euu, fioteshale, reh ethaiSht lioee, f juet the Gheeo'efuocti, o. The oext etea ie heo t, Merel, a ao aaah, ach f, h Meth ioio s the weiShte io thie caee the aie 2.1.3 RestduaFo efiotttoo aod g totmtzattoo a LIMi R(a)=v()- approx G(, a )2onip: (z)s We wtIpptch the onis to mahe R()smap Geoehal aaah, ach: Pick a eet, f teet fuo cti, oe 91, .. n, ao Mf, Ice R()t,be hth, S, oal t, the eet; pi(ar(a)ds=0 for all i Note 4 If the basie fuocti, o heaheeeot i, o haaaeoet, exactlp eaheeeot the Meo etp the the heeiMal R(a)Mefioe M, o the t, a, f the ab, re eline will be zeh, f, h all a weishte the po, t the caee, ao Mio eteaM we will thp t, aick the basie fuocti, o onie t, e u eh, w u io iu ize R(a). Ooe aaah, ach t, u ioi izo S (a)iet, u ake it, hth, S, oal t, a c, llecti,o, f teet fuo cti, oe Aeo, teM, o the b,tt, u, f the eline, eof, hcio s, hth,S,oalitp io thie caee u eao e eo euros that the io teShal, f the ah, Muict, f R(a) ao Mo(), reh the eurface ie zeh, 2.1.4 Restduafg tofmtzattoo Ustol Test Fuocttoos a IM ∫;(x)F(x)dS=0|→ ()(x15-/px(xG(x)∑n9()ds=0 We will generate different methods by choosing theφ1,…,,φn Collocation i(a)=s(r-Tt(point matching Weighted Residual Method i(r)=1 if pi(a)f0(averages) Ae o, teM, o the t, a, f the ab, re dime, eubetitutios f, h the heimdal io the S, dalit aheri, ue eline pielke sir euuati, oe each with tw, ioteSale The firet ioteShal ie, reh the eurface, f the ah, Mact, f the Sire a, teotial with a teet fuo cti, o. The eec, o MioteShaliea Mu ble io teShal, reh the eurface. The ioteShao M, f the Muble io teShal ie a ah, Mict❣P❍★❀ ●■ß✤❁❄❃❥❏r✾❲❧❥❇r❈■▼❄❣✯●✦♦■✾✹❇➐❣⑥❏r❇r❈■❁◆❧❥t❥❏➐▼❄❁◆❃❆✾✹❣➐●■ß✁￾⑥❍❆❣✐❏ê❏Pt❆✾❵❑á❇P✾✹✾❲❃❘❅ ❣✯ß➯❍★❃❆❱✠❏P❁❄●■❃✻ã➐ä✪t★✾✪❃★✾❲å✏❏➞❣✐❏P✾✹♣q❁❄❣ ❏Pt❆✾❲❃☞❏P●✇❅✰✾✹♦■✾❲▼❄●■♣❪❈❋❃❪❈❋♣❆♣★❇P●✏❈■❱❤t✆ß➯●■❇❉❅✰✾❲❏P✾❲❇r❀q❁◆❃❆❁◆❃★❧❂❏rt★✾❼Ý❯✾✹❁◆❧❥t✏❏r❣✹Ü✰❁◆❃☞❏Pt❆❁❄❣❉❱✹❈■❣P✾❵❏rt★✾✄✂➹ ❅ ❣❲ã ➠✻➡ ❝ ➡✆☎ ✝✛✏✼✜✱✳✺✯✭✯✥★✧❯❡☞✛✟✞✯✘✯✱➯✙✜✱✳✲✴✘✖✥❆✘✞✺➁✵✱✳✘✯✱✳❨❿✱✡✠■✥✰✙✜✱✳✲✴✘ ✌ ✍✏✎✒✑✔✓☞☛ ✌q➫➯➭✚➲✎✍æ➩❂➫✳➭✴➲✑✏✓✒✕✔✗✖✘✖✘✙✛✚✢✜ ✣✥✤✙✧✦★✔✁✩✫✪ ➸✇➫➯➭✞➺✐➭➻ ➲ ➚ ✬➹❄➘✞➴ ➼✤➚➹Ò➷✯➹ ➫➯➭➻ ➲⑥➽◗➾➻ ✭✛❇✁✇✱✳✧❜✧ê❩❭✱✳❴✜④ ✙✹✸✯✛❚➼➚ ➹ ☛ ✼❹✙✜✲❬❨❿✥✰④✴✛✮✌❂➫✳➭✴➲▲✼✜❨❿✥★✧❜✧ ➡ ❑á✾❲❃❆✾❲❇❤❈❋▼✮❈❋♣★♣❆❇P●✏❈■❱❤t✻➨✰✯ê❁⑤❱❤❖✆❈q❣✐✾❲❏❉●❋ß✔❏r✾✹❣✐❏✿ß➯❍★❃❆❱✠❏P❁❄●■❃❆❣✲✱➴ ➺✔✓✔✓✦✓✠➺✛✱➚ Ü✤❈❋❃❆❅✆ß➯●■❇❤❱✠✾✳✌q➫➯➭✴➲❯❏r●✇❊✤✾ ●■❇P❏Pt❆●■❧■●❥❃❆❈❋▼✤❏P●❂❏rt★✾✉❣✐✾❲❏✵✴ ✒ ✱ ➹ ➫➯➭✴➲✥✌q➫➯➭✴➲✐➽✏➾✖➳ ✵ ✶✸✷✢✹✻✺✠✣❊✣❃✩ Ô➥Õ✮ÖØ×✽✼ ➦③ß❆❏rt★✾✿❊❆❈❥❣✐❁⑤❣✞ß➯❍★❃❆❱Ø❏r❁◆●❥❃❹❇r✾❲♣❆❇P✾✜❣✐✾✹❃❥❏❭❁❄●■❃❂t❆❈❋♣★♣✴✾❲❃✤❣✯❏P●▲✾✠å★❈■❱✠❏P▼❄⑦❼❇P✾✹♣★❇r✾✹❣P✾❲❃✏❏✞❏rt★✾❉❅✰✾✹❃❆❣✐❁◆❏⑥⑦■Ü❋❏Pt★✾✹❃ ❏Pt❆✾❦❇P✾✜❣✐❁⑤❅✰❍❆❈■▼✾✌q➫➯➭✴➲➃❅✰✾✛✚✤❃★✾✹❅➎●❥❃è❏rt★✾✇❏r●■♣❬●❋ßÞ❏Pt❆✾✆❈❋❊✴●✦♦■✾✇❣P▼◆❁⑤❅✰✾❦Ý✿❁❄▼◆▼❯❊✤✾✆❑❲✾❲❇r●☞ß➯●■❇➃❈■▼◆▼➞➭✔ã ä✪t★❁⑤❣✿❁⑤❣✿❍❆❣P❍❆❈❋▼❄▼◆⑦❘❃★●❋❏✿❏rt★✾➃❱✹❈■❣P✾■Ü★❈❋❃✤❅❪❁◆❃❆❣✐❏P✾✜❈■❅✮Ü❆Ý❯✾❼Ý✿❁❄▼◆▼✻❏P❇r⑦❦❏P●✇♣❆❁❄❱❤❖✆❏Pt★✾✉❊✤❈■❣P❁❄❣✪ß➯❍❆❃❆❱Ø❏r❁◆●❥❃ ÝÞ✾❲❁❄❧■t✏❏r❣✹Ü✯❏Pt★✾❪➼➚ ➹ ❅ ❣❲Ü➐❏r●è❣P●■❀q✾❲t❆●✦Ý ❀q❁❄❃★❁❄❀❂❁✕❑❲✾✿✌q➫➯➭✚➲Øã ✰❃★✾☞❈❋♣★♣★❇r●❥❈❥❱❤t➥❏P●➥❀❂❁❄❃★❁❄❀q❁✧❑✹❁◆❃★❧ ✌q➫➯➭✚➲➞❁⑤❣Þ❏P●q❀✇❈❋❖❥✾✬❁◆❏❉●■❇P❏Pt❆●■❧■●❥❃❆❈❋▼❆❏P●✇❈q❱✠●❥▼◆▼❄✾✹❱✠❏P❁❄●■❃✆●■ß✔❏r✾✹❣✐❏Þß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣❲ã Û❣✪❃★●❋❏r✾✹❅❘●■❃❘❏Pt★✾ ❊✴●❋❏✐❏r●■❀ ●❋ß✪❏rt★✾❘❣P▼◆❁⑤❅✰✾❥Ü✯✾✹❃✰ß➯●■❇❤❱✠❁❄❃★❧❚●❥❇✐❏rt★●■❧❥●■❃❆❈■▼◆❁◆❏⑥⑦❚❁❄❃❬❏Pt❆❁❄❣❹❱❲❈■❣P✾❦❀q✾✹❈■❃❆❣✉✾✹❃❆❣P❍★❇P❁❄❃★❧s❏rt❆❈✦❏ ❏Pt❆✾❼❁◆❃✏❏P✾✹❧■❇❤❈❋▼✚●❋ß✞❏Pt★✾❼♣❆❇P●✰❅✰❍❆❱✠❏❉●❋ß✑✌❂➫✳➭✴➲✿❈■❃❆❅✿✱✞➫✳➭✴➲✪●✦♦❥✾❲❇Þ❏Pt❆✾✉❣✐❍★❇Pß✳❈■❱❲✾▲❁❄❣ ❑✹✾❲❇r●❆ã ➠✻➡ ❝ ➡❁❀ ✝✛✏✼✜✱✳✺✯✭✯✥★✧❯✵✱✳✘✯✱✳❨❿✱✡✠■✥✰✙✜✱✳✲✴✘✓❂❘✼✜✱✳✘✯✢❄❃Þ✛✏✼✹✙❆❅✔✭✯✘✞❴■✙✹✱✳✲✴✘✯✼ ✌ ✍✏✎✒✑✔✓❈❇ ➵❉✱➹ ➫➯➭✚➲❊✌q➫➯➭✚➲⑥➽◗➾ ➳✭✵ ❋ ✒●✱➹ ➫➯➭✚➲P➩❂➫✳➭✴➲⑥➽◗➾❍✏✓✒●✒■✔✗✖✘✖✘✙✛✚✢✜ ✣✥✤✙✧✦★✔✁✩✫✪ ✱ ➹ ➫✳➭✴➲✐➸✇➫➯➭✔➺P➭➻ ➲ ➚ ✬ ❏ ➘✞➴ ➼✤➚❏ ➷ ❏ ➫➯➭➻ ➲✐➽✏➾➻ ➽◗➾❬➳ ✵ ❑✿▲◆▼✾❖◗P❘P❚❙❯▲❲❱❳▲✛❨✥❩✫❬❊▲✲❭✁❖❘❪✘▲✧❨❊▲❲❱❫❬✾❴❵▲✛❬❊❛✟❜✗❭✁❝❡❞✗❢❤❣✥❛✟❜❫❜✵❝✐❖◗❱❳❙✳❬❊❛✟▲❦❥✸❧❲♠✛♥❲♥♦♥❲♠❊❥✘♣ qsr✉t✆t✆r✇✈❯①✟②♦③❁r✘④⑥⑤ ❥✘⑦❊⑧⑩⑨✉❶❸❷❍❹✁⑧⑩⑨✳❺✽⑨❼❻❾❽❊❶➀❿⑩➁✉❜❯❖❘❱❫❬✎❴✄❩✫❬❊❣✥❛✟❖❘❱✟❙✢➂ ➃❤①✟t✆➄✗➅✵➆✇③⑩④❍➇➈➄✗②➊➉✇r❚➋➌⑤ ❥⑦ ⑧⑩⑨✘❶➍❷➌➎⑦ ⑧⑩⑨✘❶➀❿⑩❞✟❩✵❝✐❖❘❝➀➏➐❬❊▲❲❝➑❬✧➂ ➒➄✁③✆➓✘➉✉②❲➄✁➋❈➔❵➄✁→♦③❾➋↔➣✇①❼t❡➇➈➄✗②➊➉❚r✇➋➐⑤ ❥⑦ ⑧⑩⑨✉❶❸❷➈↕❦➙✡➛➜➎⑦ ⑧⑩⑨✘❶◆➝❷❄➞❉❿➟❩♦➠✢▲✛❨✥❩✵❙✵▲❲❝✥➂ Ô➥Õ✮ÖØ×➢➡ Û❣✆❃★●❋❏r✾✹❅➜●❥❃➜❏Pt★✾Ú❏P●■♣✶●■ß▲❏Pt★✾➥❈❋❊✴●✦♦■✾❚❣P▼◆❁⑤❅✰✾■Ü❉❣P❍★❊❆❣✐❏P❁◆❏P❍★❏P❁❄❃★❧❿ß➯●❥❇✇❏Pt★✾è❇P✾✜❣✐❁⑤❅✰❍❆❈■▼❉❁❄❃✶❏Pt★✾ ●■❇P❏Pt❆●■❧■●❥❃❆❈❋▼❄❁◆❏⑥⑦Ú✾✹♠✏❍❆❈❋❏P❁❄●■❃❿●❥❃❬❏Pt❆✾❘♣★❇r✾❲♦◗❁❄●■❍❆❣❹❣✐▼❄❁⑤❅✰✾❘⑦◗❁❄✾❲▼⑤❅★❣✹Ü➐❧■❁❄♦■✾❲❃ ✜➜❏P✾✹❣✐❏❂ß➯❍★❃❆❱Ø❏r❁◆●❥❃❆❣✹Ü ✜ ✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣✬✾✹❈❥❱❤t❚Ý✿❁◆❏Ptè❏⑥Ý❯●☞❁◆❃✏❏r✾❲❧■❇❤❈❋▼⑤❣✹ã▲ä✪t❆✾ ✚❆❇❤❣✐❏á❁❄❃✏❏P✾❲❧❥❇r❈■▼✞❁⑤❣á●✦♦■✾✹❇❵❏rt★✾❂❣P❍★❇Pß✳❈■❱❲✾❹●■ßê❏Pt★✾ ♣★❇r●✰❅✰❍❆❱Ø❏❭●❋ß❆❏Pt★✾✪❧■❁❄♦■✾✹❃✉♣✴●❋❏P✾✹❃✏❏P❁⑤❈❋▼◗Ý✿❁é❏rt❂❈❵❏P✾✜❣⑥❏✯ß➯❍★❃❆❱✠❏P❁❄●■❃✻ã➐ä✪t★✾✪❣P✾✹❱✠●❥❃❆❅✉❁❄❃✏❏P✾✹❧■❇❤❈❋▼✏❁❄❣❭❈á❅✰●■❍✰à ❊★▼❄✾❼❁◆❃✏❏r✾❲❧■❇❤❈❋▼✮●✦♦❥✾❲❇Þ❏Pt❆✾➃❣✐❍❆❇✐ß✳❈❥❱✠✾■ã❯ä✪t★✾❼❁◆❃✏❏r✾❲❧■❇❤❈❋❃✤❅✆●■ß✯❏rt★✾✉❅✰●❥❍★❊★▼❄✾❼❁◆❃✏❏P✾✹❧■❇❤❈❋▼✮❁⑤❣❉❈q♣★❇r●✰❅✰❍❆❱Ø❏ ➤