# L22 - Heat Conduction t u(t 0 = 0 ut = a2 uxx u(t L = 0 u(0...

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Heat Conduction t x 0 L 0 o C 0 o C u t = a 2 u xx u ( t, L ) = 0 u ( t, 0) = 0 u (0 , x ) = f ( x )

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In summary, we want to ﬁnd a function u ( t,x ) , 0 t < , 0 x L , of two variables that satisﬁes the Heat Equation u t ( t,x ) = a 2 u xx ( t,x ) , 0 t < , 0 x L , ( E ) the Left and Right Boundary Conditions u ( t, 0) = 0 and u ( t,L ) = 0 , 0 t < , ( BC ) and the Initial Condition u (0 ,x ) = f ( x ) , 0 x L . ( IC ) Our strategy is going to be this: ﬁrst ﬁnd all functions that satisfy the Three Green conditions; then identify among those that function u that also satisﬁes the Red ini- tial condition . 2
Three Green con- ditions are linear: by that I mean that if u 1 ,u 2 ,...,u n are functions satisfying the Three Green conditions, then any linear combination c 1 u 1 + c 2 u 2 + ··· + c n u n also satisﬁes the Three Green conditions. This allows us to try for very special solu- tions of the Three Green conditions, namely those that have their variables separated:

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## This note was uploaded on 09/07/2010 for the course PHY 303L taught by Professor Turner during the Spring '08 term at University of Texas.

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L22 - Heat Conduction t u(t 0 = 0 ut = a2 uxx u(t L = 0 u(0...

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