2.4 Eigenvalue Problem Find non-trivial pairs(u, A) 0 in with homogeneous conditions on T From the mathematical clas sification point of view, the eigenvalue equation is a semi linear elliptic equation We shall see below that the eigenvalue problem of a given spatial operator is closely related to the temporal evolution of the solution of the associated time dependent equation. The particular eigenvalue problem shown here is close elated to the Heat equation 2.5 One Spatial variable In some cases we wil consider the above equations involuing only one spatial ariable Unknown Equation a(a 2=f u(a) 0 (u(x),入) W,z+ Au= 0 3 Fourier Analysis 3.1 Definition SLIDE 9 Let g(a)be an"arbitrary"periodic real function with period 2T➸❴➺❇➻ ➼⑨➽➏➾❨➚❃➪r➶❥➹❡➘♥➴❼➚✓➷❆➬❋➮❨➱✄➘♥➚s✃ ❐☛❒➭❮❊❰sÏÑÐ Ò➍Ó✜Ô✎Õ❯Ô❋Ö✴Ô✷×●Ø✰Ù❽Ó✜Ú✪Ó✈Û✮Ü❱Ý❋Û✴Ó✜Ù★Þ❤ß❇à➍á❽â☛ã ä✎å✖æ à✬çèâ❋à❸é✓êëÓ✶Ôíì îÓ✜Ø✰ï⑨ð❃ñ☛ò❫ñ☛ó✎ô✪õ➍ô✪ñ✎ö➍÷❼ø✼Ö✴Ô❋Õ✷Ó✜Ø✰Ó✶Ö✴Ô✎Þ❴Ö✴Ôúù ûýü✰þ✽ÿ✁￾✄✂✆☎✣ÿ✞✝✟￾✠✂✡☎✼ÿ☛✝☞￾✍✌✏✎✑✝☞✒✓✎✔✒✕✝✗✖✑✖✘✌✙✚✎✑✝✟￾✛✌✠þ✟✜✣✢❋þ✟✌✄✜✆￾ þ✥✤✧✦★✌✏☎✔✩✫✪✓￾✠✂✡☎✬☎✔✌✮✭✯☎✔✜✆✦✟✝✟✒✱✰✆☎✲☎✑✳✔✰✆✝✟￾✛✌✠þ✟✜✴✌✵✖✧✝✶✖✔☎✿ÿ✷✌✄✸ ✒✱✌✄✜✹☎✺✝✮ü✻☎✼✒✄✒✮✌✢✽￾✛✌✏✎✞☎✺✳✔✰✆✝✟￾✛✌✠þ✟✜✹✾ ✿❁❀Ö✴Ô✷×➲Ü✶Ó✜Ô✡❂✞Û✮Ù ✿❄❃✘❅Ü✜Ö➭Þ❆❂❬Ü✕❇❉❈❾Ù❊❂❬Ü✶Û✮Ø❊❂✞Õ✇Ø❽Ö✖Ö✴Ø✰ï✡❂❬Ù①Ý❄Ù❽Ö●❋❄Ü❍❂✔■✁Þ ❏❑☎✬✖❊✂✡✝✟✒✄✒▲✖✼☎✑☎✶▼✑☎✔✒✜þ✟✩◆￾✠✂✡✝☞￾❖￾✄✂✆☎✶☎✔✌✮✭✯☎✔✜✆✦✟✝✟✒✱✰✆☎✞✢☛ü❽þ●▼✔✒✕☎✿ÿ þ✥✤❑✝❑✭●✌✄✦✗☎✔✜P✖✛✢✡✝☞￾✍✌✏✝☞✒❼þ✺✢✆☎✿ü✺✝✟￾❹þ✽ü❑✌✵✖ ✎✔✒✜þ✗✖✔☎✔✒✱◗✑ü❊☎✼✒❍✝✟￾❘☎✺❙❑￾➲þ❚￾✠✂✡☎✞￾❘☎✼ÿ▲✢❋þ✽ü❊✝☞✒❯☎✼✦✽þ✟✒✱✰❉￾✛✌✠þ✟✜⑤þ✥✤✷￾✄✂✆☎✷✖✼þ✟✒✱✰❉￾✛✌✠þ✟✜èþ❆✤✷￾✄✂✆☎✧✝✗✖✑✖✼þ★✎✔✌✏✝✟￾❘☎✑❙✲￾✛✌❊ÿ☛☎✔✸ ❙●☎❘✢✆☎✼✜✹❙●☎✔✜✆￾✷☎✑✳✔✰✆✝✟￾✛✌✠þ✟✜✹✾❲❱✽✂✆☎✷✢✆✝✽ü✘￾✛✌✏✎✼✰❉✒❍✝✽ü❳☎✔✌✮✭✯☎✔✜✆✦✟✝✟✒✱✰✆☎☛✢☛ü✰þ✯▼✼✒❍☎✿ÿ✁✖❊✂❄þ✟✩❨✜❩✂✡☎✼ü❊☎❑✌✵✖❑✎✼✒✜þ✟✖✔☎✼✒✱◗ ü❊☎✔✒✕✝✟￾❘☎✑❙✞￾❹þ✞￾✠✂✡☎✓❬✻☎✑✝✟￾❨❭❪✳✔✰✆✝✟￾✛✌✠þ✟✜✽✾ ➸❴➺✍❫ ❴è➪➐➚❛❵❝❜✄➹❡❞✷➽♥➹❡➘✻❢❁➹➍➬✎➽➏➹❥➱✄➘♥➚ ❣✘✜✣✖✼þ✽ÿ✞☎❑✎✺✝✟✖✔☎✘✖✧✩❤☎✬✩❨✌✄✒✄✒❪✎★þ✟✜✡✖✘✌✏❙✯☎✿ü❚￾✄✂✆☎✬✝✯▼★þ✟✦✟☎❑☎✺✳✔✰✆✝✟￾✛✌✠þ✟✜✆✖✧✌✄✜✆✦✽þ✟✒✱✦✐✌✄✜❥✭èþ✟✜✆✒✮◗❁þ✟✜✹☎❚✖✍✢✆✝✟￾✛✌✏✝✟✒ ✦✟✝✽ü✘✌✏✝●▼✔✒✕☎✐✾ ❐☛❒➭❮❊❰sÏ❧❦ ♠➐Ô✆♥➭Ô❋ÖîÔ ♦❤♣❥q❋Û✮Ø✰Ó✶Ö✴Ô à➍ß✏r☛ãts ✉①à✽✈✐✈❤é①✇ à➍ß✏r☛ãts ②qà✹✈❤é äà✹✈✐✈ à❥ß✠r❃á❆③♥ãts à⑤④❥é äà✽✈✐✈ à❥ß✠r❃á❆③♥ãts à⑤④✆ç⑥②⑧à✽✈❜é➠ê ß❇à❥ß✠r❱ã❙á❽â☛ã⑦s à✽✈✐✈➐çèâ❋à❸é✓ê ⑧ ⑨✲⑩❷❶✲❸❯❹✔❺▲❸❼❻❾❽✲❿❖➀✼➁➃➂✫❹✼➂ ➄❴➺❆➅ ➆➠➚➈➇⑧➪❼➽✥❞✷➽♥➮r➪ ❐☛❒➭❮❊❰sÏ❧➉ ➊➋❂✼Ø➍➌❱ß✏r☛ã➍❋✽❂❜Û✮Ô ❃Û✴Ù❊❋❄Ó✜Ø✰Ù★Û✮Ù✺❇❉❈❜Ý✽❂✼Ù❽Ó✜Ö✷Õ✷Ó✈øqÙ❊❂✞Û✮Ü✹➎✠q❄Ô✎ø❙Ø✰Ó✶Ö✴Ô îÓ✜Ø✰ï✑Ý✽❂✼Ù❽Ó✜Ö✷Õ❑➏✟➐ ❀➒➑ ➓