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 <0 Ellipti Note that an equation may change type from one point to another since the oefficients may be functions of a and y. We will typically assume that, when we say that an equation is of a given type, it remains of the same type over the Consider a valid change of independent variables s=S(a, 0), n=n, y), such J≠0 pzz= p<s 52+2p<n S272+om n2+p< Sra+pnnz The transformed equation becomes AS2+BSSy+C b 2A Szz B(Szny+ Sy n2)+2C Syny An2+B n2y+C, THEOREM: This classification is invariant under valid non-singular transfor Proof: From above b2-4ac=(B2-4AC)(Gnw-Sw72)2=(B2-4AC)J12 CANONICAL FORMS HYPERBOLIC case(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