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 ➝