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