Note 14 Just as in the discretized first-kind equation, we generate a system of equations that can be used to solve for the ni s, the piecewise constant charge densities for each of the subintervals. The right-hand side of this system of equations is vector of known potentials at inter val centers(the collocation points). The it row of the matrix corresponds to unfolding the sum in the collocation equation d the entries in the jth column corresponds to how much the charge on the The major difference between the matrix in this discretized second-kind example and the first-kind example is circled on the slide. There is an additional one on the diagonal of the discretized second-kind equation that did not appear in the first-kind equation. More precisely, Ase I+A 3.4 Numerical Results with Increasing n SLIDE 15 Answers Are Improving!!! Unlike the results from discretizing the first kind equation, progressively refining the discretization of the second kind equation produces more accurate answers Once again, the plot is a little hard to decipher without looking at a color version It shows the oni s produced using n=10, n=20 and n=40 subintervals. For a point is plotted at r ints plotted for the st discretization and forty points plotted for the finest discretization, but all sets of points span the interval E[-1,1 What is clear from comparing the blue points (n=10) to the red points(n=20) and to the green points(n=40), is that the charge density seems to be approach-Ô➥Õ✮ÖØ×ÚÙ✉✼ ￾❥❍❆❣⑥❏✪❈■❣➞❁◆❃✆❏Pt★✾▲❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑❲✾✹❅ ✚✤❇r❣✐❏✐à③❖◗❁◆❃❆❅❦✾✹♠✏❍❆❈✦❏r❁◆●❥❃✻Ü✏Ý❯✾✬❧■✾❲❃❆✾❲❇❤❈✦❏P✾✬❈➃❣P⑦◗❣✐❏P✾✹❀ ●❋ß✻✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣ ❏Pt✤❈✦❏✬❱✹❈❋❃s❊✴✾➃❍❆❣P✾✹❅☞❏r●❘❣P●■▼❄♦■✾▲ß➯●■❇❵❏Pt★✾❹➼❆➚➹ ❅ ❣❲Ü❆❏Pt★✾❹♣★❁◆✾✜❱✠✾✹Ý✿❁❄❣P✾➃❱✠●❥❃❆❣✐❏r❈❋❃✏❏✬❱❤t✤❈❋❇r❧■✾❼❅✰✾❲❃✤❣✐❁◆❏P❁❄✾✹❣ ß➯●■❇➞✾✹❈■❱❤t✇●■ß✚❏Pt★✾❵❣P❍★❊★❁❄❃✏❏P✾✹❇P♦✦❈❋▼⑤❣✹ã✯ä✪t★✾❵❇P❁❄❧■t✏❏✐à③t❆❈■❃❆❅q❣✐❁⑤❅✰✾❉●■ß✮❏Pt★❁⑤❣➞❣P⑦✰❣⑥❏r✾❲❀ ●❋ß✮✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣➐❁⑤❣➞❈ ♦■✾✜❱Ø❏r●■❇✿●❋ßê❖◗❃★●✦Ý✿❃❪♣✴●❋❏r✾❲❃✏❏P❁⑤❈❋▼⑤❣❵❈❋❏❉❁◆❃✏❏P✾✹❇P♦✦❈■▼✞❱✠✾✹❃✏❏P✾❲❇❤❣ ✖➯❏Pt★✾❂❱✠●■▼❄▼❄●◗❱✹❈✦❏r❁◆●❥❃☞♣✴●■❁❄❃✏❏r❣ ✘Øã✿ä✪t★✾ ✩ Ñ❫❪ ❇r●✦Ý✶●❋ß✻❏rt★✾▲❀q❈❋❏P❇r❁éå✆❱❲●■❇r❇P✾✜❣✐♣✴●■❃✤❅★❣ê❏r●❹❍❆❃✰ß➯●■▼⑤❅✰❁❄❃★❧➃❏rt★✾❼❣✐❍❆❀✄❁◆❃❘❏Pt❆✾▲❱❲●■▼❄▼◆●✰❱✹❈✦❏P❁❄●■❃✆✾✹♠✏❍❆❈✦❏r❁◆●❥❃ ✂á➫➯➭✚Ð ❽ ➲ê➳➁➼✤➚➹ ✌ ➚ ✬ ❏ ➘✯➴ ➼❆➚❏ ✒✞ ✮ ✞ ✮✡✠ P ✴ ➭✴Ð ❽ ✏➥➭➻ ✴ ➽◗➾➻ ❈❋❃✤❅❪❏Pt★✾❂✾❲❃✏❏r❇P❁❄✾✹❣✬❁◆❃Ú❏Pt★✾ ❬ Ñ❫❪ ❱✠●■▼❄❍★❀q❃è❱❲●■❇r❇P✾✜❣✐♣✴●■❃✤❅★❣✪❏P●☞t★●✦Ý ❀❹❍❆❱❤ts❏Pt❆✾❂❱❤t❆❈■❇P❧❥✾✉●■❃Ú❏Pt★✾ ❬ Ñ❫❪ ❁◆❃✏❏r✾❲❇r♦②❈■▼✻❱✠●❥❃❥❏r❇P❁❄❊★❍✰❏r✾✹❣Þ❏r●q❏Pt★✾ ✩ Ñ❫❪ ♣✴●❋❏P✾✹❃✏❏P❁⑤❈❋▼Òã ä✪t★✾❯❀q❈✗￾⑥●■❇✯❅✰❁✁￾✚✾❲❇r✾❲❃✤❱✠✾ê❊✴✾✠❏⑥ÝÞ✾❲✾✹❃❼❏Pt★✾Þ❀✇❈✦❏r❇P❁◆åá❁❄❃✉❏rt★❁⑤❣✞❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑❲✾✜❅❼❣✐✾✜❱✠●❥❃❆❅◗à③❖✏❁❄❃❆❅❼✾❲å★❈❋❀q♣★▼❄✾ ❈❋❃✤❅✇❏Pt❆✾✯✚❆❇❤❣✐❏✐à③❖✏❁❄❃❆❅✆✾✠å★❈❋❀q♣★▼❄✾❵❁⑤❣✪❱✠❁❄❇r❱❲▼◆✾✜❅❦●■❃❦❏Pt★✾❼❣P▼◆❁⑤❅✰✾■ã➐ä✪t★✾✹❇P✾á❁⑤❣Þ❈■❃✆❈■❅❆❅✰❁é❏r❁◆●❥❃❆❈❋▼✚●❥❃★✾❵●❥❃ ❏Pt❆✾▲❅★❁❄❈■❧■●■❃✤❈❋▼✴●❋ß✔❏rt★✾❼❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑❲✾✹❅❘❣✐✾✜❱✠●❥❃❆❅◗à③❖✏❁❄❃❆❅✆✾✹♠✏❍❆❈✦❏r❁◆●❥❃❦❏rt❆❈✦❏❉❅✰❁⑤❅✆❃❆●❋❏❉❈❋♣★♣✴✾✹❈■❇✪❁◆❃❘❏Pt★✾ ✚❆❇❤❣⑥❏PàÒ❖◗❁❄❃❆❅❘✾✹♠✏❍❆❈✦❏r❁◆●❥❃✻ã ▼s●■❇r✾▲♣★❇r✾✹❱✠❁⑤❣P✾❲▼❄⑦■Ü ✁ ❮✄✂ Ð Ó➚✆☎✞✝ ➹➚✟☎á➳✡✠ ✌ ✁ ❰ ➹ ☛ ❮ Ñ☞✝ ➹➚✆☎✂✓ ✢✪➊✍✌ ￾➓✥✩✻➝❆➑P→✚➍✦✧✂✁✩✔➏◗➓✧➧➣★➏☎✄✖➑⑥➣★➟✝✆✏➔❵→✚➝✩✔➍❭➏◗➑P➔✟✞ ➔ ✌ ✍✏✎✒✑✔✓✖✕✁❇ n = 10 n = 20 n = 40 Answers Are Improving!!! Ô➥Õ✮ÖØ×ÚÙ❚➡ ☎❉❃❆▼◆❁❄❖■✾❯❏Pt★✾Þ❇r✾✹❣P❍★▼é❏❤❣✻ß➯❇r●■❀✷❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✹❁◆❃★❧❵❏Pt★✾✱✚❆❇❤❣⑥❏✯❖◗❁◆❃❆❅➃✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü✦♣★❇r●■❧❥❇P✾✜❣P❣P❁❄♦■✾❲▼❄⑦❵❇r✾✛✚✤❃★❁◆❃❆❧ ❏Pt❆✾á❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑✹❈❋❏P❁❄●■❃❦●■ß✻❏Pt❆✾▲❣P✾✹❱❲●■❃❆❅❦❖✏❁❄❃❆❅❘✾✹♠✏❍❆❈❋❏P❁❄●■❃❦♣★❇r●✰❅✰❍❆❱❲✾✹❣Þ❀q●■❇r✾✬❈■❱✹❱✠❍★❇❤❈✦❏r✾✬❈■❃❆❣✐ÝÞ✾❲❇❤❣✹ã ✰❃❆❱❲✾ê❈❋❧❥❈■❁◆❃✔Ü✠❏Pt❆✾ê♣★▼❄●❋❏✻❁⑤❣✔❈✪▼❄❁é❏P❏P▼❄✾êt❆❈❋❇❤❅✬❏P●❉❅✰✾✜❱✠❁❄♣★t★✾✹❇✻Ý✿❁é❏rt★●■❍★❏✔▼◆●◗●❥❖✏❁❄❃★❧✿❈✦❏✯❈✿❱✠●■▼❄●■❇✮♦❥✾❲❇❤❣✐❁❄●■❃✔ã ➦③❏❵❣Pt★●✦Ý❉❣❯❏Pt★✾➃➼➚ ➹ ❅ ❣✪♣★❇r●✰❅✰❍❆❱❲✾✹❅☞❍❆❣P❁◆❃★❧✗✜❚➳ ✳❉✵★Ü ✜Ú➳✡✠❈✵❂❈❋❃❆❅✏✜❚➳☞☛✂✵❂❣P❍★❊★❁❄❃✏❏P✾✹❇P♦✦❈❋▼⑤❣✹ãê❢★●■❇ ✾✹❈❥❱❤ts❅✰❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏r❁◆●❥❃✻Ü★❈✇♣✴●■❁❄❃❥❏❉❁⑤❣❉♣❆▼◆●■❏✐❏P✾✜❅s❈✦❏❵➼➚ ➹ Ü❆➭➹ ß➯●■❇✯✩ê➳ ✳■➺✦✓✧✓❄➺ ✜➐Ü✴❣✐●q❏rt★✾❲❇r✾✉❈❋❇r✾▲❏P✾✹❃ ♣✴●■❁❄❃❥❏❤❣❂♣★▼◆●■❏✐❏r✾✹❅❬ß➯●❥❇➃❏rt★✾❪❱✠●❥❈■❇r❣P✾✹❣✐❏❂❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏P❁❄●■❃❿❈■❃❆❅❬ß➯●❥❇✐❏⑥⑦➎♣✴●■❁❄❃✏❏r❣❂♣★▼❄●❋❏✐❏r✾✹❅❿ß➯●■❇❂❏Pt★✾ ✚❆❃★✾✜❣⑥❏❵❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏P❁❄●■❃✔Ü✏❊❆❍✰❏❵❈❋▼❄▼✔❣P✾✠❏r❣✪●■ß✞♣✴●■❁❄❃✏❏r❣❉❣P♣❆❈❋❃❘❏Pt❆✾❼❁◆❃✏❏P✾✹❇P♦✦❈■▼✮➭✷✶✹✸❁✏ ✳■➺✦✳✻✺③ã ■✶t❆❈❋❏✪❁⑤❣✿❱✠▼❄✾✹❈■❇Þß➯❇P●❥❀ ❱✠●❥❀❂♣✤❈❋❇r❁◆❃★❧➃❏Pt★✾❼❊❆▼◆❍★✾▲♣✴●■❁❄❃✏❏r❣ ✖➯❃✍✌ ë✏✎ ✘➞❏r●❹❏rt★✾▲❇P✾✜❅✆♣✴●■❁❄❃✏❏r❣ ✖✳❃✑✌▲ç ✎ ✘ ❈❋❃✤❅❼❏P●á❏rt★✾❯❧❥❇P✾✹✾❲❃❹♣✤●❥❁◆❃✏❏r❣ ✖✳❃✑✌❴✎ ✘ØÜ❋❁⑤❣✞❏Pt❆❈❋❏✯❏rt★✾✿❱❤t❆❈❋❇r❧■✾Þ❅✰✾❲❃✤❣✐❁◆❏⑥⑦✉❣✐✾✹✾❲❀✇❣✞❏P●á❊✤✾✿❈■♣★♣★❇r●❥❈■❱❤t★à ❁❄❃★❧✇❈❂❣P❀q●◗●❋❏Pts❣P●■▼❄❍✰❏P❁❄●■❃✔ã ë■ë