3 Normalized 1d problem 3.1 Basis Function Approach ()=/g(,x)(x)ds′x∈,1 Centroid collocated piecewise constant scheme 今+++++4 olaya)ds Note 3 Lets take a simple example in 1D. The domain is the segment [0, 1] of the real scheme, we divide the domain into n segments [==/.9 de on this le. We want to solve the integral equation shown at the top of the shi do ain. The t reen' s function is denoted by g(a, a. In o=0 and En=1. The charge density o is assumed to be piecewise constant on each of these intervals. The potential, y is then evaluated at the centroids ci This results in n equations in n variables, the collocation weights li, i=l,., n which can be written in matrix form. Our task is to first evaluate the entries entry of the matrix ss'quently solve the set of equations. Note that the i>th of the matrix and an integral of the treen's function, evaluated at the collocation point ci, over the interval [-ucull, which is the interval over which the basis function uis nonzero(recall that we have chosen a piecewise constant approximation). If, however, we decided to choose a different set of function and [0, 1.D Sral would be nonzero only on the support of the basis basis functions this inte ise, on the intersection of the support of the basis① ￾❼✂✁☎✄⑤③✝✠✟✝✆⑨☛✟✞ ￾▲② ✠✡✁✣❼✂☛♥✝✠☛☞✄ ✌✉❾❇❿ ➀⑥➁➃➂✙➄❇➂➆➅▲➇➉➈❬➊✰➋✑➄q➌➍➈➏➎➑➐➉➐❬➒✑➌❫➁✳➊✭➓ ✍✴→✥↔✰→✥↔ ↕✪✰✼✥✼✥✪✴✘✮✍❲✔✵✢✥✪✰✜✏✎☞✢✥✏✵✘❏✬✻✖❑✔✗✢✒✑✮✍❲✔✵✢✥✪✰✜❚✪✭★ ↔ ✎➳✌➉✤✧✦✣✍❲✔✵✢✥✪✰✜ ➙✰➛❑➜➞➝✴➟✔✓ ➪✖✕✥Ï✘✗✚✙✜✛✏✢ ✣✥✤ ✕rÏ✧✦❇Ï✘★✩✗❇➘☎✕rÏ✘★✪✗✬✫✮✭☎★ Ï✰✯✔✱✲✳✦✵✴✝✶ ❳❀✓❊❑●■◆❖❉❋✾❁❣♦❂✠❉✮❴❁❴❁❉✙❂✗❍✻●■❀✗❣♦▼✑✾❭❀✵❂✠❀✗❃❄✾❁❅❖❀❈❂✠❉✮❊✭❅q●◗❍❏❊❑●▲❅❖❂◗P✑❀✗❜❝❀ Normalized 1-D Problem Basis Function Approach Collocation Discretization of 1-D Equation ( ) ( ) ( ) 1 0 Ψ = x g x x, ′ ′ σ x dS′ " x∈[0,1] x0 = 0 xn =1 1 x n 1 x x2 − σ1 σ n 1c x 2c x nc x ( ) ( ) 1 1 , to be evaluated j i j x j x n c j x S σ g x x d − = Ψ = ! ′ ′ " !""#""$ ➪✖✕rÏ✘✷✹✸✺✗✚✙✼✻✾✽✂✺✿ ✢ ➘✂❀✛❂❁✵❃ ❁ ❃✬❄❆❅ ✤ ✕✥Ï✘✷✹✸❇✦❖Ï★ ✗✬✫✮✭★ ❈ ❉✝❊ ❋ ❒❍●✟■✹❏❑❏▼▲✝◆❇❖◗P❘◆◗❒❍❏❚❙ ➡➤➢✧➥➧➦❱❯ ➺➣❀✓●■❅✉●◗❍❏➴❋❀➵❍❦❅❇✾❁❜❝▼✑❴❁❀③❀✠Ñ✑❍❏❜❝▼✑❴❁❀❈✾❁❊ Ò➽➭✈✺➾✉P✑❀➵❣✑❉✮❜❦❍❏✾❁❊❢✾❁❅▲●❖P✑❀➭❅❇❀✗❩✮❜❝❀✓❊❑●❲✱✲✳✦❳✴❳✶✴❉❏❵✳●■P✑❀③◆■❀✗❍❏❴ ❴❁✾❭❊✑❀❋✈✞➸➤❀❄❃✉❍❏❊❑●✞●■❉③❅❇❉❋❴❭❨❋❀❫●■P✑❀❄✾❁❊❋●■❀✓❩❋◆■❍✮❴❑❀✵t❑❡✺❍✻●■✾❭❉❋❊➭❅❖P✑❉✻❃❄❊➫❍❏●✳●❖P✑❀✉●■❉✮▼➫❉❏❵✰●❖P✑❀❄❅❖❴❭✾❆❣❲❀❄❉✮❊➭●❖P✺✾❁❅ ❣❲❉❋❜❝❍✮✾❭❊✴✈✉➾✉P✺❀ ☎✎◆❖❀✗❀✓❊➣➻ ❅✉❵r❡✑❊✭❂➧●❖✾❁❉✮❊➩✾❁❅➉❣❲❀✓❊✑❉✮●❖❀✵❣♥♣❑❛ ✤ ✕rÏ❨✦❖Ï★ ✗➧✈✉❤✐❊✩❍❯❂✓❀✓❊❑●❖◆■❉✮✾❆❣♥❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊ ❅■❂◗P✑❀✓❜❝❀✮➚✣❃⑨❀❯❣❲✾❭❨✙✾❆❣❲❀❦●■P✑❀✆❣❲❉❋❜❝❍✮✾❭❊⑥✾❁❊❑●❖❉✩➷❚❅❖❀✓❩✮❜❝❀✗❊❋●◗❅❩✱Ï✂ ✦❇Ï✂✺❬ ✢ ✶ ✠❭✙❪✲✳✦❳❫✵❫❳❫✝✦❖➷➃➚✳❃❄✾❰●■P Ï ✣ ✙✥✲➵❍✮❊✺❣❦Ï✽❵❴ Ò ✈✳➾✉P✑❀➉❂◗P✭❍❏◆■❩✮❀▲❣❲❀✗❊✺❅❇✾❭●q❛❦➘❢✾❆❅➍❍❋❅❖❅❖❡✑❜❝❀✗❣❝●■❉➵♣✰❀❬▼✑✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀✎❂✠❉❋❊✺❅q●◗❍❏❊❑●❫❉❋❊ ❀✗❍❋❂◗P✆❉✮❵✞●■P✑❀✗❅❖❀❻✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅✗✈➃➾✉P✑❀❻▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴✸➚✭➪ ✾❁❅✉●■P✑❀✓❊☞❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣♦❍❏●⑨●■P✑❀③❂✠❀✗❊❑●❖◆■❉✮✾❆❣✑❅⑨Ï✘✷✹✸➧✈ ➾✉P✑✾❆❅✣◆❖❀✵❅❇❡✑❴❭●■❅✣✾❁❊➭➷✆❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✣✾❭❊➫➷❯❨✻❍❏◆■✾❆❍❏♣✑❴❁❀✗❅✗➚✛●❖P✑❀✉❂✠❉❋❴❭❴❁❉❲❂✓❍❏●❖✾❁❉✮❊➵❃⑨❀✓✾❁❩✮P❑●◗❅ ￾✃ ✦❖❐❀✙❛✴❜✦❳❫✵❫❳❫✝✦❖➷ ❃❄P✑✾❆❂◗P✷❂✗❍❏❊➩♣✰❀❝❃❄◆❖✾❭●❇●■❀✓❊➤✾❭❊➤❜❦❍✻●■◆❖✾❭Ñ❢❵r❉✮◆■❜♥✈ ✓❡✑◆➉●■❍❋❅❇➴❢✾❆❅✎●❖❉♦Ð✺◆◗❅q●❈❀✓❨✻❍✮❴❭❡✺❍❏●❖❀➭●❖P✑❀❝❀✗❊❋●■◆❖✾❁❀✗❅ ❉❏❵➍●❖P✑❀❦❜❦❍❏●❖◆■✾❰Ñ➩❍✮❊✺❣✷❅❖❡✑♣✺❅❖❀✗t❑❡✑❀✗❊❑●❖❴❁❛➩❅❇❉❋❴❭❨❋❀➵●■P✑❀❯❅❖❀✠●❈❉❏❵⑨❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✗✈✖❝❬❉❏●❖❀➫●■P✺❍✻●❈●■P✑❀❝❐✠ ❒✥❮ ❀✓❊❑●■◆❖❛ ❉❏❵❈●■P✑❀➩❜❦❍✻●■◆❖✾❭Ñ❚✾❁❅☞❍❏❊❚✾❁❊❑●❖❀✗❩✮◆◗❍❏❴❬❉✮❵❈●❖P✑❀ ☎✎◆❖❀✗❀✓❊➣➻ ❅✆❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚▲❀✗❨✛❍✮❴❭❡✭❍✻●❖❀✵❣❺❍✻●♦●❖P✑❀ ❂✠❉❋❴❭❴❁❉❲❂✓❍❏●❖✾❁❉✮❊❚▼✰❉✮✾❁❊❋●❢Ï ✷ ✸◗➚➉❉✻❨❋❀✓◆❯●■P✑❀➤✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❞✱Ï✂❜✦❖Ï✂✺❬ ✢ ✶✫➚❬❃❄P✑✾❆❂◗P➬✾❆❅♦●❖P✑❀➤✾❁❊❋●■❀✓◆■❨✻❍❏❴❻❉✻❨✮❀✗◆ ❃❄P✑✾❆❂◗P❢●❖P✺❀➵♣✺❍❋❅❇✾❆❅❄❵r❡✑❊✭❂➧●❖✾❁❉✮❊➤➱✂➵✾❁❅➉❊✑❉✮❊✺➲✓❀✓◆■❉✰❡r◆■❀✗❂✗❍❏❴❁❴➣●■P✺❍✻●❻❃➍❀➵P✺❍✛❨✮❀➵❂◗P✺❉❋❅❖❀✓❊✩❍❯▼✺✾❭❀✵❂✠❀✓❃❄✾❆❅❖❀ ❂✠❉❋❊✺❅❇●■❍❏❊❑●➵❍✮▼✑▼✑◆■❉✛Ñ✙✾❁❜❦❍✻●■✾❭❉❋❊❵❢➧✈♦❤✫❵q➚✞P✑❉✻❃⑨❀✓❨❋❀✓◆✵➚✴❃⑨❀❯❣❲❀✗❂✓✾❁❣✑❀✗❣➤●❖❉✷❂◗P✑❉✙❉❋❅❖❀❝❍❢❣✑✾❰➯✹❀✓◆■❀✓❊❑●➭❅❖❀✠●➵❉✮❵ ♣✺❍❋❅❇✾❆❅➍❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚✙●❖P✑✾❆❅✉✾❭❊❑●■❀✓❩✮◆◗❍❏❴✹❃⑨❉✮❡✑❴❆❣❯♣✰❀❻❊✑❉✮❊✺➲✓❀✓◆■❉➵❉❋❊✑❴❁❛❯❉✮❊✆●❖P✑❀❈❅❖❡✑▼✑▼✰❉✮◆❖●✉❉✮❵✴●■P✑❀❻♣✺❍✮❅❖✾❁❅ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊✺❅❣❡✥❉✮◆✵➚❄●❖❉ ♣✭❀➤▼✑◆■❀✗❂✓✾❁❅❖❀✮➚▲❉❋❊❚●❖P✺❀✷✾❁❊❑●❖❀✓◆◗❅❖❀✗❂➧●■✾❭❉❋❊❺❉❏❵➵●❖P✑❀⑥❅❇❡✺▼✑▼✭❉❋◆❇●☞❉❏❵③●■P✑❀➤♣✺❍✮❅❖✾❁❅ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊❢❍❏❊✭❣❭✱✲✳✦✵✴✝✶✪❢➧✈ ➼