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 △ 0❻✉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 ➩ ◆✚❖ ➩ ◗