dt SLIDE 17 For t< 1 <t →x=t+x0→a(x,t)=1 For t<z<i: dt=1-to+x=(1-)t+ro For z>1 0→x=x0→ For t<a<l, we first solve for the characteristic lines. In this case they are defined by g=u is constant along each characteristic, we know that on each line dz uo. Once we have determined the characteristic lines. we use the fact that u is constant along each line to determine the overall solutio a(a, t) At t= l the solution develops a discontinuity. This corresponds to the time at which the characteristics first cross The procedure breaks down for t>1Ñ✺Ò✫Ó❆Ô✗Õ❉Ö✫× Ø✭Ù◆ÚÜÛ➫ÝßÞ Ø✭Ù◆Ú✌à➋á❉â✉ã äà ä â➋åçæ✑è à å â✿éêà❜ë è ì✎íà✁î❱â❱ï åðæ Ø✭Ù◆Ú✌â✦ñ❉àòñ æ ã äà ä â➋åçæ✝ó à❜ë è à åôí✕æ✌ó à❜ë♣ï➏â✗éêà❜ë ì✩íà✗î❚â❱ï åçæ✝ó à❜ë å æ✌ó à æ✌ó â Ø✭Ù◆Ú✌à➋õ æ ã äà ä â å❥ö÷è à å àë è ì✎íà✁î❱â❱ï åøö ù✁ú♣û✓â✉ñüàêñ æ✷ý❧þ❳ÿ✁￾û✄✂✆☎✝✂①ú✟✞✡✠ ÿ☞☛ú♣û✌☎✎✍ ÿ✑✏✍✓✒②û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✗✕ ✏ ✞✡✕✙✘ ÿ ✂✛✚✢✜✣✘✤☎✙✍✥✕✖✂ ✏✒✦✂ ÿ ☎✎✍ ÿ✛✧ ✒♣ûÿ ★ ÿ✗￾✘ ÿ★✪✩ ✧✬✫✄✭✫✯✮ å❥ì ✚✱✰✲✕✙✘✏❿ÿ✉ì ✕✖✂ ✏ú✳✘✓✂✆☎✴✒✳✘✵☎☞✒✟✞✄ú✳✘✷✶ ÿ✒ ✏✍ ✏✍✵✒♣û✔✒ ✏ ☎ÿ û✆✕✖✂✄☎✸✕ ✏❾ý❧þ❳ÿ✺✹✘✺úþ ☎✙✍✵✒✟☎ ú✟✘ ÿ✒ ✏✍✪✞✡✕✙✘ ÿ✻✫✄✭✫✯✮ å❥ìë✷✚✺✼✝✘✏❿ÿ➵þ❳ÿ ✍✓✒✟✠ ÿ ★ ÿ ☎ ÿ û✆✽✾✕✙✘ ÿ★ ☎✎✍ ÿ✢✏✍✵✒♣û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✸✕ ✏ ✞✿✕✙✘ÿ ✂ ýtþ❳ÿ✱❀✂ ÿ ☎✎✍ ÿ❁☛✒ ✏ ☎❁☎✎✍✓✒✳☎ ì ✕✖✂ ✏ú✳✘✓✂✆☎✴✒✳✘✵☎✝✒✟✞✄ú✳✘✷✶ ÿ✒ ✏✍❂✞✡✕✙✘ ÿ ☎➏ú ★ ÿ ☎ÿ û✆✽❃✕✙✘ÿ ☎✎✍ ÿ ú✟✠ ÿ û✯✒✟✞✙✞❄✂①ú✳✞❀ ☎✗✕✣ú✳✘✲✚ Ñ✺Ò✫Ó❆Ô✗Õ❉Ö❆❅ Ø✭Ù◆ÚÜÛ❈❇ Þ ì✩íà✗î❱â❱ï å ❉ æ à➋ñ æ ö à❋❊ æ ● ☎✩â å æ ☎✎✍ ÿ ✂①ú✟✞❀ ☎✸✕✣ú✟✘ ★ ÿ ✠ ÿ ✞✜ú✯❍■✂✑✒ ★✕✖✂ ✏ú✳✘✵☎✸✕✙✘❀ ✕✙☎✧ ✚❑❏▲✍✥✕✖✂ ✏ú♣û❡ûÿ ✂✸❍✭ú✳✘★✂✢☎ú✻☎✎✍ ÿ ☎✗✕✙✽ÿ ✒✟☎ þ ✍▼✕ ✏✍◆☎✙✍ ÿ✾✏✍✓✒②û✔✒ ✏ ☎ÿ û✆✕✖✂✆☎✗✕ ✏ ✂ ￾ û✄✂✄☎ ✏ û❚ú✟✂✄✂✣✚ ❖✝P✓◗❙❘✭Ú▲Ù✥❚✣◗❱❯✓❲✭Ú✔◗❈❳✭Ú✯◗❱❨❆❩▼❬✝❯❀Ù✟❭☞❪❑❫⑤Ù◆Ú✌â❁❊ æ ❴