happening the numerical technique at fault, or is the integral equation a problem? 3.3 Example ProbLems 3.3.1 1D Secon P in Equation (x)=0(x)+/1|x-1(x)dsx∈[-1,1 The potential is gⅳen The density must be computed y(x=r-x o(x) is unknown Note 1I We are going to post pone examining what went wrong in the first-kind example nd instead look at a oecond Kind equation. For this equation, we assume that the potential, (a), is known and that the charge density a(a) is unknown Here, a is in the interval [1, 1], and the integration is over that same interval Once again, the Greend function is given by G(, r)=r-a. What makes this equation oecond-Kind instead of first is circled in the equation on the top of the slide, The unknown c s both inside and outside of he integral. In the first-kind equation, the density appeared only inside the ntegral. This seemly small difference has enormous numerical ramifications n the left plot below the equation, an example given potential, as-a is plotted function of a. On the right is a plot of a charge density as a function of hich satisfies this second kind integral equation. As we will see below, this equation is easily solved numerically 3.3.2 Collocation Discretization of 1D Equation SLIDE 13 业(x)=(x)+1x-x1(x)dsx∈[-1,1 CentroilCollocatel Piecewise Constant Sc eme (ci)=oni+2t=ant 2i _ lac: -a'ldsr■✶t◗⑦s❁❄❣✬❏rt★❁❄❣▲t✤❈❋♣★♣✴✾❲❃★❁❄❃★❧✠❂➜➦⑥❣á❏Pt★✾❂❃◗❍★❀q✾❲❇r❁❄❱✹❈❋▼✞❏P✾✜❱❤t★❃★❁⑤♠❥❍❆✾q❈❋❏✬ß✳❈■❍★▼◆❏✹Ü✮●■❇▲❁⑤❣✬❏Pt★✾❂❁◆❃✏❏r✾❲❧■❇❤❈❋▼ ✾✹♠✏❍❆❈❋❏P❁❄●■❃❪❈❂♣★❇P●❥❊★▼❄✾❲❀☎❂ ✢✪➊☛✢ ✣✒✤✿➍✦✥➛★✧✞✩✫✪è➝★↔✭✬★✧✞✩✮✥➏ ☎✔➡✆☎✔➡ ❝ ❝◗❡ ✄✻✛◗❴❥✲✤✘✯✺ ✯s✱✳✘✯✺➜✩✬✫✮✭✯✥✰✙✹✱❜✲✤✘ ✌ ✍✏✎✒✑✔✓✖✕■➤ ➩❂➫➯➭✚➲➞➳➁➼➐➫✳➭✴➲ ✌ ➵ ➴ ✳ ➴ ✴ ➭➜✏➥➭✤➻ ✴ ➼➐➫➯➭✤➻➯➲✐➽✏➾ê➻ ➭✷✶☎✸◗✏ ✳❥➺✔✳ ✺ Convergence Analysis Example Problems x∈ −[ 1,1] σ ( ) x is unknown ( ) 3 Ψ = x x x − The potential is given The density must be computed Solution = 3x ( ) ( ) ( ) 1 1 x σ σ x x x x dS − Ψ = + − ′ ′ ′ " Ô➥Õ✮ÖØ×ÚÙ ✎ ■➥✾✪❈■❇P✾Þ❧■●❥❁◆❃❆❧❵❏r●✬♣✴●❥❣✐❏P♣✴●■❃★✾✪✾❲å✰❈■❀q❁◆❃★❁❄❃★❧▲Ý✿t❆❈❋❏✯ÝÞ✾❲❃✏❏➐Ý✿❇P●❥❃★❧á❁◆❃❹❏Pt★✾◆✚❆❇❤❣✐❏✐à③❖✏❁❄❃❆❅❹✾✠å★❈❋❀q♣★▼❄✾■Ü ❈❋❃✤❅✇❁◆❃✤❣⑥❏r✾✹❈■❅❦▼◆●◗●■❖q❈✦❏✿❈ ●◗✾✜❱✠●❥❃❆❅✁▲❁❄❃❆❅❦✾✜♠❥❍✤❈✦❏P❁❄●■❃✔ã✯❢❆●■❇❯❏Pt★❁⑤❣➞✾✜♠✏❍❆❈✦❏r❁◆●❥❃✻Ü❥ÝÞ✾✬❈❥❣P❣P❍★❀q✾❉❏rt❆❈✦❏ ❏Pt❆✾s♣✤●■❏P✾❲❃✏❏r❁❄❈■▼❜Ü ✂á➫➯➭✴➲✠Ü✪❁❄❣✇❖◗❃★●✦Ý✿❃➜❈❋❃✤❅❿❏rt❆❈✦❏✇❏rt★✾s❱❤t❆❈❋❇r❧■✾❪❅✰✾❲❃✤❣✐❁◆❏⑥⑦ ➼➐➫➯➭✚➲q❁⑤❣✇❍★❃★❖◗❃★●✦Ý✿❃✻ã ❶❉✾✹❇P✾❥Ü✰➭❪❁❄❣✪❁❄❃☞❏Pt★✾▲❁❄❃❥❏r✾❲❇r♦✦❈❋▼✭✸❁✏ ✳■➺✔✳ ✺ÒÜ✰❈■❃❆❅❦❏rt★✾▲❁◆❃✏❏r✾❲❧■❇❤❈✦❏r❁◆●❥❃✆❁⑤❣✪●✦♦■✾✹❇❯❏rt❆❈✦❏❉❣r❈❋❀q✾▲❁◆❃✏❏r✾❲❇r♦②❈■▼❜ã ✰❃❆❱❲✾❹❈❋❧✏❈❋❁❄❃✻Ü✤❏Pt★✾q❑á❇P✾✹✾❲❃❘❅ ❣❉ß➯❍❆❃❆❱Ø❏r❁◆●❥❃Ú❁⑤❣❵❧❥❁◆♦❥✾❲❃❚❊◗⑦s➸✇➫➯➭✞➺✐➭✤➻✒➲❉➳ ✴ ➭✽✏➎➭✤➻ ✴ ã ■✶t✤❈✦❏✬❀✇❈■❖■✾✹❣ ❏Pt❆❁❄❣❵✾✹♠✏❍❆❈✦❏r❁◆●❥❃❍●◗✾✹❱❲●■❃❆❅◗à✂▲❁❄❃❆❅❪❁◆❃✤❣⑥❏r✾✹❈■❅❪●■ß ✚❆❇r❣✐❏❉❁⑤❣❵❱✠❁❄❇❤❱✠▼❄✾✹❅❪❁❄❃❪❏Pt★✾❹✾✹♠✏❍❆❈✦❏r❁◆●❥❃❪●■❃❪❏Pt❆✾❼❏P●❥♣ ●❋ßÞ❏Pt❆✾❦❣✐▼❄❁❄❅★✾■ã❘ä✪t❆✾✇❍★❃★❖◗❃★●✦Ý✿❃❬❱❤t❆❈❋❇r❧■✾q❅✰✾✹❃❆❣✐❁◆❏⑥⑦è❈■♣★♣✴✾✹❈❋❇❤❣▲❊✴●❋❏Pt❬❁❄❃❆❣P❁❄❅★✾❦❈❋❃❆❅➥●■❍★❏r❣P❁❄❅✰✾✇●■ß ❏Pt❆✾✆❁❄❃❥❏r✾❲❧❥❇r❈■▼❜ã❚➦➧❃ ❏Pt★✾✆✚❆❇❤❣⑥❏PàÒ❖◗❁❄❃❆❅❬✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü✯❏rt★✾☞❅★✾❲❃❆❣P❁é❏⑥⑦❬❈■♣★♣✤✾✜❈❋❇r✾✹❅➎●❥❃★▼◆⑦➎❁❄❃❆❣P❁❄❅✰✾✆❏Pt★✾ ❁❄❃❥❏r✾❲❧❥❇r❈■▼❜ã❭ä✪t★❁⑤❣❉❣✐✾✹✾❲❀q▼❄⑦✆❣✐❀✇❈■▼◆▼✻❅✰❁✚✾❲❇r✾❲❃❆❱❲✾❼t❆❈■❣✪✾✹❃★●■❇r❀q●■❍❆❣Þ❃◗❍★❀q✾❲❇r❁❄❱✹❈❋▼✮❇❤❈❋❀q❁✧✚✤❱❲❈❋❏P❁❄●■❃❆❣✹ã ➦➧❃❂❏Pt★✾❵▼◆✾❲ß✒❏➞♣★▼❄●❋❏ê❊✴✾❲▼❄●✦Ý ❏rt★✾✿✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü✏❈❋❃q✾✠å★❈❋❀q♣★▼❄✾✿❧■❁❄♦■✾✹❃❹♣✴●❋❏r✾❲❃✏❏P❁⑤❈❋▼ÒÜ■➭☎✄❸✏✇➭✇❁⑤❣➐♣★▼❄●❋❏P❏P✾✜❅ ❈■❣▲❈❘ß➯❍★❃❆❱✠❏P❁❄●■❃➥●❋ßÞ➭✔ã ✰❃Ú❏rt★✾q❇P❁❄❧■t✏❏▲❁❄❣❼❈☞♣★▼❄●❋❏❼●❋ß✪❈❘❱❤t✤❈❋❇r❧■✾❹❅★✾❲❃❆❣P❁é❏⑥⑦è❈■❣▲❈❘ß➯❍★❃❆❱Ø❏r❁◆●❥❃è●■ß ➭ÚÝ✿t❆❁❄❱❤t❚❣r❈✦❏r❁❄❣✚❆✾✹❣❉❏Pt★❁⑤❣❵❣P✾✹❱❲●■❃❆❅❪❖◗❁❄❃❆❅❪❁❄❃❥❏r✾❲❧❥❇r❈■▼✔✾✹♠✏❍❆❈❋❏P❁❄●■❃✻ã Û❣❵ÝÞ✾✉Ý✿❁❄▼◆▼✯❣✐✾✹✾✉❊✤✾✹▼◆●✦Ý❼Ü✤❏Pt❆❁❄❣ ✾✹♠✏❍❆❈❋❏P❁❄●■❃☞❁❄❣✪✾✜❈■❣P❁◆▼❄⑦✆❣P●■▼❄♦■✾✜❅❦❃◗❍★❀q✾❲❇r❁❄❱✹❈❋▼❄▼◆⑦❥ã ☎✔➡✆☎✔➡⑤➠ ①✲✴✧✳✧✳✲✔❴■✥✰✙✜✱✳✲✴✘✖❡☞✱✳✼✜❴❋✣✦✛✏✙✹✱✡✠■✥✰✙✜✱✳✲✴✘✶✲✤❫❼❝◗❡ ✩✬✫✮✭✯✥✰✙✜✱✳✲✴✘ ✌ ✍✏✎✒✑✔✓✖✕❆✝ ➩❂➫➯➭✚➲➞➳➁➼➐➫✳➭✴➲ ✌ ➵ ➴ ✳ ➴✭✴ ➭➜✏➥➭✤➻ ✴ ➼➐➫➯➭✤➻➯➲✐➽✏➾ê➻ ➭✷✶☎✸◗✏ ✳❥➺✔✳ ✺ 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➳➁➼✤➚➹ ✌ ➪➚ ❏ ➘✯➴ ➼❆➚❏ ➵✞ ✮ ✞ ✮✡✠ P ✴ ➭✴Ð ❽ ✏➥➭✤➻ ✴ ➽◗➾➐➻ ✄