Note 13 To compute the numerical solution to the one-dimensional second-kind equation at the top of the slide, once again consider using a piecewise-constant collocation scheme. Once again, we select n+ 1 points on the interval and denote the points as ao, rI,..., n, as shown in the figure in the middle of the slide For this example, o =-1 and In= 1. The corresponding basis functions, Ii(a),sp2(),.,An (a)), are once again [x;-1,x The charge density o is approximately represented by ()-an()≡∑on(x), There o i is the weight associated with the itn basis function Plugging the basis function representation of the charge density into the second kind integral equation at the top of the slide yields 4(x)=∑on/(x)+ < pi(a)dsy which can be simplified by exploiting the specific basis functions te 重(x)=∑on9()+∑a/x-x1 As shown in the middle of the slide, the collocation points are the subinterval center points, c =0.5*(ai-1+ai. When collocation is used,(9) must be satisfied exactly at the collocation points and therefore Oni pj Ic:-r'ds' Note that pi (aci=0 when i # j, and pi(aci)=1. Using this fact yields the equation on the bottom of the slide 3.3.3 Collocation Discretization of 1D Equation- The Matr SLIDE 14 乎 onn ly(=,)IÔ➥Õ✮ÖØ×ÚÙ ◗ ä✞●á❱❲●■❀q♣★❍✰❏r✾❯❏rt★✾✪❃◗❍★❀q✾❲❇r❁❄❱✹❈❋▼✰❣✐●❥▼◆❍★❏P❁❄●■❃✉❏r●á❏Pt★✾✪●■❃❆✾✠à➧❅✰❁◆❀q✾✹❃❆❣✐❁❄●■❃✤❈❋▼✰❣✐✾✜❱✠●❥❃❆❅◗à③❖✏❁❄❃❆❅➃✾✹♠✏❍❆❈✦❏r❁◆●❥❃ ❈✦❏✞❏Pt★✾❯❏P●❥♣➃●■ß✰❏Pt★✾✪❣✐▼❄❁⑤❅✰✾■Ü✦●❥❃❆❱✠✾Þ❈❋❧❥❈■❁◆❃❹❱✠●■❃✤❣✐❁⑤❅✰✾❲❇✞❍❆❣P❁◆❃★❧á❈❉♣★❁❄✾✹❱❲✾❲Ý✿❁⑤❣✐✾❲à③❱❲●■❃❆❣✐❏r❈■❃✏❏✞❱✠●❥▼◆▼❄●✰❱❲❈✦❏r❁◆●❥❃ ❣r❱❤t★✾❲❀q✾■ã ✰❃❆❱❲✾❦❈❋❧❥❈■❁◆❃✔Ü✔ÝÞ✾❦❣✐✾✹▼◆✾✜❱Ø❏ ✜✁✌ ✳✇♣✴●■❁❄❃✏❏r❣✉●❥❃➥❏Pt★✾❦❁❄❃✏❏P✾✹❇P♦✦❈❋▼❯❈❋❃❆❅❬❅✰✾✹❃★●❋❏r✾q❏Pt★●✏❣✐✾ ♣✴●■❁❄❃❥❏❤❣❪❈■❣☞✎✹➭✒✑❥➺✐➭ ➴ ➺✦✓✧✓✕✓◆➺P➭✤➚✔✓✏Ü✬❈❥❣❘❣Pt★●✦Ý✿❃➁❁❄❃æ❏rt★✾ ✚❆❧■❍★❇r✾Ú❁❄❃➶❏Pt★✾➥❀q❁❄❅★❅★▼◆✾➎●❋ß➃❏Pt★✾➎❣✐▼❄❁⑤❅✰✾■ã ❢★●❥❇➃❏rt★❁⑤❣❹✾❲å★❈❋❀q♣★▼❄✾■Üê➭✑❚➳ ✏ ✳❘❈■❃❆❅❿➭✴➚ ➳ ✳❥ã❬ä✪t★✾☞❱❲●■❇r❇P✾✜❣✐♣✴●■❃✤❅✰❁◆❃❆❧❚❊❆❈■❣P❁⑤❣✉ß➯❍★❃✤❱Ø❏P❁❄●■❃✤❣❲Ü ✎ ➷ê➴ ➫➯➭✴➲✠➺ ➷ ✑ ➫➯➭✴➲✠➺✔✓✕✓✧✓❄➺ ➷ ➚✞➫➯➭✴➲✕✓❥Ü❆❈■❇P✾▲●❥❃❆❱✠✾❼❈❋❧✏❈❋❁❄❃ ➷➹ ➫➯➭✚➲➞➳✴✳ ➭✷✶✹✸➭ ➹ ✳ ➴ ➺✐➭➹ ✺ ✖ ✼ ✘ ➷❭➹ ➫➯➭✚➲➞➳✭✵ ✷✗✖✙✘✒✚❯✹✗✛✲✩✢✜✣✚✼✓ ✖ ✦ ✘ ä✪t★✾❼❱❤t❆❈■❇P❧❥✾á❅✰✾✹❃❆❣P❁é❏⑥⑦✆➼➥❁⑤❣✿❈❋♣★♣❆❇P●②å✰❁❄❀q❈❋❏P✾✹▼◆⑦✇❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏r✾✹❅❦❊◗⑦ ➼➐➫➯➭✚➲✁￾✶➼➚ ➫➯➭✚➲✎✍ ➚ ✬➹◆➘✯➴ ➼➚ ➹ ➷➹ ➫➯➭✚➲Ø➺ ✖ ☛✘ Ý✿t★✾✹❇P✾❼➼➹ ❁⑤❣✪❏rt★✾❼Ý❯✾✹❁◆❧❥t✏❏✿❈❥❣P❣P●✰❱✠❁⑤❈✦❏r✾✹❅❦Ý✿❁◆❏Pt☞❏rt★✾ ✩ Ñ❫❪ ❊❆❈■❣P❁❄❣Þß➯❍❆❃❆❱Ø❏r❁◆●❥❃✻ã ✯ê▼❄❍★❧■❧❥❁◆❃❆❧❼❏Pt★✾✬❊❆❈■❣P❁⑤❣❭ß➯❍★❃❆❱✠❏P❁❄●■❃❦❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏❤❈✦❏P❁❄●■❃q●■ß✮❏Pt★✾á❱❤t❆❈❋❇r❧■✾✿❅✰✾✹❃❆❣✐❁◆❏⑥⑦q❁◆❃✏❏r●❼❏Pt★✾á❣✐✾✜❱✠●❥❃❆❅ ❖◗❁◆❃✤❅✆❁❄❃✏❏P✾❲❧❥❇r❈■▼✮✾✹♠✏❍❆❈❋❏P❁❄●■❃❪❈✦❏✪❏Pt★✾▲❏r●■♣❪●❋ß✞❏Pt★✾✉❣P▼❄❁❄❅✰✾❼⑦◗❁❄✾❲▼⑤❅★❣ ✂á➫➯➭✚➲➞➳ ➚ ✬ ❏ ➘✞➴ ➼➚ ❏ ➷ ❏ ➫➯➭✚➲ ✌ ✒ ➴ ✳ ➴ ✴ ➭➜✏➥➭➻ ✴ ➚ ✬➹◆➘✞➴ ➼➚ ➹ ➷➹ ➫➯➭➻ ➲✐➽✏➾➻ ➺ ✖ ✒ ✘ Ý✿t★❁⑤❱❤t☞❱✹❈❋❃☞❊✤✾✉❣P❁❄❀❂♣❆▼◆❁✧✚❆✾✹❅☞❊◗⑦✆✾❲å✰♣★▼◆●❥❁é❏r❁◆❃❆❧❹❏rt★✾✉❣P♣✤✾✜❱✠❁✧✚✤❱á❊❆❈■❣P❁❄❣Þß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣✪❏P● ✂á➫✳➭✴➲❯➳ ➚ ✬ ❏ ➘✯➴ ➼❆➚❏ ➷ ❏ ➫✳➭✴➲ ✌ ➚ ✬ ❏ ➘✯➴ ➼❆➚❏ ✒✞ ✮ ✞ ✮✡✠ P ✴ ➭➜✏➥➭➻ ✴ ➽◗➾➻ ✓ ✖ ✄ ✘ Û❣á❣Pt★●✦Ý✿❃❚❁❄❃❚❏Pt❆✾❹❀q❁⑤❅★❅✰▼❄✾❹●■ß➐❏Pt★✾q❣✐▼❄❁⑤❅✰✾■Ü✚❏rt★✾❂❱❲●■▼❄▼◆●✰❱✹❈✦❏P❁❄●■❃s♣✴●■❁❄❃✏❏r❣á❈❋❇r✾✉❏rt★✾❂❣P❍★❊★❁❄❃✏❏P✾✹❇P♦✦❈❋▼ ❱✠✾✹❃✏❏P✾❲❇❼♣✴●■❁❄❃✏❏r❣✹Ü✞➭✚Ð ❽ ➳ ✵▲✓ ✁✄✂ ➫✳➭➹ ✳ ➴ ✌✶➭➹ ➲Øã✏■✶t★✾❲❃❿❱❲●■▼❄▼◆●✰❱❲❈❋❏P❁❄●■❃➥❁❄❣✉❍❆❣✐✾✜❅✮Ü ✖ ✄ ✘á❀❹❍❆❣✐❏➃❊✤✾ ❣r❈✦❏P❁⑤❣✚❆✾✹❅❘✾✠å★❈■❱✠❏P▼❄⑦✆❈✦❏✪❏rt★✾✉❱✠●❥▼◆▼❄●✰❱❲❈✦❏r❁◆●❥❃✆♣✴●■❁❄❃✏❏r❣❉❈■❃❆❅❦❏rt★✾❲❇r✾✠ß➯●❥❇P✾ ✂á➫➯➭ Ð ❽ ➲ê➳ ➚ ✬➹◆➘✯➴ ➼➚ ➹ ➷ ❏ ➫✳➭Ð ❽ ➲ ✌ ➚ ✬ ❏ ➘✞➴ ➼➚ ❏ ✒✞ ✮ ✞ ✮✡✠ P ✴ ➭ Ð ❽ ✏➥➭➻ ✴ ➽✏➾➻ ✓ ✖ ë✏✎ ✘ ✔❉●■❏P✾➃❏Pt❆❈❋❏ ➷ ❏ ➫➯➭ Ð ❽ ➲✿➳ ✵✆Ý✿t★✾❲❃ ✩✄✂➳✰❬✤Ü✮❈❋❃❆❅ ➷➹ ➫➯➭ Ð ❽ ➲✿➳ ✳❥ã ☎✬❣✐❁❄❃★❧❦❏rt★❁⑤❣❉ß✳❈❥❱Ø❏▲⑦✏❁❄✾❲▼⑤❅★❣❵❏Pt★✾ ✾✹♠✏❍❆❈❋❏P❁❄●■❃☞●■❃❘❏Pt★✾❼❊✴●❋❏P❏P●❥❀ ●❋ß✞❏Pt★✾✉❣P▼❄❁❄❅✰✾❥ã ☎✔➡✆☎✔➡✆☎ ①✲✴✧✳✧✳✲✔❴■✥✰✙✜✱✳✲✴✘✖❡☞✱✳✼✜❴❋✣✦✛✏✙✹✱✡✠■✥✰✙✜✱✳✲✴✘✶✲✤❫❼❝◗❡ ✩✬✫✮✭✯✥✰✙✜✱✳✲✴✘✝✆❲❃✉✸✞✛Ú✵✷✥✰✙❲✣✦✱✳❳ ✌ ✍✏✎✒✑✔✓✖✕❫☛ SMA-HPC ©1999 MIT Convergence Analysis Example Problems Collocation Discretization of 1-D Equation - The Matrix ( ) ( ) 1 1 1 0 1 1 1 0 1 1 1 1 n n n n n n n x x c c x x c x x n c c c x x x x dS x x dS x x x x dS x x dS σ σ − − $ % + − ′ ′ − ′ ′ & '$ % $ % Ψ & ' & ' = & ' & ' & ' ( ) & ' & ' Ψ − + ′ ′ − ′ ′ ( ) & ' ( ) " " " " ' ( ) ( ( ( ' n1 σ nn σ ë✏✎