n the next several slides we will investigate the convergence properties of these discretization methods. How these methods converge depends on what kind of Itegral equation is being solving. Examining this issue will introduce one of o begin, consider the example one-dimensional first-kind integral equation on the top of the above slide. For this equation, we assume that the potential, 4(a), is known and that the charge density o(a) is unknown. Here, a is in the terval [-1, 1, and the integration is over that same interval. Note that fo this example, the Green's function is giv n the left plot below the equation, an example given potential, a 3-a is plotted as a function of a. On the right is a plot of a charge density as a function of x ich might be a solution to the this problem the ques tion of what is the solution is not so easy to ans wer 3.1.2 Collocation Discretization of 1D Equation 业(x)=广1|x-xp(x)dsx∈[-1,1 Centroid Collocated Piecewise Constant Scheme |+++ To compute the numerical solution to this one-dimensional problem, conside solving the integral equation at the top of the slide using a piecewise-constant allocation scheme. In such a scheme, we first select n+l points on the interval, in this case [-1, 1]. We denote those points as ao, Ii, Inb, as shown in the figure in the middle of the slide. For this example, ao=-1 and n=1 Then, we can define a set of basis functions on the subintervals, pi(e),2(a),,Pn(a)l [x;-1,xi]y;(x) The charge density a can then be represented approximatelyÔ➥Õ✮ÖØ×✁￾ ➦➧❃✇❏rt★✾✬❃❆✾✠å◗❏Þ❣P✾❲♦❥✾❲❇❤❈❋▼❆❣P▼◆❁⑤❅✰✾✜❣➞ÝÞ✾❵Ý✿❁❄▼◆▼✚❁❄❃◗♦■✾✹❣✐❏P❁❄❧❥❈❋❏P✾✪❏rt★✾á❱❲●■❃◗♦■✾✹❇P❧❥✾❲❃❆❱❲✾✿♣★❇r●■♣✴✾❲❇P❏P❁❄✾✹❣ê●■ß✮❏Pt★✾✜❣✐✾ ❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈❋❏P❁❄●■❃☞❀❂✾❲❏Pt★●✰❅★❣✹ã✪❶❉●✦Ý➶❏Pt★✾✜❣✐✾✉❀q✾❲❏Pt★●✰❅★❣✬❱✠●■❃◗♦❥✾❲❇r❧■✾▲❅✰✾❲♣✴✾❲❃❆❅❆❣❉●■❃❪Ý✿t❆❈❋❏❉❖◗❁◆❃❆❅❪●■ß ❁❄❃❥❏r✾❲❧❥❇r❈■▼➐✾✹♠✏❍❆❈✦❏r❁◆●❥❃➎❁❄❣➃❊✤✾✹❁◆❃★❧è❣✐●❥▼◆♦◗❁❄❃★❧❆ã ❖➐å★❈❋❀q❁◆❃❆❁◆❃★❧❪❏rt★❁⑤❣✉❁⑤❣P❣P❍★✾❦Ý✿❁❄▼◆▼❯❁◆❃✏❏r❇P●✰❅✰❍❆❱❲✾✇●■❃★✾✇●■ß ❏Pt❆✾✉❣✐❍★❊★❏P▼❄✾❼♣✤●❥❁◆❃✏❏r❣✿❈■❊✤●❥❍✰❏✿❁◆❃✏❏r✾❲❧■❇❤❈❋▼✮✾✜♠✏❍❆❈✦❏r❁◆●❥❃❆❣❲ã ä✞●✇❊✤✾✹❧■❁❄❃✻Ü✤❱❲●■❃❆❣P❁❄❅★✾❲❇✪❏Pt❆✾✉✾✠å★❈❋❀q♣★▼❄✾❼●■❃❆✾✠à➧❅✰❁◆❀q✾✹❃❆❣✐❁❄●■❃✤❈❋▼ ✚❆❇❤❣⑥❏PàÒ❖◗❁❄❃❆❅❘❁◆❃✏❏P✾✹❧■❇❤❈❋▼✻✾✜♠✏❍❆❈✦❏r❁◆●❥❃❘●❥❃ ❏Pt❆✾✆❏r●■♣ ●■ß❉❏Pt❆✾❪❈■❊✤●✦♦❥✾✆❣P▼◆❁⑤❅✰✾■ã❿❢❆●■❇❂❏Pt❆❁❄❣❂✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü➞Ý❯✾☞❈■❣r❣✐❍★❀q✾✆❏Pt❆❈❋❏❂❏rt★✾☞♣✤●■❏P✾❲❃✏❏r❁❄❈■▼❜Ü ✂á➫➯➭✚➲ØÜ✤❁❄❣❵❖✏❃❆●✦Ý✿❃☞❈■❃❆❅❘❏Pt❆❈❋❏❉❏Pt★✾❹❱❤t❆❈❋❇r❧■✾▲❅✰✾❲❃✤❣✐❁◆❏⑥⑦❘➼➐➫✳➭✴➲✿❁⑤❣✿❍★❃❆❖✏❃❆●✦Ý✿❃✻ã❯❶❵✾❲❇r✾■Ü★➭❚❁⑤❣✿❁❄❃❪❏Pt★✾ ❁❄❃❥❏r✾❲❇r♦✦❈❋▼ ✸❁✏ ✳■➺✦✳✻✺③Ü✯❈■❃❆❅➥❏Pt★✾❘❁◆❃✏❏r✾❲❧■❇❤❈✦❏r❁◆●❥❃➎❁❄❣➃●✦♦■✾❲❇▲❏rt❆❈✦❏q❣P❈■❀q✾✇❁◆❃✏❏P✾✹❇P♦✦❈■▼❜ã ✔❵●❋❏P✾❦❏Pt❆❈❋❏➃ß➯●■❇ ❏Pt❆❁❄❣✿✾❲å★❈❋❀q♣★▼❄✾■Ü◗❏Pt❆✾➃❑á❇P✾✹✾❲❃❘❅ ❣Þß➯❍★❃❆❱✠❏P❁❄●■❃❪❁❄❣✪❧❥❁◆♦❥✾❲❃☞❊◗⑦✆➸✇➫➯➭✞➺✐➭✤➻➯➲ê➳ ✴ ➭➜✏è➭✴➻ ✴ ã ➦➧❃❂❏Pt★✾❵▼◆✾❲ß✒❏➞♣★▼❄●❋❏ê❊✴✾❲▼❄●✦Ý ❏rt★✾✿✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü✏❈❋❃q✾✠å★❈❋❀q♣★▼❄✾✿❧■❁❄♦■✾✹❃❹♣✴●❋❏r✾❲❃✏❏P❁⑤❈❋▼ÒÜ■➭☎✄❸✏✇➭✇❁⑤❣➐♣★▼❄●❋❏P❏P✾✜❅ ❈■❣✿❈❂ß➯❍❆❃❆❱Ø❏r❁◆●❥❃s●❋ß❭➭✔ã ✰❃☞❏rt★✾❼❇P❁❄❧■t✏❏❉❁⑤❣❉❈❂♣★▼❄●❋❏❵●■ß❭❈q❱❤t❆❈■❇P❧❥✾▲❅★✾❲❃❆❣P❁é❏⑥⑦☞❈■❣✿❈❂ß➯❍★❃✤❱Ø❏P❁❄●■❃❪●■ß❭➭ Ý✿t★❁⑤❱❤t❚❀q❁◆❧❥t✏❏✬❊✴✾❂❈❘❣✐●❥▼◆❍✰❏r❁◆●❥❃❪❏P●❘❏Pt★✾❹❁◆❃✏❏r✾❲❧■❇❤❈❋▼✞✾✹♠✏❍❆❈❋❏P❁❄●■❃✻ã Û❣✬Ý❯✾❹Ý✿❁◆▼❄▼❭❣✐✾✹✾❹❣Pt★●■❇P❏P▼❄⑦■Ü✴ß➯●■❇ ❏Pt❆❁❄❣✿♣❆❇P●❥❊★▼◆✾✹❀ ❏Pt❆✾✉♠❥❍❆✾✹❣✐❏P❁❄●■❃☞●❋ß✞Ý✿t✤❈✦❏❉❁⑤❣Þ❏Pt★✾✉❣P●■▼❄❍✰❏P❁❄●■❃☞❁⑤❣✿❃★●❋❏❉❣P●q✾✹❈■❣P⑦✇❏r●q❈■❃❆❣PÝ❯✾✹❇✹ã ☎✔➡ ❝ ➡⑤➠ ①✲✴✧✳✧✳✲✔❴■✥✰✙✜✱✳✲✴✘✖❡☞✱✳✼✜❴❋✣✦✛✏✙✹✱✡✠■✥✰✙✜✱✳✲✴✘✶✲✤❫❼❝◗❡ ✩✬✫✮✭✯✥✰✙✜✱✳✲✴✘ ✌ ✍✏✎✒✑✔✓✝✆ ➩❂➫✳➭✴➲➞➳➶➵ ➴ ✳ ➴ ✴ ➭➜✏➥➭➻ ✴ ➼➐➫✳➭➻ ➲⑥➽◗➾➻ ➭✷✶✹✸❁✏ ✳■➺✦✳✻✺ ①✛✏✘✚✙✹✣✦✲✤✱❜✺ ①✲✴✧✳✧❜✲✻❴■✥✰✙✜✛◗✺ ￾✱❜✛◗❴■✛✂✁✇✱✳✼✜✛ ①✲✴✘✞✼✹✙✹✥❆✘✚✙☎✄✔❴②✸✞✛◗❨❿✛ SMA-HPC ©1999 MIT Convergence Analysis Example Problems Collocation Discretization of 1-D Equation ( ) ( ) 1 1 x x x σ x dS − Ψ = − ′ ′ ′ " x∈ −[ 1,1] x0 = −1 xn =1 1 x n 1 x x2 − 1c x 2c x nc x ( ) 1 1 j i i j x n c j c j x x σ x x dS − = Ψ = ! − ′ ′ " n1 σ nn σ ➩❂➫➯➭✴Ð ❽ ➲ê➳➶➪➚ ❏ ➘✞➴ ➼✤➚❏ ➵✟✞ ✮ ✞ ✮✡✠ P ✴ ➭✴Ð ❽ ✏➎➭✤➻ ✴ ➽✏➾➐➻ Ô➥Õ✮ÖØ×☞☛ ä✞●❚❱✠●■❀q♣★❍★❏P✾q❏Pt❆✾❦❃◗❍★❀q✾❲❇r❁❄❱✹❈❋▼ê❣P●■▼❄❍✰❏P❁❄●■❃➥❏P●❪❏rt★❁⑤❣✉●■❃❆✾✠à➧❅✰❁◆❀q✾✹❃❆❣✐❁❄●■❃✤❈❋▼➐♣★❇r●■❊★▼❄✾❲❀❪Ü✯❱✠●■❃✤❣✐❁⑤❅✰✾❲❇ ❣P●■▼❄♦✏❁❄❃★❧✇❏rt★✾➃❁❄❃✏❏P✾✹❧■❇❤❈❋▼✻✾✜♠❥❍✤❈✦❏P❁❄●■❃è❈✦❏❉❏rt★✾➃❏r●■♣❚●■ß➐❏Pt★✾❂❣✐▼❄❁⑤❅✰✾➃❍❆❣P❁❄❃★❧✆❈❦♣❆❁◆✾✜❱✠✾❲Ý✿❁⑤❣P✾✠à➧❱✠●■❃✤❣⑥❏❤❈❋❃✏❏ ❱✠●❥▼◆▼❄●✰❱❲❈❋❏P❁❄●■❃❹❣r❱❤t★✾✹❀❂✾❥ã✞➦➧❃✇❣P❍❆❱❤tq❈á❣r❱❤t★✾✹❀❂✾❥Ü❋ÝÞ✾✱✚✤❇r❣✐❏➞❣✐✾✹▼◆✾✜❱Ø❏ ✜✍✌ ✳❯♣✴●■❁❄❃✏❏r❣❭●■❃❂❏rt★✾✿❁◆❃✏❏r✾❲❇r♦②❈■▼❜Ü ❁❄❃❪❏Pt★❁⑤❣❵❱✹❈■❣P✾ ✸◗✏ ✳❥➺✔✳✻✺③ã✢■➥✾➃❅✰✾✹❃★●❋❏r✾❼❏Pt❆●❥❣P✾❼♣✤●❥❁◆❃✏❏r❣✬❈■❣✏✎✜➭✒✑❥➺✐➭ ➴ ➺✔✓✕✓✕✓◆➺P➭✤➚✔✓✏Ü❆❈■❣✬❣✐t★●✦Ý✿❃❪❁❄❃❪❏Pt★✾ ✚❆❧❥❍★❇P✾➞❁◆❃➃❏Pt★✾Þ❀q❁❄❅★❅★▼◆✾❯●❋ß★❏rt★✾❯❣P▼◆❁⑤❅✰✾❥ã✯❢❆●■❇✻❏rt★❁⑤❣✞✾❲å★❈❋❀q♣★▼❄✾■Ü②➭✒✑á➳ ✏ ✳➞❈■❃❆❅❼➭✤➚❦➳ ✳➞ä✪t★✾✹❃✻Ü✦Ý❯✾ ❱❲❈■❃❦❅✰✾✛✚✤❃★✾✬❈❹❣✐✾❲❏❯●❋ß✻❊✤❈■❣P❁❄❣êß➯❍★❃❆❱Ø❏r❁◆●❥❃❆❣❯●■❃✇❏rt★✾✬❣P❍★❊★❁❄❃✏❏P✾❲❇r♦✦❈❋▼⑤❣❲Ü✒✎ ➷ê➴ ➫✳➭✴➲✠➺ ➷ ✑ ➫➯➭✴➲✠➺✔✓✕✓✧✓❄➺ ➷ ➚✻➫✳➭✴➲✕✓❥Ü Ý✿t★✾✹❇P✾ ➷➹ ➫➯➭✴➲❯➳ ✳ ➭✷✶☎✸➭ ➹ ✳ ➴ ➺✐➭➹ ✺ ➷➹ ➫✳➭✴➲➞➳✭✵ ✷✗✖✙✘✒✚❯✹✗✛✲✩✢✜✣✚✂✓ ✖ ë ✘ ä✪t★✾❼❱❤t❆❈■❇P❧❥✾á❅✰✾✹❃❆❣P❁é❏⑥⑦✆➼➎❱✹❈❋❃❘❏Pt★✾✹❃☞❊✴✾❼❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏P✾✜❅✆❈■♣★♣★❇r●②å◗❁❄❀✇❈✦❏r✾❲▼❄⑦❦❈■❣ ➼➐➫➯➭✚➲✥✤✶➼✤➚✻➫➯➭✚➲✎✍ ➚ ✬➹◆➘✯➴ ➼✤➚➹③➷✯➹ ➫➯➭✚➲Ø➺ ✦