kontsyslos090109

kontsyslos090109 - LUNDS TEKNISKA HÖGSKOLA MATEMATIK SVAR...

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Unformatted text preview: LUNDS TEKNISKA HÖGSKOLA MATEMATIK SVAR OCH ANVISNINGAR KONTINUERLIGA SYSTEM 7.5 hp 2009–01–09 1. Lägg in ett koordinatsystem så att den 100 gradiga cylindern ligger i intervallet < x < 1 och den 0 gradiga i intervallet 1 < x < 2 . Med u ( x, t ) som temperaturen i x vid tiden t får vi följande modell ∂u ∂t- ∂ 2 u ∂x 2 = 0 , < x < 2 , t > u x (0 , t ) = u x (2 , t ) = 0 , t > u ( x, 0) = 100( θ ( x )- θ ( x- 1)) , < x < 2 . Lösningen blir u ( x, t ) = 50 + 200 π ∞ summationdisplay k =1 sin( kπ/ 2) k e − k 2 π 2 t/ 4 cos( kπx/ 2) . 2. a) Låt u och v vara två funktioner i D A . En partialintegration ger att ( u |A v ) = integraldisplay π/ 2 − π/ 2 u ( x ) A v ( x ) cos x dx = integraldisplay π/ 2 − π/ 2 u ′ ( x ) v ′ ( x ) cos x dx (1) Då u = v får vi att ( u |A u ) = integraldisplay π/ 2 − π/ 2 u ′ ( x ) u ′ ( x ) cos x dx = integraldisplay π/ 2 − π/ 2 | u ′ ( x ) | 2 cos x dx ≥ ....
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This note was uploaded on 10/07/2011 for the course FMA 021 taught by Professor Pellepettersson during the Spring '11 term at Lund.

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kontsyslos090109 - LUNDS TEKNISKA HÖGSKOLA MATEMATIK SVAR...

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