L24 - Heat Conduction II t x L o C o C u t = a 2 u xx u (...

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Unformatted text preview: Heat Conduction II t x L o C o C u t = a 2 u xx u ( t, L ) = 0 u ( t, 0) = 0 u (0 , x ) = f ( x ) We want to find a function u ( t,x ) , t < , x L , of two variables that satisfies the Heat Equation u t ( t,x ) = a 2 u xx ( t,x ) , t < , x L , ( E ) the Left and Right Boundary Conditions u ( t, 0) = 0 and u ( t,L ) = 0 , t < , ( BC ) and the Initial Condition u (0 ,x ) = f ( x ) , 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 The clue here is that the Three Green Conditions are Linear . By that I mean that if u 1 ,u 2 ,...,u n are functions satisfy- ing the Three Green conditions, then any linear combination c 1 u 1 + c 2 u 2 + + c n u n also satisfies Three Green conditions. This allows us to try for very special solutions of the Three Green conditions, namely those that have their variables separated: u is the product of a function of t alone with a function of x alone: u ( t,x ) = T ( t ) X ( x ) . ( S ) 3 Separation of Variables On a function u ( t,x ) = T ( t ) X ( x ) , the Three Green conditions read as follows: T ( t ) X ( x ) = a 2 T ( t ) X 00 ( x ) , t < , x L , T ( t ) X (0) = T ( t ) X ( L ) = 0 , t < . Henceforth let us go only after nonzero solutions u . The Three Green conditions imply T ( t ) a 2 T ( t ) = X 00 ( x ) X ( x ) , t < , < x < L , ( E ) X (0) = X ( L ) = 0 . ( BC ) 4 ( E ) implies that T ( t ) a 2 T ( t ) does not depend on t and X 00 ( x ) X ( x ) does not depend on x : both quo- tients are constant, in fact the same con- stant. To summarize: the Three Green conditions on u ( t,x ) = T ( t ) X ( x ) say that There exists a constant such that T ( t ) a 2 T ( t ) = X 00 ( x ) X ( x ) = ( E ) and X (0) = X ( L ) = 0 , t < . ( BC ) Let us go after the information concerning X first: let us find all functions X and constants that satisfy 5 X 00 ( x ) X ( x ) = = X 00 ( x ) = X ( x ) , ( EX ) X (0) = X ( L ) = 0 , ( BCX ) ....
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L24 - Heat Conduction II t x L o C o C u t = a 2 u xx u (...

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