Finite difference discretization of Elliptic equations: ID Problem ecture 2 and 3
✂✁☎✄✆✁✞✝✠✟☛✡☞✁✍✌✎✟✑✏✒✟✓✄✕✔✑✟☛✡☞✁☎✖✗✔✑✏✒✟✘✝✗✁☎✙✛✚✜✝✗✁✣✢✤✄ ✢✦✥★✧✪✩☎✩☎✁☎✫✬✝✠✁✣✔✭✧✯✮✱✰✲✚✜✝✠✁✣✢✤✄✆✖✛✳✵✴✜✡ ✶✷✏✒✢✤✸✕✩✣✟✓✹ ✺✻✟✑✔✼✝✠✰✕✏✒✟✓✖✾✽✿✚❀✄❂❁❄❃
1 Model problem 1.1 Poisson Equation in 1D Boundary Value Problem(BVP) (x)=∫(x) (0,1),u(0)=(1)=0,f Describes many simple physical phenomena(e.g) Deformation of an elastic bar Deformation of a string under tension Temperature distribution in a bar The Poisson equation in one dimension is in fact an ordinary differ tion. When dealing with ordinary differential equations we Poisson equation will be used here to illastrate numerical techniques for elliptic PDE's in multi-dimensions. Other techniques specialized for ordinary differen tial equations could be used if we were only interested in the one dimension Note 1 Poisson equation The Poisson equation(in R)is elliptic, per our classification. It is also coercive, nd symmetric(these concepts will be defined more precisely in the Finite Element lectures). These attributes are very important as regards umerical treatment. These properties are reflected in the fact(see first lecture) that the eigenvalues of -V-v are real and positive We denote by cm, more precisely, cm([0, 1]), the set of functions f(a): [0, 1]+ IR with continuous m derivat ives. Thus. cu denotes the set of continuous func Isly, Ck CC for k>
❅ ❆❇✬❈❊❉✛❋❍●❏■✼❇▲❑❊❋▼❉✛◆ ❖✼P◗❖ ❘❂❙❯❚◗❱❲❱❳❙❯❨✾❩❊❬❪❭✜❫✠❴❵❚❛❙❯❨❜❚◗❨❝❖✒❞ ❡❣❢❳❤❥✐❧❦✂♠ ♥▲♦❣♣✠q✗r✠s✉t✇✈✂①✤s❵②③♣✗④✎⑤✦t✇♦⑦⑥⑧②⑨④❲⑩❷❶❛♥▲①✆⑤❹❸ ❺❼❻❣❽❾❽❵❿➁➀➃➂❯➄✯➅⑧❿➁➀❣➂ ➆➈➇ ➀❊➉➊❿③➋❵➌ ➇ ➂☎➌➍❻⑧❿⑨➋➎➂✘➄✪❻⑧❿ ➇ ➂✼➄✷➋❵➌➏➅✎➉➑➐➓➒ ➆→➔ ↔❀↕➛➙✞➜✣➝✞➞➠➟❣↕➛➙❪➡➤➢➦➥❲➧➑➙➨➞➩➡✱➫➯➭➠↕✤➫❵➲❲➧✉➙➨➞➩➜➛➢➦➭➃➫❵➲❵↕➛➥❵➳➵➡➸↕▼➥⑦➢ ➆❀➣ ❿ ↕➵➺ ➻➯➺ ➂☎➼ ➽ ↔❀↕✣➾➁➳➎➝➨➡➤➢➦➚➨➞➩➳➵➥✕➳➦➾✼➢➦➥✕↕▼➭➪➢➵➙◗➚➨➞➪➜❀➟⑦➢➦➝ ➆✛➶ ➽ ↔❀↕✣➾➁➳➎➝➨➡➤➢➦➚➨➞➩➳➵➥✕➳➦➾✼➢✱➙◗➚➨➝✞➞➩➥❵➻➸➹❵➥➯➘✉↕➛➝❪➚➨↕▼➥⑦➙◗➞➩➳➵➥ ➆→➴ ➽➬➷↕▼➡➸➫❣↕▼➝✍➢➮➚✞➹❵➝➨↕✤➘✉➞➪➙◗➚➨➝✞➞➠➟❵➹❵➚➨➞➩➳➵➥❂➞➠➥❊➢✱➟➯➢➵➝ ➆❀➱ ✃➃❐❵❒→❮❼❰➮Ï➪Ð✍Ð▼❰➮ÑÒ❒✍Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ❊Ï❥Ñ✎❰➦Ñ❣❒➈Ø➦Ï❥Ù➸❒▼Ñ❵Ð☎Ï⑨❰➮Ñ✎Ï➪Ð→Ï❥Ñ➈Ú✣Õ➎Û✣Ö✓Õ➦Ñ✎❰➮Ü➨Ø➦Ï❥Ñ❣Õ➦Ü☎Ý➑Ø➮ÏÞ✛❒▼Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮ß❧❒✍Ó✣Ô➯Õ➦à Ö×Ï⑨❰➮Ñ❣á❝â✬❐❵❒▼Ñ★Ø➎❒✍Õ➮ßãÏ❥Ñ❲ä❍å✼Ï❥Ö➁❐✯❰➮Ü➨Ø➦Ï❥Ñ❣Õ➮Ü☎ÝæØ➮ÏÞ✛❒▼Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮ß✦❒✞Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ➯Ð❊å❯❒✎å✼Ï❥ß❥ß✦Õ➮ßçÐ▼❰❍Ô❲Ð▼❒✲Ö➁❐❵❒ èêéÜ☎Ï❥Ù➤❒❳ë✛Ñ❣❰➦ÖìÕ➦Ö③Ï⑨❰➦Ñ✆Öí❰❹Ï❥Ñ❣Ø➮Ï⑨Û✍Õ➦Öì❒✦Ø➮ÏÞ❀❒✣Ü➨❒▼Ñ➯Ö×Ï⑨Õ➮Ö×Ï⑨❰➮Ñ➃á➤✃❣❐✉Ô❲Ð✍î ❻❽❹ï ❻➃ð î ❻❽❾❽ ➄ñ❻➃ð ð î✼❒✣ÖíÛ➛á✱✃➃❐❵❒ ❮❼❰➦Ï➪Ð✞Ð▼❰➦Ñò❒✍Ó✣Ô➯Õ➦Ö③Ï⑨❰➦Ñ✲å✼Ï❥ß❥ß✠ó✍❒➈Ô❲Ð✣❒✍Ø❹❐➯❒✣Ü✞❒✤Öí❰➸Ï❥ß❥ßãÔ❲Ð☎Ö③Ü✞Õ➮Öí❒✤Ñ➯Ô✉Ù➤❒▼Ü☎Ï⑨Û✞Õ➦ß✒Öí❒✞Û✍❐❲Ñ⑦Ï⑨Ó✣Ô➯❒✣Ð✼Ú✣❰➮Ü❹❒▼ß❥ßãÏéÖ×Ï⑨Û ❮✼ô✦õ➸öÐ➈Ï❥ÑòÙ▲Ô✉ßãÖ×Ï❥à◗Ø➮Ï❥Ù➤❒▼Ñ❵Ð☎Ï⑨❰➮Ñ➯Ð✣á✬÷❪Ö❥❐➯❒✣Ü➈Öí❒✍Û✞❐✉Ñ➯Ï⑨Ó▼Ô➯❒☎Ð✤Ðé❒✍Û✣Ï⑨Õ➦ßãÏ➠ø❾❒✞Ø✦Ú✣❰➦Ü▲❰➮Ü✞Ø➮Ï❥Ñ❣Õ➦Ü☎Ý❂Ø➦ÏÞ✛❒▼Ü➨❒✣Ñ⑦à Ö×Ï⑨Õ➮ß✑❒✍Ó▼Ô➯Õ➮Ö×Ï⑨❰➮Ñ➯Ð✬Û✍❰➦Ô✉ß➠ØÒó✍❒➑Ô❲Ð▼❒✞Ø✎ÏÚ➤å❯❒➑å❯❒✣Ü✞❒✕❰➦Ñ➯ßãÝ✎Ï❥Ñ➯Öí❒✣Ü➨❒✣Ð☎Öì❒✍Ø✲Ï❥Ñ❍Ö➁❐❵❒❂❰➮Ñ➃❒✆Ø➦Ï❥Ù➸❒▼Ñ❵Ð☎Ï⑨❰➮Ñ➃Õ➮ß Û✍Õ✇Ð▼❒❾á ùòú➓û☎ü✎ý þ➈ú❲ÿ✁✂➵ú☎✄✯ü☎✆✞✝✠✟➓û◗ÿ❛ú☎✄ ➷➲❵↕☛✡⑧➳➵➞➪➙➨➙➨➳➵➥✤↕✌☞➎➹⑦➢➮➚➨➞➩➳➵➥ ❿ ➞➠➥✎✍✏✒✑ ➂ ➞➩➙✗↕▼➭➩➭➩➞➠➫✉➚✞➞➩➜✔✓➦➫❣↕▼➝✠➳➵➹❵➝⑧➜✣➭➪➢➵➙✞➙◗➞✖✕⑦➜➛➢➮➚➨➞➩➳➵➥❧➺✗✍ì➚✠➞➪➙✠➢➦➭➪➙◗➳✛➜▼➳❲↕▼➝✍➜✣➞✙✘➵↕✔✓ ➳➵➝❧➫⑦➳❳➙◗➞➠➚➨➞✙✘➵↕✑➘✉↕✚✕➯➥➯➞ê➚✞↕✔✓➦➢➦➥⑦➘✤➙◗➧❲➡➸➡➸↕✣➚➨➝✞➞➪➜ ❿ ➚✞➲❵↕➛➙➨↕✓➜▼➳➵➥➯➜▼↕▼➫✉➚✍➙✗✛❼➞➩➭➠➭❳➟⑦↕✑➘✉↕✚✕⑦➥❵↕➛➘✤➡➸➳➎➝➨↕✘➫➯➝➨↕❾➜✣➞➪➙◗↕➛➭➠➧ ➞➩➥✬➚➨➲➯↕✢✜✠➞➩➥❵➞ê➚✞↕✤✣✘➭➩↕▼➡➸↕▼➥❳➚✓➭➩↕➛➜✣➚➨➹❵➝✞↕➛➙ ➂ ➺ ➷➲❵↕❾➙◗↕✦➢➮➚➨➚➨➝✞➞➠➟➯➹✉➚➨↕❾➙✑➢➵➝➨↕✥✘➵↕➛➝➨➧✱➞➠➡➸➫❣➳➵➝➨➚✞➢➵➥➎➚❼➢➵➙✓➝✞↕▼➻➎➢➵➝✞➘➯➙ ➥❲➹❵➡➸↕▼➝✞➞➩➜➛➢➦➭❳➚➨➝✞↕➛➢➦➚➨➡➸↕▼➥❳➚❾➺ ➷➲❵↕❾➙◗↕❪➫❵➝✞➳➵➫❣↕▼➝➨➚➨➞➩↕➛➙✼➢➵➝➨↕✑➝✞↕✚✦➯↕❾➜☎➚➨↕❾➘❹➞➩➥✱➚➨➲❵↕✑➾⑨➢➎➜☎➚ ❿ ➙➨↕▼↕☛✕➯➝✍➙◗➚⑧➭➩↕➛➜✣➚➨➹❵➝✞↕ ➂ ➚➨➲⑦➢➮➚❼➚➨➲➯↕✤↕▼➞➩➻➵↕▼➥☎✘➮➢➦➭➩➹❵↕❾➙✓➳➵➾ ❺★✧✑✪✩ ➢➦➝✞↕→➝✞↕➛➢➵➭✒➢➦➥➯➘✆➫❣➳➎➙➨➞ê➚✞➞✖✘➎↕➵➺ ùòú➓û☎ü✬✫ ➐✗✭ ✚✮✯✟✠✰➎ü✱ ✲ò↕✦➘❵↕▼➥❵➳➵➚➨↕✦➟❲➧ ➐✳✭ ✓✉➡➸➳➵➝✞↕❀➫➯➝➨↕❾➜✣➞➪➙◗↕➛➭➠➧✴✓ ➐✗✭✱❿✶✵➋❵➌ ➇✚✷ ➂ ✓❳➚➨➲❵↕✤➙➨↕✣➚❼➳➦➾✗➾➁➹❵➥➯➜✣➚➨➞➩➳➵➥➯➙ ➅⑧❿⑨➀❣➂❪➼✸✵➋❵➌ ✍✏✺✛❼➞➠➚➨➲❊➜▼➳➵➥❳➚➨➞➩➥❲➹❵➳➵➹➯➙☛✻ ➘❵↕▼➝✞➞✖✘➮➢➮➚✞➞✖✘➎↕➛➙➛➺ ➇✚✷✠✹ ➷➲❳➹⑦➙✪✓ ➐➓➒ ➘✉↕▼➥➯➳➦➚➨↕❾➙✑➚➨➲❵↕➈➙➨↕✣➚✜➳➵➾✠➜▼➳➵➥❳➚➨➞➩➥❲➹❵➳➵➹➯➙✑➾➁➹➯➥➯➜✽✼ ➚➨➞➩➳➵➥⑦➙▼➺✿✾❀➟☎✘❲➞➠➳➎➹➯➙◗➭➩➧✔✓ ➐❁❀❃❂❍➐✗✭ ➾➁➳➎➝★❄❆❅❇✻ò➺ ➇
Note 3 Green's function For this problem, the solution u can be written explicitly as (, y)f(y) where G(a, y) is the Green's function given by G(0)={01-) y o show this, we start by recalling that for any function which is twice differ entiable, there are constants CI C2, such that u()=G1+/u'(y)dy C If u satisfies the one dimensional Poisson equation, then C u(a)=C1+C2a f(z)dz dy f(a)dxdy Ly F(y)l F(y)dy F() yf(y)d g (a-y)f(y)dy, by proper attention to dummy variables. Finally, we obtain the general solution
❈❊❉✳❋✽●■❍ ❏▲❑✴●☎●✱▼❖◆P▲◗❙❘❚▼✯❯☎❋✶❱❲❉☎▼ ❳❩❨✴❬❪❭❴❫❩❵❜❛❞❝❩❬❡❨✔❢✸❣✖❤✂✐✬❥☎❭❡❫❩❤❦❛❴❨✔❣✙❧♠❭❴❵✙❨✔♥✬♦q♣✪r✔♥s❢✠❤✤t❞❬❴❵✖❭✶❭❡❤✪♥✉❤✚✈♠❝❩❣✙❵❜♣✚❵✖❭❴❣✙✇sr✔❛ ♦②①④③✠⑤✿⑥⑧⑦⑩⑨ ❶❸❷①❹③❁❺❴❻❩⑤✶❼❽①❹❻❩⑤✶❾✔❻ t❞❫❩❤✂❬❴❤ ❷ ①❹③✞❺✶❻❩⑤☛❵❜❛❪❭❴❫❩❤❃❿✢❬❡❤✪❤✂♥✗➀ ❛☛➁❹❧❩♥✸♣✽❭❡❵✖❨✴♥✉➂✔❵✙➃✔❤✪♥✬❢☎✇ ❷ ①❹③❁❺❴❻❩⑤✿⑥➅➄ ❻✳①❲➆❞➇❊③✠⑤ ➈❲❼ ➉▲➊➋❻s➊⑩③ ③②①❲➆❞➇❊❻♠⑤ ➈❲❼ ③■➊❇❻❆➊➌➆ ➍✞❨s❛❴❫❩❨➎t❸❭❴❫❩❵❜❛✂❥✳t➏❤➐❛❲❭❙r✱❬❴❭✥❢☎✇✉❬❴❤✌♣✪r✱❣✙❣✙❵✖♥❩➂❆❭❴❫✸r✱❭★➁❹❨✴❬❖r✔♥☎✇❆➁❹❧❩♥❚♣✽❭❴❵✙❨✔♥❊t❞❫❩❵❜♣❙❫q❵❜❛★❭❲t❞❵❜♣✚❤➐➑♠❵✖➒✠❤✂❬✶➓ ❤✪♥➔❭❡❵✙r✔❢❩❣✖❤✴❥☎❭❴❫❩❤✂❬❴❤❦r✔❬❴❤❖♣✚❨✴♥✸❛✶❭❡r✱♥➔❭❙❛✒→ ⑨ r✔♥✸➑✉→➏➣✴❥✸❛❴❧✸♣❙❫s❭❴❫✸r✱❭ ♦❽①❹③↔⑤↕⑥ → ⑨✯➙ ⑦➋➛ ❶ ♦✠➜➝①❹❻❩⑤❲❾✴❻ ♦➜ ①❹❻❩⑤➞⑥ →➣ ➙ ⑦➋➟ ❶ ♦➜ ➜ ①✁➠➔⑤✶❾✴➠❚➡ ➢➤➁❽♦q❛❴r✱❭❴❵❜❛❲➥✸❤✌❛☛❭❴❫❩❤✤❨✴♥❩❤✤➑♠❵✙✐➐❤✂♥✸❛❴❵✖❨✴♥✸r✱❣✗➦❽❨✴❵✙❛❡❛✶❨✴♥❆❤✌➧✴❧❚r➎❭❴❵✙❨✔♥❁❥☎❭❴❫❩❤✂♥ ♦➜ ①❹❻❩⑤✯⑥➌→➣ ➇ ⑦➋➟ ❶ ❼❽①✁➠➔⑤✶❾✴➠❚➡ ➍❪❫❩❤✂❬❴❤✪➁❹❨✔❬❡❤✔❥ ♦②①④③✠⑤✿⑥✺→ ⑨ ➙ →➣ ③➨➇ ⑦➋➛ ❶➫➩⑦➋➟ ❶ ❼❽①④➠☎⑤❲❾➔➠✔➭✉❾✴❻✠➡ ➯✥❤✚➥✸♥❩❵✙♥❩➂ ➲ ①❹❻❩⑤✯⑥⑧⑦➳➟ ❶ ❼❽①④➠☎⑤❲❾➔➠❚❺ t☛❤✢❨✴❢✸❛❴❤✪❬❡➃✔❤✥❭❴❫✸r✱❭ ⑦➛ ❶ ➩⑦➟ ❶ ❼❽①✁➠➔⑤✶❾✴➠✴➭■❾✔❻ ⑥ ⑦➛ ❶ ➲ ①❹❻❩⑤✶❾✔❻ ⑥ ➵❻➲ ①❹❻❩⑤➤➸ ➛❶ ➇ ⑦➳➛ ❶ ❻➲ ➜➝①❹❻❩⑤❲❾✴❻ ⑥ ③➲ ①❹③✠⑤✯➇ ⑦➋➛ ❶ ❻❩❼❽①❹❻❩⑤✶❾✔❻ ⑥ ⑦➛ ❶ ①④③➨➇➺❻❩⑤❴❼❽①❹❻❩⑤❲❾✴❻↔❺ ❢☎✇❦❝❩❬❡❨✔❝✠❤✪❬✯r✱❭✶❭❴❤✂♥➔❭❴❵✙❨✔♥➐❭❴❨❦➑♠❧✸✐➐✐✎✇❃➃➎r✱❬❡❵❜r✱❢❩❣✙❤✂❛✂➻✞❳②❵✙♥✸r✱❣✙❣✖✇✴❥✔t☛❤❞❨✔❢♠❭❙r✱❵✙♥➐❭❴❫❩❤❞➂✴❤✪♥❩❤✂❬❡r✔❣❩❛✶❨✴❣✖❧♠❭❡❵✖❨✴♥ ❵✙♥s❭❡❫❩❤❖➁❹❨✔❬❡✐ ➼
(a-uf(y)dy For our particular problem we can now impose the boundary conditions u(0) a(1)=0 to determine the constants CI and C2. Thus, after some arithmetic, u(a) (1-x)f(y)y+/x(1-y)f(y)d u(a)=G(a, g)f(y)dy We note that G(a, y) has the following properties IS ·G(x,y)≥0 for all z,y∈(0,1 G is a piecewise linear function of a for fixed y and vice versa The particular form of expressing the solution, in terms of the green function will be revisited when we address the topic of integral equation Note 4 Consider an elastic bar of unit length which is fixed at both ends and subjected to a tangential load per unit length p(a) p() u(a) da
➽❽➾❹➚✠➪➏➶➘➹❞➴✯➷⑩➹➏➬✂➚➨➮➳➱➳✃ ❐ ➾❹➚➨➮❊❒❩➪✶❮❽➾④❒♠➪✶❰✔❒↔Ï Ð❩Ñ✴Ò➏Ñ✴Ó❩Ò☛Ô✸Õ✱Ò❴Ö❴×❜Ø✚Ó❩Ù❜Õ✱Ò➏Ô❩Ò❡Ñ✔Ú❩Ù✙Û✪Ü➅Ý☛Û✢Ø✂Õ✱Þ➨Þ❩Ñ➎Ýß×✙Ü▲Ô❚Ñ➔à✶Û✥Ö❴á✸Û✥Ú✠Ñ✔Ó❩Þ❚â❩Õ✱Ò❡ãäØ✚Ñ✔Þ❚â♠×åÖ❡×✖Ñ✴Þ✸à ➽②➾✁æ✴➪ç➶ ➽❽➾❲è✌➪ç➶➌æ Ö❴Ñ➨â♠Û✪Ö❴Û✪Ò❡Ü▲×✖Þ✸Û✥Ö❡á❩Û❦Ø✚Ñ✴Þ✸à✶Ö❡Õ✱Þ➔Ö❙à ➹❪➴ Õ✔Þ✸â ➹☛➬✱é✯êá☎Ó✸à✂ë✸Õ➎ìíÖ❡Û✪Ò✒à❴Ñ✔Ü▲Û✤Õ✱Ò❡×åÖ❡á❩Ü▲Û✚Ö❴×❜Ø✱ë ➽②➾④➚✠➪✿➶ ➱✃ ❐ ❒✳➾✶è❞➮➺➚↔➪✶❮❽➾❹❒❩➪✶❰✔❒❦➷ ➱➋î ✃ ➚✞➾✶è✒➮➺❒❩➪❴❮❽➾❹❒❩➪❲❰✴❒↔ï Ñ✔Ò ➽❽➾❹➚↔➪✯➶ ➱ ➴ ❐⑧ð➾④➚❁ï✶❒❩➪❴❮❽➾❹❒❩➪❲❰✴❒↔Ï ñ❊Û✤Þ❩Ñ✱Ö❡Û✥Ö❡á✸Õ➎Ö ð ➾④➚❁ï❴❒♠➪ á✸Õ✔à☛Ö❡á❩Û❖ì❹Ñ✔Ù✙Ù✖Ñ➎Ý❞×✙Þ❩ò➐Ô❩Ò❡Ñ✔Ô✠Û✪Ò❴Ö❴×✙Û✂à✂ó ô ð ×❜à✒Ø✚Ñ✴Þ✴Ö❡×✖Þ☎Ó❩Ñ✴Ó✸à✂ë ô ð ×❜à✒à✶ã☎Ü▲Ü▲Û✚Ö❡Ò❴×❜Ø✢Û é ò é ð ➾❹➚✞ï✶❒❩➪✿➶ ð ➾❹❒↔ï✶➚↔➪ ë ô ð ➾④➚❁ï❴❒♠➪❞õ❇æ ì❹Ñ✔Ò✒Õ✔Ù✖Ù ➚✞ï✶❒✬ö❊➾✁æ❩ï✂è✌➪ ë ô ð ×❜à✒Õ➐Ô❩×✖Û✌Ø✚Û✂Ý❞×✙à❴Û❖Ù✖×✙Þ❩Û✂Õ✔Ò❪ì❹Ó❩Þ✸Ø✚Ö❴×✙Ñ✔Þ✉Ñ✱ì ➚ ì❹Ñ✔Ò❪÷❩ø♠Û✌â ❒ Õ✱Þ✸âsù☎×✙Ø✪Û✤ù✔Û✪Ò❙à❡Õ é êá❩Û✤Ô✸Õ✔Ò✶Ö❡×✙Ø✪Ó❩Ù❜Õ✱Ò❪ì❹Ñ✔Ò❡ÜúÑ✔ì②Û✪ø♠Ô❩Ò❴Û✌à❴à❴×✙Þ❩ò➐Ö❴á❩Û❦à❴Ñ✔Ù✙Ó♠Ö❴×✙Ñ✔Þ❁ë❩×✖Þ✉Ö❡Û✪Ò❡Üäà❞Ñ✱ì②Ö❴á❩Û➐û✢Ò❴Û✂Û✪Þsì❹Ó❩Þ✸Ø✽Ö❡×✖Ñ✴Þ✗ë Ý❞×✙Ù✖Ù✗Ú✠Û✤Ò❡Û✪ù☎×✙à❴×✖Ö❴Û✂âsÝ❞á❩Û✂ÞsÝ☛Û❦Õ✔â❩â❩Ò❴Û✌à❴à➏Ö❴á✸Û❖Ö❴Ñ✔Ô✸×✙Ø❖Ñ✔ì✞×✙Þ✴Ö❡Û✪ò✴Ò❡Õ✔Ù✠Û✌ü➔Ó✸Õ➎Ö❡×✖Ñ✴Þ✸à é ý❊þ✳ÿ✁✄✂ ☎✝✆✟✞✡✠✴ÿ☞☛✍✌✏✎✑✞✓✒ ✔Ñ✔Þ❚à✶×❜â♠Û✪Ò☛Õ✱ÞäÛ✂Ù✙Õ✴à❲Ö❡×✙Ø★Ú✸Õ✱Ò➏Ñ✱ì✗Ó❩Þ❩×✖Ö➏Ù✙Û✪Þ✸ò✱Ö❴á❆Ý❞á❩×✙Ø❙á❆×✙à✿÷❩ø♠Û✂â➨Õ✱Ö➏Ú✠Ñ✱Ö❡á➨Û✪Þ✸â✸à➏Õ✔Þ✸â➨à❴Ó❩Ú✡✕❲Û✂Ø✚Ö❴Û✌â Ö❴ÑäÕ✎Ö❡Õ✔Þ❩ò✔Û✂Þ➔Ö❴×❜Õ✱Ù✳Ù✙Ñ✴Õ✔âsÔ✠Û✪Ò✒Ó❩Þ✸×åÖ✒Ù✙Û✪Þ❩ò✔Ö❴á✗✖➾④➚✠➪✽é ✘
tively. From horizontal equilibrium we have gential displacement at a, respec- Let o(a) and u(a) be the axial stress and tan Aco -Ac(o +do Under the assumption of small displacements and a linearly elastic material we have where e is the modulus of elasticity and Ac is the area of the bar cross section Differentiating the constitutive equation and combining the two equations to eliminate o' we obtain the Poisson equation with f=p/(EAc) Note 5 String under transversal load Consider a string of unit length under tension T, which is subjected to a trans verse distributed load of magnitude p(a) per unit length p(a) da 6+d8 Let u(a) denote the transverse displacement at point a. Assuming small dis ements, so that the tension T can be taken as constant over the whole string and considering vertical equilibrium we have T(6+d6)-T6=pd The angle 8 can be related to the displacement u simply as dau Note the minus sign which is due to the fact that a positive u corresponds to a downwards displacement. Combining the two equations to eliminate the variable 8 we obtain the Poisson equation with f= p/T
✙✛✚✢✜✤✣✦✥★✧✑✩✤✪✬✫✮✭✰✯✱✥✲✧✑✩✴✳✑✚✵✜☞✶✓✚✷✪✹✸✡✺✻✪✬✼✾✽✿✜☞❀❁✚❂✽❁✽✴✪❃✫✮✭✰✜❁✪✬✫✮❄❃✚✢✫❅✜❁✺❆✪❃✼❇✭✓✺❆✽☞❈✓✼✻✪❃❉❊✚❂❋●✚✢✫❅✜✤✪✹✜❍✧❏■✓❀❁✚❂✽☞❈✑✚❂❉✁❑ ✜☞✺❆▲❃✚❂✼◆▼P❖✦◗✓❀❁❘❃❋❙✶✮❘❃❀❁✺◆❚❂❘❃✫❅✜❁✪❃✼✑✚❱❯❅❲✓✺◆✼❆✺❆✳✓❀☞✺❆❲✓❋❨❳✴✚✵✶✮✪❩▲❃✚ ❬✝❭ ✣✄❪ ❬❫❭ ✥✲✣✰❴❛❵P✣✛✩❝❜❡❞❢❵❃✧ ❣✤✫❤✭✡✚✢❀✴✜☞✶✮✚✵✪P✽☞✽☞❲✓❋●❈✡✜❁✺◆❘P✫✐❘✬❥❦✽✿❋❧✪✬✼❆✼❇✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❁✽♠✪❃✫✮✭✰✪♥✼◆✺❆✫✓✚❱✪✬❀❁✼◆▼●✚✢✼✻✪❃✽✿✜☞✺✻❉✝❋●✪✬✜☞✚❂❀☞✺✻✪✬✼✑❳♦✚ ✶✮✪❩▲P✚ ✣❢❜q♣ ✥★✯✗❴❛❵P✯❇✩❦❪r✯ ❵P✧ s ✥✲t✉❘✈❘❃✇❃✚P① ✽♠✼✻✪❩❳✉✩ ❳❍✶✓✚❂❀☞✚❫♣②✺✻✽③✜❁✶✓✚✝❋●❘✡✭✡❲✓✼❆❲✮✽✴❘✬❥④✚✢✼✻✪❃✽✿✜☞✺✻❉❊✺◆✜⑤▼●✪❃✫✮✭ ❬❫❭ ✺❆✽❝✜☞✶✮✚❫✪❃❀☞✚❱✪⑥❘✬❥✾✜❁✶✓✚❫✳✮✪✬❀♠❉❊❀❁❘P✽❁✽❝✽☞✚❂❉❊✜☞✺❆❘❃✫✛❖ ⑦✝✺⑨⑧❇✚✢❀❁✚✢✫❅✜☞✺✻✪✹✜❁✺◆✫✮❄❢✜☞✶✮✚✄❉✢❘❃✫✮✽✿✜☞✺◆✜☞❲✡✜❁✺◆▲P✚⑩✚❱❯❅❲✮✪✹✜❁✺◆❘P✫❛✪✬✫✮✭❛❉❊❘❃❋♥✳✓✺❆✫✓✺◆✫✮❄❢✜☞✶✮✚✰✜⑤❳✴❘❢✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✮✽❶✜☞❘ ✚✢✼❆✺❆❋✗✺❆✫✮✪✬✜☞✚✷✣✾❷✛❳✴✚✵❘❃✳✡✜❸✪✬✺❆✫✰✜❁✶✓✚⑥❹❦❘❃✺✻✽☞✽☞❘❃✫⑩✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✄❳❍✺⑨✜❁✶✏❺❻❜❼❞✾❽✡✥❾♣❬✝❭ ✩✁❖ ❿r➀✾➁✁➂❢➃ ➄❝➁✿➅❩➆❾➇❏➈➊➉❤➇③➋✱➂✬➅❢➁✿➅✹➌✡➇✱➍❅➎✡➂✬➅❩➍❱➌✡➏❍➏⑤➀✡➌❇➋ ➐❘❃✫❤✽✿✺✻✭✡✚✢❀♠✪✗✽⑤✜❁❀☞✺❆✫✓❄❶❘❃❥❏❲✓✫✓✺◆✜❍✼◆✚❂✫✓❄✬✜❁✶✰❲✓✫❤✭✡✚✢❀✴✜☞✚❂✫✮✽☞✺◆❘P✫✰➑⑥■✈❳❍✶✓✺✻❉❸✶✰✺✻✽♠✽✿❲✓✳✓➒⑤✚❂❉✁✜❁✚❂✭✐✜☞❘●✪⑥✜❁❀❁✪❃✫✮✽⑤❑ ▲❃✚❂❀❁✽☞✚✵✭✡✺❆✽✿✜☞❀❁✺❆✳✓❲✡✜☞✚❱✭✄✼◆❘❅✪❃✭⑩❘✬❥④❋❧✪❃❄❃✫✓✺◆✜☞❲✮✭✓✚❍❞④✥★✧❇✩✴❈❤✚❂❀✤❲✓✫✓✺◆✜✤✼❆✚✢✫✓❄❃✜☞✶✛❖ ✙✛✚✢✜⑥✯✱✥✲✧✑✩✷✭✓✚✢✫✓❘❃✜☞✚●✜☞✶✮✚❧✜☞❀❸✪✬✫✮✽☞▲❃✚❂❀❁✽☞✚●✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❶✪✹✜⑥❈✑❘❃✺❆✫❅✜❶✧❏❖✄➓✉✽❁✽✿❲✮❋✗✺❆✫✓❄❢✽☞❋❧✪✬✼❆✼❝✭✡✺✻✽⑤❑ ❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❁✽❂■✹✽✿❘✉✜☞✶✮✪✬✜❏✜❁✶✓✚❝✜❁✚✢✫✮✽☞✺❆❘❃✫⑥➑❛❉❂✪✬✫⑥✳✑✚③✜❸✪✬✇P✚✢✫❶✪P✽④❉✢❘❃✫✮✽✿✜❁✪❃✫P✜④❘✹▲❃✚✢❀❏✜☞✶✓✚✴❳❍✶✓❘P✼◆✚✴✽⑤✜❁❀☞✺❆✫✓❄✮■ ✪✬✫❤✭⑩❉✢❘❃✫✮✽☞✺❆✭✓✚✢❀❁✺◆✫✓❄✗▲❃✚❂❀✿✜❁✺❆❉❂✪✬✼✾✚❱❯P❲✮✺◆✼❆✺◆✳✮❀☞✺❆❲✓❋❙❳✴✚✵✶✮✪❩▲❃✚ ➑❶✥★➔✉❴❛❵❧➔P✩❦❪r➑✤➔⑥❜❡❞✏❵P✧❏→ ➣♠✶✓✚✷✪✬✫✮❄❃✼❆✚❫➔❧❉❂✪✬✫✄✳❤✚✷❀❁✚✢✼✻✪✹✜❁✚❂✭✰✜☞❘●✜☞✶✮✚⑥✭✡✺❆✽☞❈✓✼✻✪❃❉✢✚✢❋●✚✢✫❅✜❍✯❢✽✿✺❆❋●❈✓✼❆▼⑩✪❃✽ ➔❶❜②❪ ❵❃✯ ❵❃✧ → ↔✤❘❃✜☞✚✏✜❁✶✓✚❢❋●✺❆✫❅❲❤✽⑩✽☞✺◆❄P✫↕❳❍✶✓✺❆❉❸✶➙✺✻✽✰✭✓❲✓✚❢✜❁❘❡✜☞✶✓✚❢❥✲✪❃❉✁✜✰✜☞✶✮✪✬✜✄✪❡❈✑❘P✽☞✺⑨✜❁✺◆▲P✚✏✯➛❉✢❘❃❀❁❀☞✚❱✽✿❈✑❘❃✫✮✭✮✽ ✜☞❘❢✪✏✭✓❘✹❳❍✫❅❳♠✪✬❀❸✭✓✽✵✭✡✺✻✽✿❈✮✼❆✪P❉❊✚✢❋●✚❂✫P✜❱❖ ➐❘❃❋♥✳✓✺❆✫✓✺◆✫✮❄❻✜☞✶✓✚●✜⑤❳✴❘✏✚❂❯❅❲✮✪✬✜☞✺❆❘❃✫✮✽❫✜❁❘✏✚✢✼❆✺◆❋●✺❆✫✮✪✹✜❁✚❧✜☞✶✓✚ ▲✹✪✬❀❁✺❆✪❃✳✓✼❆✚✝➔●❳✴✚❫❘P✳✡✜❁✪❃✺◆✫✄✜❁✶✓✚⑥❹❦❘❃✺✻✽☞✽☞❘❃✫✰✚❂❯❅❲✮✪✹✜❁✺◆❘P✫⑩❳❍✺◆✜☞✶❢❺❻❜❡❞✾❽❩➑⑥❖ ➜
Temperature distribution Let u(a) and q(a) denote the temperature and heat flux in a homogeneous heat conducting bar of unit length. The bar is subjected to a distributed volumetric heat source p(a)and the temperature is maint ained at zero at the end points; the sides of the bar are assumed insulated so that the heat flow is one-dimensional cT(e) Acp da (q dq The stationary temperature distribution can be obt ained by considering the energy balance Ac(q+dq)-Acq= Acp d and the empirical relation between the temperature and the heat flux In the above equations, k is the heat conductivity and Ac is the bar cross ectional area. Defining f=p/k and eliminating q from the above equations 1. 1.1 Solution Properti SLIDE 2 The solution u(a) always exists ·u(x) is always“ smoother"than the data f(x) (see first lecture). In particular, if f has m continuous derivatives, u will have n+2 continuous derivatives. Thus, if Eco, thenuEC2 ·If(x)≥0 for all ax, then a(x)≥0 for all az Follows from the positivity of Greens function us≤(1/8川fls Given f(a) the solution u(a)is unique Continuous stability estimate ecall that for a function u: Q-R
➝r➞✾➟✁➠➢➡ ➤③➠✬➥r➦✉➠✬➧✹➨✾➟❂➩❤➧P➠✏➫❇➭❾➯P➟☞➧❱➭☞➲✱➩④➟✿➭⑤➞✈➳ ➵✛➸✢➺♦➻❦➼★➽❇➾③➚✬➪❤➶✐➹✡➼★➽❇➾❝➶✡➸❂➪✓➘✬➺❁➸❍➺☞➴✮➸✤➺☞➸❂➷●➬❤➸❂➮❁➚✬➺☞➱✓➮❁➸✝➚❃➪✮➶●➴✓➸❂➚✬➺❝✃✮➱✓❐●❒❆➪✰➚✷➴✮➘❃➷●➘❃❮P➸✢➪✓➸❂➘❃➱✮❰✦➴✮➸❂➚✹➺ Ï➘P➪✮➶✡➱Ï➺☞❒❆➪✓❮✗Ð✮➚✬➮✴➘✬Ñ❏➱✮➪✓❒⑨➺❍Ò❆➸✢➪✓❮❃➺☞➴✛Ó③Ô♠➴✮➸✝Ð✮➚❃➮♦❒✻❰♠❰✿➱✮Ð✡Õ⑤➸Ï➺☞➸❂➶✰➺☞➘●➚❶➶✓❒❆❰✿➺☞➮❁❒◆Ð✮➱✡➺☞➸❱➶✰ÖP➘❃Ò❆➱✓➷●➸❊➺☞➮❁❒ Ï ➴✓➸❱➚✹➺④❰☞➘❃➱✓➮ Ï➸✾×✱➼★➽❇➾✛➚✬➪❤➶❫➺☞➴✓➸❝➺☞➸✢➷●➬✑➸✢➮❸➚✹➺❁➱✓➮☞➸✦❒✻❰✛➷❧➚❃❒◆➪❅➺❁➚❃❒◆➪✮➸❂➶✷➚✹➺④Ø✢➸✢➮❁➘✉➚✬➺✛➺☞➴✓➸❝➸✢➪✮➶✷➬✑➘❃❒❆➪❅➺❁❰❂Ù❩➺☞➴✓➸ ❰☞❒❆➶✡➸❱❰✴➘✬Ñ✛➺❁➴✓➸✝Ð❤➚✬➮❍➚✬➮❁➸✝➚P❰☞❰☞➱✓➷●➸❂➶✰❒◆➪✮❰☞➱✓Ò✻➚✹➺❁➸❂➶✰❰☞➘⑥➺❁➴✮➚✹➺♠➺❁➴✓➸✵➴✓➸❂➚✬➺♦✃✮➘✹Ú➙❒✻❰✴➘❃➪✓➸✢ÛÜ➶✓❒◆➷●➸✢➪❤❰✿❒❆➘❃➪✮➚❃Ò❾Ó Ô♠➴✓➸➢❰✿➺❁➚✬➺☞❒❆➘❃➪✮➚❃➮☞ÝÞ➺☞➸✢➷●➬✑➸✢➮❸➚✹➺❁➱✓➮☞➸❢➶✓❒❆❰✿➺☞➮❁❒◆Ð✮➱✡➺☞❒❆➘❃➪ Ï➚✬➪➙Ð✑➸❢➘❃Ð✡➺❸➚✬❒❆➪✓➸❂➶↕Ð✈Ý Ï➘❃➪❤❰✿❒✻➶✡➸✢➮❁❒❆➪✓❮❼➺☞➴✓➸ ➸✢➪✮➸✢➮❁❮❃Ý❧Ð✮➚❃Ò❆➚❃➪Ï➸ ß✝à ➼❾➹❍á❛â❅➹✬➾❦ã ß❫à ➹✵ä ß✝à×❢âP➽❏å ➚✬➪❤➶✐➺❁➴✓➸✷➸✢➷●➬✓❒❆➮☞❒ Ï➚❃Ò✾➮☞➸❂Ò❆➚✬➺☞❒❆➘❃➪✄Ð❤➸✢➺⑤Ú♦➸❂➸✢➪⑩➺☞➴✓➸✵➺❁➸✢➷●➬✑➸✢➮❸➚✹➺☞➱✮➮☞➸⑥➚❃➪✮➶✐➺❁➴✓➸✷➴✓➸❱➚✹➺♠✃✮➱✡❐ ➹✷äæã✉ç✈è✝éëê ➼✲ì✮➘❃➱✓➮❁❒◆➸❂➮❂í ❰✴Ò❆➚❩Ú✉➾ î✍➪q➺❁➴✓➸❼➚✬Ð✑➘✹Ö❃➸➢➸❱ï❅➱✮➚✹➺❁❒◆➘P➪✮❰✢ð❫ç➙❒✻❰✰➺❁➴✓➸r➴✓➸❂➚✬➺ Ï➘❃➪✮➶✡➱Ï➺☞❒❆Ö✈❒⑨➺⑤Ýq➚❃➪✮➶ ßà ❒❆❰⑩➺☞➴✮➸ñÐ✮➚❃➮ Ï➮❁➘P❰❁❰ ❰☞➸Ï➺❁❒◆➘P➪✮➚✬Ò③➚❃➮☞➸❱➚✓Ó❧ò✝➸❊ó✮➪✓❒❆➪✓❮➢ôõäö×❇÷❃ç➢➚❃➪✮➶r➸✢Ò❆❒◆➷●❒❆➪✮➚✹➺❁❒◆➪✓❮❢➹✰Ñ★➮❁➘❃➷ø➺❁➴✓➸✐➚✬Ð✑➘✹Ö❃➸●➸❂ï❅➱✮➚✬➺☞❒❆➘❃➪✮❰ Ú✴➸❫➘PÐ✡➺❁➚❃❒◆➪⑩➺☞➴✮➸⑥ù❦➘❃❒✻❰☞❰☞❒◆➘P➪✰➸❱ïP➱❤➚✹➺☞❒❆➘❃➪❏Ó ú✑û✲ú✑û✲ú ü❏ý❤þ❾ÿ✁✄✂✲ý✆☎✞✝✠✟❩ý☛✡✌☞✍✟✎✏✂✑☞✍✒ ✓ ✔✍✕✗✖✁✘✚✙ ✛ Ô♠➴✓➸⑥❰☞➘❃Ò❆➱✡➺☞❒❆➘❃➪✄➻✱➼✲➽✑➾♠➚❃Ò◆Ú♠➚❩Ý✡❰ ☞✍✜✢✂✑✒✣✄✒ ✛ ➻✱➼✲➽✑➾✴❒✻❰✤➚✬Ò❆Ú♠➚❩Ý✈❰✥✤✒✄✦Þý✛ý☛✏✧★☞✩✟✫✪ ➺❁➴✮➚✬➪⑩➺☞➴✮➸⑥➶✓➚✹➺❸➚●ô❦➼✲➽✑➾ ✬✮✭✣✯✰✯✲✱✴✳✵✭✷✶✹✸✺✯✵✻✼✶✮✽✾✳✿✯❁❀❃❂❅❄✷❆❈❇❊❉✫✳✷✶✮❋●✻✣✽✾✸✺❉✫✳✵❍✹❋■ ô❑❏ ❉✫✭▼▲◆✻✵❖P❆❊✶✮❋✗❆❊✽❊❖✫✽◗✭✥❘✩✯✼✳✷❋✗❙✫❉✫✶✮❋✗❙✎✯✼✭✵❍ ➻❯❚❋✗✸✗✸ ❏ ❉✫❙✫✯ ▲ á❲❱ ✻✵❖✫❆☛✶✑❋✗❆☛✽❊❖✫✽◗✭❳❘❃✯✣✳✷❋✗❙✎❉P✶✑❋✗❙✫✯✷✭✣❂❩❨❏✽◗✭✰❍✲❋■ ô❭❬❫❪❵❴ ❍✴✶❏ ✯✣❆ ➻❛❬❜❪✢❝ ❂ ✛ îÜÑ✦ô❦➼✲➽✑➾❡❞❯❢♥Ñ★➘❃➮✤➚✬Ò❆Ò✛➽④ð✡➺☞➴✓➸❂➪❻➻✱➼✲➽✑➾❡❞❯❢✗Ñ★➘❃➮✤➚✬Ò❆Ò✾➽ ❣❖✫✸✗✸✺❖❚✭✌■✵✳✿❖✫❤✐✶❏ ✯❡❇❊❖✎✭✷❋✗✶✮❋✗❙✏❋✗✶✮❥❫❖❦■❈❧❡✳✿✯✵✯✼❆❵♠✭♥■✵✽✾❆✆✻✣✶✑❋●❖P❆✆❂ ✛❜♦✺♦ ➻ ♦✺♦ ♣rq ➼❁s✹÷✫t❅➾ ♦✉♦ ô ♦✉♦ ♣ ✈✠✇ ✛②①❒◆ÖP➸✢➪✏ô❦➼★➽❇➾✴➺☞➴✓➸⑥❰☞➘❃Ò❆➱✡➺❁❒◆➘P➪⑩➻❦➼★➽❇➾✴❒❆❰ ÿ★☎③✂✑④✾ÿ③☞ ✈t ➝r➞✾➟✁➠⑥⑤ ⑦❶➞✡➳✦➟✿➭❾➳③➩✾➞✛➩✮➯❧➯❅➟☞➨✛➲❤➭✑⑧✲➭✿➟●⑨➊➠✬➯❅➟✿➭❾➥❛➨✾➟✁➠ ⑩r➸✷➮☞➸ Ï➚✬Ò❆Ò✑➺☞➴✮➚✬➺❍Ñ★➘❃➮✤➚♥Ñ★➱✓➪Ï➺☞❒❆➘❃➪✏➻❛❶✩❷✞❸ î❹ ❺
where Q is the domain of definition. For example, the -norm of the functions z, z(1-x), eva and sin(T), in the interval Q=[0, 1] is 1, 1/4, e and 1, respectively. non-negative we have u()≤/G(x,y)f()y≤川fGx,y)4=12(1-) l=sup,(x)≤ll This estimate is a consequence of the fact that the solution u depends cor tinuously on the data f. In other words, we can say that if f is small so is Note 8 Solution uniqueness Uniqueness of the solution follows directly from the above estimate. If we have two solutions u, and u2 which satisfies the Poisson problem for a given f, we ave that uf-u2=(u1-u2"=0. This implies that u1-u2 sat isfy the Poisson problem for f =0. Thus, we can use the above stability estimate to show that J01-u2llo0=0. Therefo (We note that the be reached by integrating(u1 -u2)"=0 twice and imposing the appropriate boundary conditions.) 2 Numerical solution 2.1 Finite differences 2.1. 1 Discretization Subdivide interval(0, 1)into n+1 equal subintervals △ 0
❻✉❻ ❼♥❻✺❻ ❽❿❾➁➀❦➂➄➃ ➅❃➆❃➇ ❻ ❼★➈●➉✆➊✣❻➌➋ ➍❅➎➄➏✏➐✿➏❭➑➓➒ ➀❈➔➎➄➏❭→✾➣✩↔❈↕P➒✉➙❯➣❃➛✠→➄➏✼➜❊➙➄➒➔ ➒✉➣❃➙✁➝➟➞➄➣✩➐➠➏✼➡➄↕❃↔➃❊➢➏✩➤ ➔➎➄➏ ❻✉❻➄➥✢❻✉❻ ❽❜➦ ➙➄➣✩➐✿↔➧➣P➛ ➔➎➄➏ ➛➂ ➙❊➨➔ ➒✉➣❃➙➀✲➉ ➤ ➉★➈❁➩✴➦➫➉➭➊ ➤✾➯P➲➅ ↕P➙❊→ ➀ ➒✉➙ ➈➵➳❵➉➭➊ ➤✾➒✉➙ ➔➎➄➏➸➒✉➙➔ ➏✏➐✿➺✫↕➢ ➑❯➻➽➼➾ ➋✣➩✣➚ ➒ ➀➪➩ ➤ ➩✎➶✎➹ ➤✩➯✠↕P➙☛→ ➩ ➤✾➐✰➏➀✿➃ ➏✏➨➔ ➒✺➺✩➏➢✉➘➝ ➴ ➒✉➙❊➨✼➏❳➷➽➒ ➀ ➙➄➣❃➙✾➬➮➙➄➏✏➱✩↕➔ ➒✺➺✩➏➸➍✹➏✠➎❊↕✎➺❃➏ ❻ ❼★➈●➉✆➊✣❻✾✃❒❐❯❮ ❰ ➷➈➵➉③➋❦Ï➄➊✣❻ ÐÑ➈➵Ï➄➊✏❻ Ò✩Ï❫✃Ó❻✺❻ Ð♥❻✉❻ ❽ ❐❯❮ ❰ ➷➈➵➉③➋❦Ï➄➊❁Ò✩ÏÔ❾Õ❻✉❻ Ð♥❻✉❻ ❽ ➩ Ö ➉③➈❦➩❅➦×➉➭➊✷Ø Ù➎➄➏✏➐✿➏✣➛➵➣❃➐✰➏ ❻✉❻ ❼♥❻✺❻ ❽ ❾ ➀✿➂➄➃ ➅❃➆✾Ú❰✣Û ❮ÝÜ ❻ ❼★➈●➉✆➊✏❻◗✃ ➩ Þ ❻✺❻ Ð♥❻✺❻ ❽ Ø Ù➎➄➒ ➀ ➏ ➀❦➔ ➒✉↔❈↕➔ ➏❜➒ ➀ ↕⑥➨✣➣❃➙➀ ➏✏ß➂ ➏✏➙❊➨✼➏➫➣P➛ ➔➎❊➏➫➛●↕❃➨➔❈➔➎❊↕➔❈➔➎➄➏ ➀➣➢✉➂✾➔ ➒✉➣❃➙ ❼ →✾➏➃ ➏✏➙❊→➀ ➨✣➣❃➙✾➬ ➔ ➒✉➙➂ ➣➂☛➀❦➢✉➘ ➣❃➙ ➔➎➄➏❜→❊↕➔ ↕ Ð ➝✚àÝ➙á➣➔➎➄➏✣➐②➍✹➣❃➐✵→➀ ➤♥➍✹➏❫➨✏↕P➙ ➀ ↕➘✚➔➎❊↕➔ ➒✺➛ Ð ➒ ➀Ô➀↔❈↕➢✉➢❡➀➣×➒ ➀ ❼ ➝ â✚ã❵ä✷å❛æ ç♥ã✾è✿é③ä❦êÝã✾ëìé❊ë★êÝí③é✢åPë✴åPî✄î ï➙❊➒✉ß➂ ➏✏➙➄➏➀✰➀ ➣P➛ ➔➎➄➏ ➀➣➢✉➂✾➔ ➒✉➣❃➙❫➛➵➣➢✉➢➣✫➍➀ →✾➒✉➐✰➏✏➨➔✰➢✺➘ ➛➵➐✿➣✩↔ ➔➎➄➏ð↕❃ñ☛➣✫➺✩➏➸➏ ➀❁➔ ➒✺↔❈↕➔ ➏❃➝♥à➮➛★➍✲➏✠➎❊↕✎➺✩➏ ➔➍✹➣ ➀➣➢✉➂✾➔ ➒✉➣❃➙➀➪❼ ❮ ↕P➙❊→ ❼✆ò ➍❅➎➄➒ó➨✵➎ ➀ ↕➔ ➒ ➀➜❊➏➀➸➔➎➄➏❈ôÑ➣❃➒ ➀✿➀➣❃➙ ➃ ➐✿➣✩ñ➢➏✏↔õ➛➵➣❃➐ð↕❫➱❃➒✉➺❃➏✣➙ Ð ➤❵➍✲➏ ➎❊↕✎➺✩➏ ➔➎❊↕➔♥❼➭ö ö ❮ ➦÷❼✆öòö ❾➽➈➵❼ ❮ ➦✥❼➭ò✏➊❦ö ö✆❾ ➾❊➝ Ù➎➄➒ ➀ ➒✉↔➃❊➢ ➒✉➏➀★➔➎❊↕➔♥❼ ❮ ➦÷❼✆ò❡➀ ↕➔ ➒ ➀ ➛➘ð➔➎❊➏❡ôÑ➣✩➒ ➀✰➀➣❃➙ ➃ ➐✰➣❃ñ➢➏✣↔r➛➵➣❃➐ Ð➫❾ ➾❊➝ Ù➎➂❊➀ ➤➄➍✲➏❳➨✏↕P➙ ➂❊➀ ➏ ➔➎➄➏❳↕❃ñ☛➣✫➺✩➏ ➀❦➔ ↕❃ñ➄➒ ➢ ➒➔❁➘ ➏ ➀❁➔ ➒✺↔❈↕➔ ➏ ➔➣ ➀➎➄➣✫➍ ➔➎❊↕➔ ❻✉❻ ❼ ❮ ➦❲❼ò ❻✺❻ ❽ ❾ ➾❊➝ Ù➎➄➏✣➐✰➏✼➛➵➣✩➐✿➏✩➤ ❼ ❮ ❾Õ❼ò ➈➵ø➏❈➙➄➣➔ ➏ ➔➎❊↕➔ð➔➎➄➏ ➀ ↕P↔Ô➏❈➨✼➣❃➙☛➨➢✉➂❊➀ ➒✉➣❃➙✚➨✣↕❃➙ ñ✆➏②➐✰➏✏↕✩➨✵➎➄➏✏→ùñ➘ ➒✺➙➔ ➏✣➱❃➐✵↕➔ ➒✺➙❊➱ ➈●❼ ❮ ➦❑❼ò ➊❁ö ö✴❾ ➾ ➔➍❅➒✉➨✣➏❈↕P➙❊→×➒✺↔➃ ➣➀ ➒✺➙❊➱ ➔➎➄➏➠↕➃➄➃➐✿➣➃ ➐✿➒ó↕➔ ➏ ñ✆➣➂ ➙❊→❊↕P➐➘ ➨✼➣✩➙❊→✾➒➔ ➒✉➣❃➙➀ ➝ ➊ ú ûýü❛þÿ✁✄✂✆☎✞✝✠✟☛✡✌☞✍✟ü✏✎ ✂✆☞✒✑ ✓✕✔✗✖ ✘✚✙✗✛✜✙✣✢✥✤✧✦★✙✣✩✠✤✫✪✬✤✭✛✯✮✰✤✲✱ ✳✫✴✶✵✷✴✶✵ ✸✺✹✶✻✽✼✿✾❁❀❃❂❄✹❆❅❈❇✥❂✽✹✶❉✷❊ ❋✷●❃❍❏■✲❑▼▲ ➴➂ ñ➭→✾➒✉➺◗➒✉→✾➏ð➒✉➙➔ ➏✏➐✿➺✫↕➢✴➈ ➾ ➋✏➩✄➊ ➒✺➙➔ ➣✚◆✚❖ ➩ ➏✏ß➂ ↕➢✢➀✿➂ñ➄➒✉➙➔ ➏✏➐✿➺✫↕➢ó➀ P➉❜❾ ➩ ◆✚❖ ➩ ◗
du w J F yy n FR5se ut, W vel owe ke-cC w1 Wuy wl ew-IR5se We 51 yeg lodf W=1yz-W: e7Wgw Toss erWgt'ydomy, maw nffifn FI FoFVE mO/// 4 A 7e(Cqr-18Wn5171 W=lRze Tol= Wow f Edu. tIo cs ea w T r,2 1 y j
❘✬❙✯❚❱❯❳❲❨❘✭❩ ❭❪❙✜❫❴❭❪❙✯❵☛❭❜❛✶❘✥❙✽❝ ❬ ❞❢❡❈❣✐❤✒❥ ❯ ❥❧❦✚♠☛♥ ♦q♣✬rts✈✉❙①✇③②❁④⑤④r⑤⑥⑧⑦②s✷⑨❁⑥✒⑩② ⑩❢♣✬r❷❶❁❸❁❹❻❺❪r✆⑥ ②❽❼ ❸ ✇③②s❪⑩⑧❾❏s❪❺② ❺❳⑥ ❼❺✥s✇ ⑩⑧❾ ②s✈✉ ❛❢❘✷❝⑤❿ ❸❁⑩➀⑦② ❾❏s❪⑩ ❘✬❙❈❿ ➁r ➁❾❏❹❏❹➂s ② ⑩➀➃✚❸❄➄➅r❨❸❁s➇➆❷⑨✿❾➈⑥⑤⑩❆❾ ✇ ⑩❆❾ ② s➊➉③r✆⑩➁r③r✆s❷✉❙ ❸❁s✰⑨➋✉ ❛✶❘✥❙➌❝❄➍ ♦✈r ➁❾❏❹❏❹➂❺❳⑥tr ❭❪❙ ❬ ⑩② ⑨❈rts② ⑩✣r ⑩❢♣✬r❷❸➎⑦➅⑦④➏②❄➐❾❏➃✚❸❁⑩⑧❾ ②s➑⑩②✍❭❪❙❈➍ ♦✈r ➁❾❏❹❏❹➒❺❳⑥✆r✍⑩❢♣✬r✒❺✥s✰⑨❈r ④⑥ ✇③②✿④r✍⑩② ❾❏s✷⑨✿❾ ✇❸❁⑩✣r❷❶➌r✇ ⑩②✿④t➍✈➓♣✥❺❳⑥ ❿ ✉ ⑨❈rts② ⑩✣r✆⑥➋⑩❢♣✬r❨❶❁r✇ ⑩②❁④➋➔ ✉ ❙✿→✿➣③↔❪❙✆↔✷↕✫➍ ➙✫➛✶➜✷➛➈➙ ➝✌➞✭➞❜➟❁➠❳➡✫➢✶➤➦➥✥➧✽➢✶➠✷➨ ➩✷➫❃➭❏➯✲➲➵➳ ➸❡➅❣✞➺✆➻✬➼✿➽❷➾✬➚➪➺❷➶t➶❄➶ ✉❳➹ ➹ ❛❢❘✥❙❁❝ ❫ ♥ ❲✍❘ ❛✉❳➹ ❛❢❘❙➏➘✭➣✗➴➎➷ ❝➒➬ ✉❃➹ ❛❢❘❙✆➮➂➣➏➴➎➷ ❝➏❝ ❫ ♥ ❲✍❘ ❛ ✉ ❙➏➘❜➣ ➬ ✉ ❙ ❲✍❘ ➬ ✉ ❙ ➬ ✉ ❙✆➮✫➣ ❲✍❘ ❝ ❚ ✉ ❙➏➘✭➣ ➬✃➱✉ ❙ ♠ ✉ ❙✆➮✫➣ ❲✍❘➷ ❞❢❡➅❣ ❲❨❘➊❐➽✚➼✿➚➪➚ ❒ ➃②❁④r ❼✆②❁④➃✚❸❁❹✕⑨❈r ④❾❏❶➌❸✿⑩❆❾ ② s ②✗❼ ⑨❁❾❮✯r ④rts✇r✺❸➎⑦❈⑦ ④➏②t➐❾❏➃✚❸❁⑩⑧❾ ②s❪⑥q⑩②❨❼❺✥s✇ ⑩❆❾ ② s❱⑨❈r ④❾❏❶➌❸✿⑩❆❾❏❶❁r✆⑥ ➁❾❏❹❏❹❜➉③r ✇③②s✬⑥⑤❾✶⑨➅r ④ ❹❰❸✿⑩Ïr ④t➍ ➙✫➛✶➜✷➛➪Ð ÑÓÒ➂Ô✭➥✬➧❄➢✶➠✷➨❜Õ ➩✷➫❃➭❏➯✲➲▼Ö ➬✞❭✰×❄×Ø❚★Ù ❐➏Ú✬Û❈Û➺ ❐✗Ü➎❐ ➶t➶❄➶ ➬ ❭❬ ❙➏➘✭➣ ➬Ý➱✭❬❭❙ ♠ ❭❬ ❙✆➮➂➣ ❲✍❘➷ ❚★Ù➒❛❢❘❙ ❝ ♥➋❥ ❯ ❥❱❦ ❭✷Þß❚à❬ ❬ ❭✰↕✿➘✭➣✞❚ ❤ ❚✫á âã❬❭ ❚✧Ù ä
&du i b P aui) 2aU2) a WW盏 a(能 u aqui csthn2trint ∈Rl 66 for an≠,( n is SPD) arAxi Matrix Properties We give below the definition of some matrix classes and their main properties w sthn2trin Positiv2 D2finit2 (SPDt We sa that a matrix A sitive definite if M> for an non-zero vector For s mmetric matrices this condition is e Divalent to rehiring that all the elg es of the matrix be positive To show this we note that if n is s, mmetric and has real coet cients, it can bwwwritten as M=e6e, where 6 is the diagonal matrix of eigenvalues and e is an orthonormal transformation Iwe_A sinve e is non-Singtlar), impies br>, for an -#, or an. all the greater than zero bviousl, an matrix which is i s b is also non-sing ular and therefore invertiblE Ci=e 6 ele t Dieronel Dominent ft atv W that a matrix M diagonal dominant if
å✷æ❃ç❏è✲é▼ê ë★ì í î❨ï✷ð ñò ò ò ò ò ò òó ô õ í ö ÷❄÷t÷ ö õ í ô õ í ø ø ø ø ø ø ö ø ø ø ø ø ø ø ø ø ö ø ø ø ø ø ø õ í ô õ í ö ÷t÷t÷ ö õ í ô ùtúú ú ú ú ú ú ûýü ÿþ ì ñò ò ò ò òó ÿ✁ þÿþ ð ø ø ÿ✄✂✆☎ þ ø ÿ✄✂þ ùtúú ú ú ú û ü ✝ ì ñò ò ò ò òó ✝✟✞ï ✡✠ ✝✟✞ï ð ✠ ø ø ✝✟✞ï ✂✆☎ ø ✠ ✝✟✞ï ✂☛✠ ùtúú ú ú ú û ☞✍✌✏✎✁✑✒✑✔✓✖✕✘✗✚✙✜✛✆✢ ë✤✣✦✥✧✂✩★✪✂ ÿþ ü ✝ ✣✦✥✧✂ ✫✁✬✜✭✩✬✯✮ ✌✱✰✳✲✵✴✱✕✶✙✜✰✩✷ å✷æ❃ç❏è✲é✹✸ ✺✘✻ ë ✷✼✰✩✷✱✽ ✻ ✙✜✷✿✾✳✴✿✲✜❀✪✗❂❁ ❃❅❄❇❆❉❈❋❊✆● ❍ ì❏■ ❍ ü ❍ ð ü ❑✘❑✡❑ ü ❍ ✂✩▲✶▼ ❍▼ ë ❍ ì í î✍ïð ✞❍ ð ❖◆ ✂ P◗❘ ð ✞❍ ◗ õ ❍ ◗ ☎ ✘✠ ð ◆ ❍ ð ✂ ✠ ❙❉❚✡❊☛❯❱❚ ❍▼ ë ❍ ❲ ö❅❳✆❨❄❇❆❉❈❋❊✆●❩❍ ❭❬ ö ✞ë❫❪❵❴ ✌✱❛❝❜✠ ❞❢❡ ë ÿþ ì ✝ ❣ ÿþ ✓✖❤✁✙ ✻✕ ✻✐❈❥❊☛❦ ❪❵❴ ✴✿✷✿✙✜❧✏✴✿✓ ❞ í❄ö ♠♦♥✏♣rqts ✉✇✈✏♣②①④③⑥⑤⑧⑦❝①❇♥❥⑨❢q✚①✆♣⑩③❶q❋❷ ❸❖❹❅❚✐❺❻❈✚❼⑩❆ ❪✯❽✚ë ❹☛❈❴ ❈❾❊✖❿☛❺❾➀✩❚✘❆❖❄❨✱➁❆⑩❄➁ ❚✡❆②❼ ❪❚ ❴ ❼⑩❹✳❈✚❼❖➂❪✯➃❵➃ ➀✳❚➄❚❽ ➁ ➃❄❪❼⑩❚✶❦ ❪❊❩❼⑩❹☛❚✐❈❥❊☛❈➃●❴②❪➅❴ ➆♦❚❝➇ ø ❪❵➈❚✐➀✳❚➃❄✚➂➉❼⑩❹❅❚➊❦❅❚❱➋☛❊❪❼ ❪❄❥❊✦❄❨ ❴❄❥❺➌❚➄❺➌❈❋❼⑩❆ ❪➍❽ ❯➃❈❴✍❴❚ ❴ ❈❥❊☛❦❂❼⑩❹❅❚❪❆➎❺❻❈❪❊ ➁ ❆✍❄➁ ❚✘❆⑩❼ ❪❚❴ ø ✌✁✎✁✑✔✑✔✓✖✕✘✗✚✙✜✛➏❛➐✰✻ ✙✜✕✡✙⑥➑✩✓➒❜➓✓❇➔✟✷✿✙⑥✕✡✓✔☞✍✌✁❛❝❜→✢ ➆♦❚ ❴❈④●✔❼⑩❹☛❈❋❼❂❈➣❺❻❈✚❼⑩❆ ❪✯❽⑧↔ ❪❵❴ ➁ ❄❴⑩❪❼ ❪❵➈❚➒❦✪❚✘➋☛❊❪❼⑩❚ ❪❨ ❍▼↔❍ ❲ ö♦❨❄❥❆➏❈❥❊✆●✒❊❅❄❇❊✪↕➛➙✡❚✘❆✍❄ ➈❚✶❯r❼✍❄❥❆➎❍ ø ❃☛❄❥❆ ❴●✆❺➌❺➜❚✘❼⑩❆ ❪❯➄❺❻❈❋❼⑩❆ ❪❯✘❚❴ ❼✍❹❪❵❴ ❯❱❄❇❊☛❦❪❼ ❪❄❥❊ ❪➅❴ ❚✡➝✖❿❪❵➈❈➃❚✘❊✖❼❉❼⑩❄❻❆✍❚✡➝✖❿❪ ❆ ❪❊❅➇➌❼✍❹☛❈✚❼ ❈➃❵➃ ❼✍❹❅❚➊❚❪➇❥❚✡❊➈❈➃❿☛❚❴ ❄ ❨ ❼✍❹❅❚➊❺❻❈✚❼✍❆❪✯❽ ➀✩❚ ➁ ❄❴⑩❪❼ ❪✯➈❚ ø ❸✱❄ ❴❹❅❄✚➂➉❼⑩❹❪➅❴ ➂➞❚➊❊❅❄❋❼✍❚➄❼⑩❹☛❈❋❼ ❪❨ ë❫❪❵❴ ❴●✖❺➌❺➌❚❱❼✍❆❪ ❯❝❈❋❊✳❦❂❹✳❈❴ ❆✍❚✡❈➃ ❯❱❄✆❚❱➟❩❯❪❚✡❊❇❼ ❴ ❳ ❪❼❢❯✡❈❋❊✦➀✩❚➊➂➎❆❪❼②❼✍❚✘❊➒❈❴➐↔ ì➉➠▼✱➡➠ ❳ ➂➎❹❅❚✡❆⑩❚ ➡ ❪➅❴ ❼⑩❹☛❚➢❦❪❈❋➇❇❄❥❊☛❈➃ ❺❻❈✚❼⑩❆ ❪✯❽ ❄ ❨ ❚❪➇❥❚✘❊➈❈➃❿❅❚ ❴ ❈❋❊☛❦ ➠➤❪❵❴ ❈❋❊→❄❥❆⑩❼⑩❹☛❄❥❊❅❄❇❆⑩❺❻❈➃ ❼⑩❆➥❈❋❊❴ ❨❄❥❆✍❺❻❈✚❼ ❪❄❥❊ ❳ ❪ ø ❚ ø ➠ ☎ ì➦➠▼ ø ❸❖❹❅❚✘❊ ❳ ❍ ▼↔❍ ì ❍ ▼ ➠▼➡➠❍ ì➉➧▼ ➡➧ ❲ ö❻❨❄❥❆❉❈❋❊✆●❂❍ ❭❬ ö ✞❄❇❆➐❈❋❊✆● ➧ ì❫➠❍ ❭❬ ö ❴②❪❊☛❯❱❚ ➠➨❪❵❴ ❊❅❄❥❊❅↕ ❴⑩❪❊☛➇❥❿➃❈❋❆ ✠ ❳ ❪❺ ➁ ➃❵❪❚ ❴ ❼⑩❹✳❈✚❼✐❈➃✯➃ ❼⑩❹❅❚❾❚✘❊✖❼⑩❆ ❪❚ ❴❉❪❊ ➡ ❺➢❿❴❼✐➀✳❚ ➇❥❆✍❚✡❈❋❼⑩❚✡❆➩❼⑩❹☛❈❥❊➏➙✡❚✘❆✍❄ ø✟➫➀➈✆❪❄❥❿❴⑩➃● ❳ ❈❥❊✆●➢❺❻❈✚❼✍❆❪✯❽ ➂➎❹❪ ❯➥❹ ❪❵❴❖➭✪➯➳➲➉❪➅❴ ❈➃➅❴❄➊❊☛❄❥❊✪↕ ❴②❪❊❅➇❇❿➃❈❥❆➵❈❋❊✳❦ ❼⑩❹☛❚✘❆✍❚❨❄❥❆✍❚ ❪❊➈❚✡❆②❼ ❪➀➃❚ ❳ ↔☎ ì➉➠➡ ☎ ➠▼ ø ❜➓✙✜❀❅✾✩✰✳✷✿❀❅✲➞❜❂✰✩✑✔✙✵✷✼❀❅✷✄✕ ➆♦❚ ❴❈④●➌❼✍❹☛❈✚❼➐❈➜❺❻❈✚❼✍❆❪✯❽➓↔ ì✤■✶➸◗➻➺ ▲ r➼ ◗ ➽ ➺ ➽ ➼ ✂ ❪➅❴ ❦❪❈❥➇❥❄❇❊☛❈➃❵➃●❩❦❅❄❥❺❪❊✳❈❋❊✖❼ ❪❨ ➾
uP rat ictl b. Fire iww pint d s ira vidp ictl b- leys t we ra p pat m dix iw wpPidple di g/c. e dmic.cpl c t wafs c Ka pypdlidple di g/c Ie d/mic.cp Cidt we f wt dlt ra pfbdm pixf iwc pvpidple di gfc. Ie df m ofd ds wytxdt ppat ndyp cdF vpyrat tl b- He a/Edw,.pid wra p dt di gfc.He dmie·cp: cd vbda ra p w ofo· pwI.vp fct afs rat ict1面 e aiww pivntd ic Tt Ge Fa: pf iw. di g/c He fmic cpm pix ic iadtdbdi E ofdnl o I -matrix Cm函 xj iwC班d·c,- mix io ip w pint w up dc. t vafs c 101 1 pdm·x· FFpat tcpditw c/c-ctg pilt cbm tdl pAixyvic dt rat. fIt vidp ictI b Hire iwe p wpivntd fdt Ttde ds1H/ s dyip d c·Rf· t vafs c ra pio Fat F yp ictl b.ire… fft iwat fat d e. c tl b Fireycd j iw. c iadt dbd. F di g/c. Fe d/mic.cpm pixyratc.F c /c-ctg' pi Tire fo Fat dftffiditcpw fof -fi ur Nit ww tis ys t s iiF aflt pat rat dtffiditcpw dt cfc-ctg pill Thomas’ algorithn G bvi c tFimic pic dc. t tffiditcple cJc-vic gbFa pidi e ptm Jo pat bvicg pat ofE s icg. Igfdiram
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