→r ❮➛➣ ➅ ↕ ➉ ×Ö ä➜➙ ♦➇ Ö❞➎ ➅ å r ➏ ➝
Where g is an arbitrary function to be determined by the initial and boundary A linear second order partial differential equation can be written as AOxz Boxy Cpyy =F(, g,,%x,pu) Phere A, B and C may be functions of r and y. Based on the local value of the coefficients the equations are classified as follows B2-4AC>0 Hyperbolic B4-4AC=0 Parabolic B2-4AC n this case it is always possible to choose s, n so that a=c=0, 1.e
➞➠➟✆➡✢➢☛➡✺➤❞➥❵➦✁➧✌➨●➧✌➢❏➩✆➥✕➫❏➢❏➧❙➢☛➭✜➯✞➲✛➨✛➳✵➫❏➥✑➵✠➨❢➫❏➵✜➩✄➡❡➸✓➡✦➫☛➡✢➢☛➺✜➥✣➨✆➡✢➸❢➩✏➭❣➫☛➟✆➡❉➥✣➨✆➥✑➫☛➥❵➧✌➻➋➧❙➨✛➸❢➩✄➵✠➲✆➨✛➸✆➧❙➢☛➭ ➳✍➵✠➨✛➸✓➥✑➫☛➥✣➵❙➨✛➦✢➼ ➽➋➾✚➚✠➪✄➶✈➹❅➘✜➴❜➹✈➾✚➴★➷❉➬❢➮❬➱✑✃ ❐ ➻✣➥✑➨✆➡✙➧✌➢▲➦☛➡✢➳✦➵❙➨✛➸❢➵❙➢❤➸✓➡✦➢♥❒✛➧❙➢▼➫❏➥✣➧❙➻✫➸✓➥✕❮✗➡✦➢❏➡✦➨✚➫❏➥✣➧❙➻✮➡✢❰✚➲✛➧✌➫☛➥✣➵❙➨●➳✦➧✌➨✷➩❯➡❉Ï✁➢☛➥✑➫▼➫❏➡✦➨❘➧❙➦ Ð❴Ñ❯Ò✢Ò✺Ó➐Ô❛Ñ✄Ò✙Õ✁Ó×Ö❉Ñ❯Õ✍Õ❡Ø➠Ù❣Ú❱Û➋Ü☛Ý❯Ü❤Ñ❳Ü❏Ñ❯Ò✛Ü❤Ñ❯Õ✌Þ Ï✁➟✆➡✢➢☛➡ Ð❧ß✓Ô ➧✌➨✄➸ Ö ➺❣➧❂➭❬➩❯➡❥➯✞➲✛➨✛➳✵➫❏➥✑➵✠➨✛➦②➵❙➯ Û ➧❙➨✛➸ Ý ➼❭à◗➧✠➦▼➡✙➸✜➵❙➨✜➫❏➟✆➡❴➻✣➵✓➳✦➧❙➻✛á❜➧✌➻✣➲✆➡❥➵❙➯✫➫☛➟✆➡ ➳✍➵✏➡✦â❣➳✦➥✑➡✢➨✚➫❏➦◗➫☛➟✛➡❉➡✢❰✚➲✛➧❜➫❏➥✑➵✠➨✛➦✁➧✌➢❏➡❪➳✍➻❵➧❙➦❏➦▼➥✑ã✛➡✢➸✷➧❙➦◗➯✞➵✠➻✑➻✣➵❜Ï▲➦✢ä Ô❧å✁æ✲ç✠Ð✺Öéèëê ì●í✫î ➾✠➴❂ï②➪❯ð❱ñ✼➚ Ôå æ✲ç✠Ð✺ÖòØ➊ê ➷❴ó✓➴❜ó✓ï②➪❯ð❱ñ✼➚ Ôå æ✲ç✠Ð✺Öéôëê ➮✺ð❱ð✼ñî✈õ ñ✼➚ ö▲➵❙➫☛➡★➫☛➟✛➧✌➫❛➧✌➨×➡✢❰✚➲✛➧❜➫❏➥✑➵✠➨➐➺❣➧❂➭✲➳❤➟✄➧✌➨✆÷✠➡❣➫◆➭✏❒✄➡❢➯✞➢☛➵✠➺ø➵✠➨✆➡★❒❯➵❙➥✣➨✚➫❧➫❏➵✲➧❙➨✆➵✌➫❏➟✆➡✦➢❛➦▼➥✣➨✛➳✍➡★➫☛➟✆➡ ➳✍➵✏➡✦â❣➳✦➥✑➡✢➨✚➫❏➦❴➺✜➧❂➭✷➩❯➡❧➯✞➲✆➨✄➳✵➫☛➥✣➵❙➨✄➦❥➵❙➯ Û ➧❙➨✛➸ Ý ➼❥➞✲➡❛Ï✁➥✑➻✣➻➋➫◆➭✏❒✆➥❵➳✦➧❙➻✑➻✣➭❘➧❙➦❏➦▼➲✛➺❛➡❧➫☛➟✛➧✌➫ ß Ï✁➟✆➡✢➨ Ï◗➡❴➦❏➧❂➭❬➫❏➟✛➧❜➫✁➧✌➨❢➡✢❰✚➲✛➧✌➫☛➥✣➵❙➨❞➥❵➦◗➵✌➯❳➧❬÷❙➥✣á❙➡✢➨❣➫◆➭✏❒✄➡ ß ➥✑➫✁➢☛➡✢➺❣➧✌➥✣➨✛➦②➵❙➯✮➫☛➟✛➡✺➦❏➧✌➺✜➡▲➫◆➭✏❒❯➡❪➵❜á❙➡✦➢②➫☛➟✆➡ Ï✁➟✆➵✠➻✑➡❉➸✓➵✠➺✜➧❙➥✑➨➋➼ ù➵❙➨✄➦▼➥❵➸✓➡✦➢❴➧❘ú❂û✌üþý❱ÿ❬➳❤➟✛➧✌➨✆÷✠➡❉➵✌➯❭➥✣➨✛➸✓➡✦❒❯➡✦➨✄➸✓➡✦➨✚➫❴á❜➧❙➢☛➥❵➧✌➩✆➻✣➡✢➦✁ Ø Ú✞Û❳Ü▼Ý✆Þ✵ß✄✂★Ø☎✂✫Ú✞Û❳Ü▼Ý✆Þ✵ß ➦▼➲✄➳❤➟ ➫☛➟✄➧❜➫ ✆ Ø✞✝ Ò Õ ✂Ò ✂Õ✠✟ Ü ✡ ✆ ✡☞☛Ø➊ê✍✌ ✎♥➟✆➡✢➨ ß ÑÒ Ø Ñ☞✏ Ò Ó Ñ☞✑✒✂Ò ÑÒ✢Ò Ø Ñ☞✏✓✏ Òå Ó✕✔❙Ñ☞✏✖✑ Ò ✂Ò Ó Ñ☞✑✗✑✘✂Òå Ó Ñ✙✏ Ò✙Ò Ó×Ñ☞✑✘✂Ò✙Ò ➼ ➼ ➼ ✎♥➟✆➡✺➫☛➢❤➧✌➨✄➦◆➯✞➵✠➢☛➺✜➡✢➸❢➡✢❰✚➲✛➧✌➫☛➥✣➵❙➨✷➩✄➡✙➳✍➵✠➺❛➡✙➦ ✚Ñ✏✖✏ Ó✜✛✍Ñ✏✖✑ Ó✠✢✢Ñ✑✣✑ Ø✥✤❖Ú Ü✦✂✗Ü❏Ñ❳Ü❤Ñ✏ Ü❤Ñ✑ Þ Ï✁➥✑➫☛➟ ✚ Ø Ð Òå Ó➐Ô Ò Õ Ó×Ö Õå ✛ Ø ✔❙Ð Ò ✂Ò Ó×Ô Ú Ò ✂Õ Ó Õ ✂Ò Þ❳Ó✜✔✠Ö Õ ✂Õ ✢ Ø Ð✧✂Òå Ó×Ô★✂Ò ✂Õ Ó×Ö✩✂Õå ✪✬✫✮✭✍✯✱✰✲✭✴✳✶✵✷✎♥➟✆➥❵➦◗➳✍➻❵➧❙➦❏➦▼➥✑ã✄➳✦➧✌➫☛➥✣➵❙➨❞➥❵➦❁➥✑➨✏á❜➧✌➢❏➥✣➧❙➨✚➫❁➲✆➨✛➸✆➡✦➢◗á❂➧❙➻✑➥❵➸❞➨✛➵❙➨✹✸❃➦▼➥✣➨✆÷✠➲✆➻✣➧❙➢ ➫❏➢❏➧❙➨✛➦◆➯✞➵✠➢☛➺❣➧✺✸ ➫☛➥✣➵❙➨✄➦✦➼ ✻✽✼✿✾❀✾❂❁✗❃❅❄✆➢❏➵❙➺ ➧✌➩❯➵❜á❙➡ ✛å æ✲ç✚✢♥Ø➄Ú❱Ôå æ✲ç✠Ð❴Ö❉Þ❪Ú Ò ✂Õ æ Õ ✂Ò Þ å Ø➄Ú❱Ôå æ✲ç✠Ð❴Ö❉Þ✬✡ ✆ ✡ å ➼ ❆✶❇❉❈❻➘❊❈●❋❍❆✶❇❉■✧❏◗➘❊❑❉▲☞➽ ▼❖◆◗P✷❘❅❙▲à❯❚❲❱✮❳ù ➳✢➧❙➦☛➡ Ú✼Ô❧å✁æ✲ç✠Ð✺Öéè ê✠Þ ä ❳❃➨✷➫☛➟✆➥❵➦✁➳✦➧✠➦▼➡❪➥✑➫▲➥✣➦❥➧✌➻✣Ï◗➧❂➭✓➦❁❒✄➵✚➦☛➦☛➥✑➩✛➻✑➡❪➫❏➵✜➳❤➟✛➵✚➵✚➦▼➡✬ ß✙✂ ➦☛➵❬➫❏➟✛➧❜➫ ✚ Ø❨✢▲Ø➊ê✆ß ➥➓➼ ➡✠➼ ç
+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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2.2 Parabolic Equations Heat Equation at 2u+f in Q “ Smooth” solutions even when the initial, boundary conditions, or f, are not smooth. The domain of dependence of u (r, g, T)is(, y, t<T) In this course we will address the solution of parabolic equations using Finite Difference and Finite Element Methods 2.3 Hyperbolic Equations SLIDE 5 Wave Equation(First order) U·Vu=fin9 t Non-smooth solutions The domain of dependence of u(a, T)is(c(t),t<T) U·Vu=fing Characteristics are streamlines of U, e.g. =U The domain of dependence of u(a)is(ac(s),8<0 We will present Finite Difference and Finite Volume Methods for solving hyper bolic equations. In particular, Finite volume Methods will be extended to deal th non-linear hyperbolic equations
æ♣ç✷æ è➣é✘ê✍é✘ë➍ì✇í❷î✹ï④ð✄ñ✲ò②é✴ó✞î❷ì✇ô②õ ö✯÷✔ø❨ù❢ú❱û ü✄ý✆þ✞ÿ✁✄✂✆☎✴þ✞ÿ✞✝✠✟☛✡ ☞☛✌☞✎✍✑✏✓✒☛✔✖✕ ✌✖✗✙✘ ✚✜✛✣✢ ✤✦✥★✧✪✩✬✫✪✫✮✭✰✯✎✱✖✲✰✫✴✳✶✵✷✭ ✚✫✛ ✲ ✸✺✹✼✻✽✹✿✾❁❀❃❂❄✹✼✾❆❅❇❂❄✹❉❈❊✾❋❈❊❅●❈✠❍✽■❑❏▼▲★◆✮❖✷✾☛P✴❍✮◗❙❘❯❚★◆✽✾❱P✽❈❊❅●❈✠◆✽✾❄❲★❏▼◆✮◗ ✘ ❏❳❍✽◗✰✹❉✾☛◆✮❅❨❲❙❩✁◆❬◆✮❅❊❂❋❭ ✤❫❪❴✯❋❵❜❛✷✫✴✩✁❝✚✶✛ ✫✮❞❡❛✷❵✼❢☛❵✛ ❛❄❵✛❋❣❵❤✫✮❞ ✌❥✐❇❦❃❧♥♠❱❧♥♦q♣r✚ ✲ ✐✠❦s❧♥♠❱❧✰✍❳t✙♦✄♣ ✉✾✙❅❊❂✷❈✈❲❯❚★◆✽❖✷◗★❲✼✹✇❀r✹❯❀❡❈❊■❊■①❍✴P②P✽◗✰✹✿❲★❲③❅❊❂❋✹③❲✿◆✮■④❖✷❅●❈✠◆✽✾⑤◆♥⑥⑧⑦❋❍✽◗✰❍②▲★◆✽■④❈✠❚⑨✹★⑩✼❖❋❍✽❅●❈✠◆✽✾❄❲✇❖✪❲❙❈❊✾✪❶❸❷r❈❊✾❋❈❊❅❹✹ ❺ ❈❻❼✹✿◗❽✹✿✾❱❚❽✹❾❍✽✾❱P✖❷r❈❊✾❋❈❊❅❹✹⑧❿▼■✜✹✼❩✬✹✼✾❋❅❳➀❆✹✿❅❇❂❄◆✞P➁❲✼❭ æ♣ç➃➂ ➄✓➅❴➆❤➇❢ê✍ë➍ì✇í❷î✹ï④ð✄ñ♣ò⑥é✴ó✍î✽ì✇ô✤õ ö✯÷✔ø❨ù❢ú➉➈ ➊❈þ②➋✶ý❁✄✂✆☎✴þ✞ÿ❬✝➃✟✎✡ ✐➃➌➍✚✜➎ ✲➏✭➐✫➎❛❄❵➎❙♣ ☞✎✌ ☞☛✍ ✗✙➑➓➒ ✔ ✌ ✏ ✘ ✚✶✛✑✢ ✤❫➔❼✫✛✷→ ✲✰✩✖✫✪✫✴✭✰✯✑✲♥✫②✳✜✵✷✭ ✚✫✛ ✲ ✤✙➣❳✯✎❝➎❝❣ ✭❽❵➎❽✚ ✲♥✭ ✚✈❣✲q↔ ↕★➙❡➛ ↕❽➜ ✏ ➑➉✐➙ ➛ ✐❇✍♥♣✰♣ ✤❫❪❴✯❋❵❜❛✷✫✴✩✁❝✚✶✛ ✫✮❞❡❛✷❵✼❢☛❵✛ ❛❄❵✛❋❣❵❤✫✮❞ ✌❥✐➙ ❧✰♦✄♣❳✚ ✲ ✐➙❡➛ ✐✠✍♥♣❙❧✰✍▼t✙♦✄♣ ö✯÷✔ø❨ù❢ú➉➝ ➞✟✎✡❱➋✯ý✪➟✢ÿ❬✝➃✟✎✡➠q✂✆☎✴þ✞ÿ❬✝➃✟✎✡ ➑➡➒ ✔ ✌ ✏ ✘ ✚✶✛✣✢ ✤❫➔❼✫✛✷→ ✲✰✩✖✫✪✫✴✭✰✯✑✲♥✫②✳✜✵✷✭ ✚✫✛ ✲ ✤✙➣❳✯✎❝➎❝❣ ✭❽❵➎❽✚ ✲♥✭ ✚✈❣✲❴❝➎❵❤✲♥✭ ➎❵❬❝✮✩✬✳ ✚✜✛❵❬✲▼✫✮❞ ➑❁➢ ❵②➤ ➥✎➤ ↕★➙❡➛ ↕❙➦ ✏ ➑ ✤❫❪❴✯❋❵❜❛✷✫✴✩✁❝✚✶✛ ✫✮❞❡❛✷❵✼❢☛❵✛ ❛❄❵✛❋❣❵❤✫✮❞ ✌❥✐➙ ♣▼✚ ✲ ✐➙ ➛ ✐➃➧➁♣✿❧❽➧❉t✙➨②♣ ➩✹❼❀❡❈❊■❊■✴⑦☛◗❽✹❙❲✼✹✼✾❋❅s❷r❈❊✾❋❈❊❅❹✹ ❺ ❈❻❼✹✼◗✰✹✼✾☛❚★✹⑧❍✽✾❱Pq❷r❈❊✾❋❈❊❅❹✹❸➫✎◆✮■④❖✷❩✬✹❼➀❆✹✼❅❊❂❋◆❬P✽❲❃⑥✿◆✽◗❼❲✿◆✮■④✻✞❈❊✾➭❶❜❂✪❘❽⑦❋✹✼◗❙➯ ▲★◆✽■④❈✠❚❸✹★⑩✿❖❋❍✮❅➃❈✠◆✮✾❄❲✼❭ ✉✾❸⑦❋❍✽◗❙❅●❈✠❚✼❖✷■✜❍✽◗★❏❴❷r❈❊✾✎❈❊❅➲✹❫➫✎◆✽■④❖✷❩✁✹✬➀❆✹✼❅❊❂❋◆❬P✽❲✖❀❡❈❊■❊■▼▲❽✹❯✹✰➳②❅❹✹✿✾❱P✴✹★P❸❅❹◆✣P✴✹★❍✽■ ❀❡❈❊❅❇❂✑✾☛◆✮✾❋➯❹■④❈❊✾❱✹★❍✽◗✄❂✪❘❽⑦❋✹✼◗✼▲❽◆✮■④❈✠❚✁✹★⑩✿❖❋❍✮❅➃❈✠◆✮✾❄❲✼❭ ➵
2.4 Eigenvalue Problem Find non-trivial pairs(u, A) 0 in with homogeneous conditions on T From the mathematical clas sification point of view, the eigenvalue equation is a semi linear elliptic equation We shall see below that the eigenvalue problem of a given spatial operator is closely related to the temporal evolution of the solution of the associated time dependent equation. The particular eigenvalue problem shown here is close elated to the Heat equation 2.5 One Spatial variable In some cases we wil consider the above equations involuing only one spatial ariable Unknown Equation a(a 2=f u(a) 0 (u(x),入) W,z+ Au= 0 3 Fourier Analysis 3.1 Definition SLIDE 9 Let g(a)be an"arbitrary"periodic real function with period 2T
➸❴➺❇➻ ➼⑨➽➏➾❨➚❃➪r➶❥➹❡➘♥➴❼➚✓➷❆➬❋➮❨➱✄➘♥➚s✃ ❐☛❒➭❮❊❰sÏÑÐ Ò➍Ó✜Ô✎Õ❯Ô❋Ö✴Ô✷×●Ø✰Ù❽Ó✜Ú✪Ó✈Û✮Ü❱Ý❋Û✴Ó✜Ù★Þ❤ß❇à➍á❽â☛ã ä✎å✖æ à✬çèâ❋à❸é✓êëÓ✶Ôíì îÓ✜Ø✰ï⑨ð❃ñ☛ò❫ñ☛ó✎ô✪õ➍ô✪ñ✎ö➍÷❼ø✼Ö✴Ô❋Õ✷Ó✜Ø✰Ó✶Ö✴Ô✎Þ❴Ö✴Ôúù ûýü✰þ✽ÿ✁✄✂✆☎✣ÿ✞✝✟✠✂✡☎✼ÿ☛✝☞✍✌✏✎✑✝☞✒✓✎✔✒✕✝✗✖✑✖✘✌✙✚✎✑✝✟✛✌✠þ✟✜✣✢❋þ✟✌✄✜✆ þ✥✤✧✦★✌✏☎✔✩✫✪✓✠✂✡☎✬☎✔✌✮✭✯☎✔✜✆✦✟✝✟✒✱✰✆☎✲☎✑✳✔✰✆✝✟✛✌✠þ✟✜✴✌✵✖✧✝✶✖✔☎✿ÿ✷✌✄✸ ✒✱✌✄✜✹☎✺✝✮ü✻☎✼✒✄✒✮✌✢✽✛✌✏✎✞☎✺✳✔✰✆✝✟✛✌✠þ✟✜✹✾ ✿❁❀Ö✴Ô✷×➲Ü✶Ó✜Ô✡❂✞Û✮Ù ✿❄❃✘❅Ü✜Ö➭Þ❆❂❬Ü✕❇❉❈❾Ù❊❂❬Ü✶Û✮Ø❊❂✞Õ✇Ø❽Ö✖Ö✴Ø✰ï✡❂❬Ù①Ý❄Ù❽Ö●❋❄Ü❍❂✔■✁Þ ❏❑☎✬✖❊✂✡✝✟✒✄✒▲✖✼☎✑☎✶▼✑☎✔✒✜þ✟✩◆✠✂✡✝☞❖✄✂✆☎✶☎✔✌✮✭✯☎✔✜✆✦✟✝✟✒✱✰✆☎✞✢☛ü❽þ●▼✔✒✕☎✿ÿ þ✥✤❑✝❑✭●✌✄✦✗☎✔✜P✖✛✢✡✝☞✍✌✏✝☞✒❼þ✺✢✆☎✿ü✺✝✟❹þ✽ü❑✌✵✖ ✎✔✒✜þ✗✖✔☎✔✒✱◗✑ü❊☎✼✒❍✝✟❘☎✺❙❑➲þ❚✠✂✡☎✞❘☎✼ÿ▲✢❋þ✽ü❊✝☞✒❯☎✼✦✽þ✟✒✱✰❉✛✌✠þ✟✜⑤þ✥✤✷✄✂✆☎✷✖✼þ✟✒✱✰❉✛✌✠þ✟✜èþ❆✤✷✄✂✆☎✧✝✗✖✑✖✼þ★✎✔✌✏✝✟❘☎✑❙✲✛✌❊ÿ☛☎✔✸ ❙●☎❘✢✆☎✼✜✹❙●☎✔✜✆✷☎✑✳✔✰✆✝✟✛✌✠þ✟✜✹✾❲❱✽✂✆☎✷✢✆✝✽ü✘✛✌✏✎✼✰❉✒❍✝✽ü❳☎✔✌✮✭✯☎✔✜✆✦✟✝✟✒✱✰✆☎☛✢☛ü✰þ✯▼✼✒❍☎✿ÿ✁✖❊✂❄þ✟✩❨✜❩✂✡☎✼ü❊☎❑✌✵✖❑✎✼✒✜þ✟✖✔☎✼✒✱◗ ü❊☎✔✒✕✝✟❘☎✑❙✞❹þ✞✠✂✡☎✓❬✻☎✑✝✟❨❭❪✳✔✰✆✝✟✛✌✠þ✟✜✽✾ ➸❴➺✍❫ ❴è➪➐➚❛❵❝❜✄➹❡❞✷➽♥➹❡➘✻❢❁➹➍➬✎➽➏➹❥➱✄➘♥➚ ❣✘✜✣✖✼þ✽ÿ✞☎❑✎✺✝✟✖✔☎✘✖✧✩❤☎✬✩❨✌✄✒✄✒❪✎★þ✟✜✡✖✘✌✏❙✯☎✿ü❚✄✂✆☎✬✝✯▼★þ✟✦✟☎❑☎✺✳✔✰✆✝✟✛✌✠þ✟✜✆✖✧✌✄✜✆✦✽þ✟✒✱✦✐✌✄✜❥✭èþ✟✜✆✒✮◗❁þ✟✜✹☎❚✖✍✢✆✝✟✛✌✏✝✟✒ ✦✟✝✽ü✘✌✏✝●▼✔✒✕☎✐✾ ❐☛❒➭❮❊❰sÏ❧❦ ♠➐Ô✆♥➭Ô❋ÖîÔ ♦❤♣❥q❋Û✮Ø✰Ó✶Ö✴Ô à➍ß✏r☛ãts ✉①à✽✈✐✈❤é①✇ à➍ß✏r☛ãts ②qà✹✈❤é äà✹✈✐✈ à❥ß✠r❃á❆③♥ãts à⑤④❥é äà✽✈✐✈ à❥ß✠r❃á❆③♥ãts à⑤④✆ç⑥②⑧à✽✈❜é➠ê ß❇à❥ß✠r❱ã❙á❽â☛ã⑦s à✽✈✐✈➐çèâ❋à❸é✓ê ⑧ ⑨✲⑩❷❶✲❸❯❹✔❺▲❸❼❻❾❽✲❿❖➀✼➁➃➂✫❹✼➂ ➄❴➺❆➅ ➆➠➚➈➇⑧➪❼➽✥❞✷➽♥➮r➪ ❐☛❒➭❮❊❰sÏ❧➉ ➊➋❂✼Ø➍➌❱ß✏r☛ã➍❋✽❂❜Û✮Ô ❃Û✴Ù❊❋❄Ó✜Ø✰Ù★Û✮Ù✺❇❉❈❜Ý✽❂✼Ù❽Ó✜Ö✷Õ✷Ó✈øqÙ❊❂✞Û✮Ü✹➎✠q❄Ô✎ø❙Ø✰Ó✶Ö✴Ô îÓ✜Ø✰ï✑Ý✽❂✼Ù❽Ó✜Ö✷Õ❑➏✟➐ ❀➒➑ ➓
g(a)=∑ (k integer) d = 2T Skk,(orthogonality We recall that Skk, is the Kronocker symbol and is equal to 1 if k= kpand a otherwiser being the above relations hipl the coei cients gk can be computed rectly g(a 一ik逻 c ote that when g(a)is reall gk=gH l where u denotes complet corrugater He ha fe a tari 6(c)] dc <no aldp note that the de p"potion precented aboe van me written in an valent for“ 0)=1中 with k=9k I g-k and uk.=e(gk-9-k)
➔➈→✠➣✹↔❯↕ ➛➙ ➜✼➝✫➞ ➙ ➔➜❷➟✗➠ ➜✼➡ →✍➢❚➤✕➥❥➦✺➧✔➨●➧★➩✑↔➭➫ ➯✹➲✡➳✓➵✼➸✟➺✄➺❍➸✟➻❨➼✄➽❥➾✣➸✟➚✘➪✠➲✡➸✺➾✯➸☞➽✽➶✟➺✮➼✄➪✍➹❧➚❊➳✔➺✕➶✟➪✛➼✏➸✟➽✆➘❆➲❉➼➴P➘✼➶☞➪✍➼✵➘✠➷❝➳✺➬❧➮✔➹✶➪✄➲✆➳✞➱✃➸☞❐❉➚✘➼✏➳✼➚✧❒✞➸✐➬●➳✼➘❚❮✺➶☞➽❩➮✑➳ ➳✑➶✗➘✘➼✄➺✱➹✧❰✟➳✔➚✘➼➷✚➳✑➬ Ï❁Ð✺Ñ Ò ➟ ➠ ➜✘➡ ➟ ➞ ➠ ➜✼Ó❍➡☛Ô➣✬↕ÖÕ☞×❧Ø➜✼➜ Ó →✏Ù●➩❊➦❊Ú✡Ù✯➨●Ù✯➥✆Û☞Ü❍➤✮➦✥Ý✡↔ Þ❑➳✲➚❊➳✑❮✑➶✟➺✄➺▲➪✠➲✡➶☞➪ Ø➜✼➜ Ó ➼✵➘✬➪✠➲✡➳❚ß❷➚❊➸✟➽✹➸★❮✑à●➳✔➚✬➘✘➹✟❒☛➮✑➸✟➺á➶☞➽✽➬â➼✵➘✶➳✑ã✼❐✆➶☞➺❪➪ä➸❧åæ➼➵ ➢P↕ç➢❉è ➶☞➽✽➬ é ➸✟➪✠➲✡➳✔➚✘➻❨➼✵➘✔➳★êÖë✃➘✘➼✄➽ì➾❑➪✠➲✡➳➃➶✯➮✑➸✟❰✟➳➃➚✺➳✼➺✕➶☞➪✍➼✏➸☞➽✡➘❊➲ì➼➴✽í➒➪✠➲✡➳❚❮✺➸✐➳✍î✴❮✔➼✏➳✼➽⑤➪✏➘ ➔➜ ❮✑➶✟➽✴➮✺➳❚❮✑➸✟❒á➴✽❐❉➪❘➳✺➬ ➬✟➼✄➚✺➳✺❮✔➪✛➺✱➹➃➶✗➘ ➔➜ ↕ å Õ✟× Ï✣Ð❊Ñ Ò ➔➈→✠➣✹↔ ➟ ➞ ➠ ➜✘➡ Ô➣ ï✻➸☞➪ä➳❖➪✄➲✆➶✟➪❝➻ð➲✡➳✔➽ ➔➈→✠➣✹↔ ➼✵➘ñ➚❊➳✑➶✟➺✱í ➔➜ ↕❩➔✡ò➞✹➜ í❯➻ð➲✆➳✼➚✺➳ ò ➬●➳✔➽✽➸☞➪ä➳✼➘✻❮✺➸☞❒▲➴✹➺✕➳✺ó❑❮✺➸☞➽✔ô✘❐●➾❥➶✟➪❘➳★ê õ❧ö➈÷✘ø❑ù ú❨ö➋û✆ü✗ý❘ø☞ü✧þ●ø☞ü✗ý❘ø☞þ ÿÙ✁✡➩✺➤✕➧★➩✄✂❊➧✔➩✺➤✕➧☎✂ðÛ●➩❊➧❯Ù✯➥✡Ü❍Ý✝✆✡➧✟✞✆➥✡➧☎✆✡✠✠Ù✯➩☛✠☞✡➥✍✌✘➦❊➤❍Ù●➥✍✂☛✎❪Ú✡➤✏✌✑Ú✑✂✺Û✟➦✺➤✒✂✓✠✠Ý✓➦❊Ú✡➧✔✠✠Ù●Ü❍Ü❍Ù✕✎❪➤✕➥✡➨á➩❊➧★➨✖✡Ü✵Û☞➩✺➤✮➦✥Ý ✌✼Ù✯➥✗✆❉➤✕➦❊➤❍Ù●➥ Ï❁Ð✺Ñ Ò ✘ ➔➈→✠➣✹↔ ✘ Ð Ô➣✚✙✜✛✜✢ ✣➧ Û☞Ü✏✂❊Ù✣➥✡Ù●➦❊➧✶➦✺Ú✆Û✟➦❚➦❊Ú✡➧✤✆❉➧☎✌✼Ù✖✥✧✦✽Ù✁✂❊➤✮➦✺➤✕Ù✯➥✜✦✡➩✺➧★✂❊➧✔➥❥➦❊➧☎✆ÖÛ✪✩✽Ù✕✫●➧✬✌✔Û☞➥✭✩⑤➧✬✎❪➩❊➤✕➦❆➦✺➧✔➥①➤❍➥①Û●➥ ➧★✮✯✡➤✒✫✟Û☞Ü❍➧✔➥❥➦✰✠✠Ù●➩✱✥ Û✖✂ ➔➈→✠➣✹↔❯↕ ➛➙ ➜✼➝Ò ✍✲➜ ✌✼Ù✯✂✆➢ì➣✴✳✶✵➜ ✂❊➤✕➥á➢ì➣✄✷ ✎❪➤✕➦❊Ú ✲ ➜ ↕❩➔➜ ✳â➔ ➞➈➜ Û●➥✗✆✸✵➜ ↕✜✹✘→✄➔➜✻✺ ➔ ➞✹➜ ↔✘➫ ✼
