Temperature distribution Let u(a) and q(a) denote the temperature and heat flux in a homogeneous heat conducting bar of unit length. The bar is subjected to a distributed volumetric heat source p(a)and the temperature is maint ained at zero at the end points; the sides of the bar are assumed insulated so that the heat flow is one-dimensional cT(e) Acp da (q dq The stationary temperature distribution can be obt ained by considering the energy balance Ac(q+dq)-Acq= Acp d and the empirical relation between the temperature and the heat flux In the above equations, k is the heat conductivity and Ac is the bar cross ectional area. Defining f=p/k and eliminating q from the above equations 1. 1.1 Solution Properti SLIDE 2 The solution u(a) always exists ·u(x) is always“ smoother"than the data f(x) (see first lecture). In particular, if f has m continuous derivatives, u will have n+2 continuous derivatives. Thus, if Eco, thenuEC2 ·If(x)≥0 for all ax, then a(x)≥0 for all az Follows from the positivity of Greens function us≤(1/8川fls Given f(a) the solution u(a)is unique Continuous stability estimate ecall that for a function u: Q-R➝r➞✾➟✁➠➢➡ ➤③➠✬➥r➦✉➠✬➧✹➨✾➟❂➩❤➧P➠✏➫❇➭❾➯P➟☞➧❱➭☞➲✱➩④➟✿➭⑤➞✈➳ ➵✛➸✢➺♦➻❦➼★➽❇➾③➚✬➪❤➶✐➹✡➼★➽❇➾❝➶✡➸❂➪✓➘✬➺❁➸❍➺☞➴✮➸✤➺☞➸❂➷●➬❤➸❂➮❁➚✬➺☞➱✓➮❁➸✝➚❃➪✮➶●➴✓➸❂➚✬➺❝✃✮➱✓❐●❒❆➪✰➚✷➴✮➘❃➷●➘❃❮P➸✢➪✓➸❂➘❃➱✮❰✦➴✮➸❂➚✹➺ Ï➘P➪✮➶✡➱Ï➺☞❒❆➪✓❮✗Ð✮➚✬➮✴➘✬Ñ❏➱✮➪✓❒⑨➺❍Ò❆➸✢➪✓❮❃➺☞➴✛Ó③Ô♠➴✮➸✝Ð✮➚❃➮♦❒✻❰♠❰✿➱✮Ð✡Õ⑤➸Ï➺☞➸❂➶✰➺☞➘●➚❶➶✓❒❆❰✿➺☞➮❁❒◆Ð✮➱✡➺☞➸❱➶✰ÖP➘❃Ò❆➱✓➷●➸❊➺☞➮❁❒ Ï ➴✓➸❱➚✹➺④❰☞➘❃➱✓➮ Ï➸✾×✱➼★➽❇➾✛➚✬➪❤➶❫➺☞➴✓➸❝➺☞➸✢➷●➬✑➸✢➮❸➚✹➺❁➱✓➮☞➸✦❒✻❰✛➷❧➚❃❒◆➪❅➺❁➚❃❒◆➪✮➸❂➶✷➚✹➺④Ø✢➸✢➮❁➘✉➚✬➺✛➺☞➴✓➸❝➸✢➪✮➶✷➬✑➘❃❒❆➪❅➺❁❰❂Ù❩➺☞➴✓➸ ❰☞❒❆➶✡➸❱❰✴➘✬Ñ✛➺❁➴✓➸✝Ð❤➚✬➮❍➚✬➮❁➸✝➚P❰☞❰☞➱✓➷●➸❂➶✰❒◆➪✮❰☞➱✓Ò✻➚✹➺❁➸❂➶✰❰☞➘⑥➺❁➴✮➚✹➺♠➺❁➴✓➸✵➴✓➸❂➚✬➺♦✃✮➘✹Ú➙❒✻❰✴➘❃➪✓➸✢ÛÜ➶✓❒◆➷●➸✢➪❤❰✿❒❆➘❃➪✮➚❃Ò❾Ó Ô♠➴✓➸➢❰✿➺❁➚✬➺☞❒❆➘❃➪✮➚❃➮☞ÝÞ➺☞➸✢➷●➬✑➸✢➮❸➚✹➺❁➱✓➮☞➸❢➶✓❒❆❰✿➺☞➮❁❒◆Ð✮➱✡➺☞❒❆➘❃➪ Ï➚✬➪➙Ð✑➸❢➘❃Ð✡➺❸➚✬❒❆➪✓➸❂➶↕Ð✈Ý Ï➘❃➪❤❰✿❒✻➶✡➸✢➮❁❒❆➪✓❮❼➺☞➴✓➸ ➸✢➪✮➸✢➮❁❮❃Ý❧Ð✮➚❃Ò❆➚❃➪Ï➸ ß✝à ➼❾➹❍á❛â❅➹✬➾❦ã ß❫à ➹✵ä ß✝à×❢âP➽❏å ➚✬➪❤➶✐➺❁➴✓➸✷➸✢➷●➬✓❒❆➮☞❒ Ï➚❃Ò✾➮☞➸❂Ò❆➚✬➺☞❒❆➘❃➪✄Ð❤➸✢➺⑤Ú♦➸❂➸✢➪⑩➺☞➴✓➸✵➺❁➸✢➷●➬✑➸✢➮❸➚✹➺☞➱✮➮☞➸⑥➚❃➪✮➶✐➺❁➴✓➸✷➴✓➸❱➚✹➺♠✃✮➱✡❐ ➹✷äæã✉ç✈è✝éëê ➼✲ì✮➘❃➱✓➮❁❒◆➸❂➮❂í ❰✴Ò❆➚❩Ú✉➾ î✍➪q➺❁➴✓➸❼➚✬Ð✑➘✹Ö❃➸➢➸❱ï❅➱✮➚✹➺❁❒◆➘P➪✮❰✢ð❫ç➙❒✻❰✰➺❁➴✓➸r➴✓➸❂➚✬➺ Ï➘❃➪✮➶✡➱Ï➺☞❒❆Ö✈❒⑨➺⑤Ýq➚❃➪✮➶ ßà ❒❆❰⑩➺☞➴✮➸ñÐ✮➚❃➮ Ï➮❁➘P❰❁❰ ❰☞➸Ï➺❁❒◆➘P➪✮➚✬Ò③➚❃➮☞➸❱➚✓Ó❧ò✝➸❊ó✮➪✓❒❆➪✓❮➢ôõäö×❇÷❃ç➢➚❃➪✮➶r➸✢Ò❆❒◆➷●❒❆➪✮➚✹➺❁❒◆➪✓❮❢➹✰Ñ★➮❁➘❃➷ø➺❁➴✓➸✐➚✬Ð✑➘✹Ö❃➸●➸❂ï❅➱✮➚✬➺☞❒❆➘❃➪✮❰ Ú✴➸❫➘PÐ✡➺❁➚❃❒◆➪⑩➺☞➴✮➸⑥ù❦➘❃❒✻❰☞❰☞❒◆➘P➪✰➸❱ïP➱❤➚✹➺☞❒❆➘❃➪❏Ó ú✑û✲ú✑û✲ú ü❏ý❤þ❾ÿ✁￾✄✂✲ý✆☎✞✝✠✟❩ý☛✡✌☞✍✟✎￾✏✂✑☞✍✒ ✓ ✔✍✕✗✖✁✘✚✙ ✛ Ô♠➴✓➸⑥❰☞➘❃Ò❆➱✡➺☞❒❆➘❃➪✄➻✱➼✲➽✑➾♠➚❃Ò◆Ú♠➚❩Ý✡❰ ☞✍✜✢✂✑✒✣￾✄✒ ✛ ➻✱➼✲➽✑➾✴❒✻❰✤➚✬Ò❆Ú♠➚❩Ý✈❰✥✤✒✄✦Þý✛ý☛￾✏✧★☞✩✟✫✪ ➺❁➴✮➚✬➪⑩➺☞➴✮➸⑥➶✓➚✹➺❸➚●ô❦➼✲➽✑➾ ✬✮✭✣✯✰✯✲✱✴✳✵✭✷✶✹✸✺✯✵✻✼✶✮✽✾✳✿✯❁❀❃❂❅❄✷❆❈❇❊❉✫✳✷✶✮❋●✻✣✽✾✸✺❉✫✳✵❍✹❋■ ô❑❏ ❉✫✭▼▲◆✻✵❖P❆❊✶✮❋✗❆❊✽❊❖✫✽◗✭✥❘✩✯✼✳✷❋✗❙✫❉✫✶✮❋✗❙✎✯✼✭✵❍ ➻❯❚❋✗✸✗✸ ❏ ❉✫❙✫✯ ▲ á❲❱ ✻✵❖✫❆☛✶✑❋✗❆☛✽❊❖✫✽◗✭❳❘❃✯✣✳✷❋✗❙✎❉P✶✑❋✗❙✫✯✷✭✣❂❩❨❏✽◗✭✰❍✲❋■ ô❭❬❫❪❵❴ ❍✴✶❏ ✯✣❆ ➻❛❬❜❪✢❝ ❂ ✛ îÜÑ✦ô❦➼✲➽✑➾❡❞❯❢♥Ñ★➘❃➮✤➚✬Ò❆Ò✛➽④ð✡➺☞➴✓➸❂➪❻➻✱➼✲➽✑➾❡❞❯❢✗Ñ★➘❃➮✤➚✬Ò❆Ò✾➽ ❣❖✫✸✗✸✺❖❚✭✌■✵✳✿❖✫❤✐✶❏ ✯❡❇❊❖✎✭✷❋✗✶✮❋✗❙✏❋✗✶✮❥❫❖❦■❈❧❡✳✿✯✵✯✼❆❵♠✭♥■✵✽✾❆✆✻✣✶✑❋●❖P❆✆❂ ✛❜♦✺♦ ➻ ♦✺♦ ♣rq ➼❁s✹÷✫t❅➾ ♦✉♦ ô ♦✉♦ ♣ ✈✠✇ ✛②①❒◆ÖP➸✢➪✏ô❦➼★➽❇➾✴➺☞➴✓➸⑥❰☞➘❃Ò❆➱✡➺❁❒◆➘P➪⑩➻❦➼★➽❇➾✴❒❆❰ ÿ★☎③✂✑④✾ÿ③☞ ✈t ➝r➞✾➟✁➠⑥⑤ ⑦❶➞✡➳✦➟✿➭❾➳③➩✾➞✛➩✮➯❧➯❅➟☞➨✛➲❤➭✑⑧✲➭✿➟●⑨➊➠✬➯❅➟✿➭❾➥❛➨✾➟✁➠ ⑩r➸✷➮☞➸ Ï➚✬Ò❆Ò✑➺☞➴✮➚✬➺❍Ñ★➘❃➮✤➚♥Ñ★➱✓➪Ï➺☞❒❆➘❃➪✏➻❛❶✩❷✞❸ î❹ ❺