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 equationsæ♣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❈✠◆✮✾❄❲✼❭ ➵