Unformatted text preview: x âˆˆ (0 , l ) and t âˆˆ (0 , T ] . Then v ( x , t ) = u ( x , t ) + Â± Â· ( Tt ) = m + Â± Â· ( Tt ) â‰¥ m > M + Â±T, so v achieves its maximum in R at some point other than initially or on the boundary, say at x 1 âˆˆ (0 , l ) and t 1 âˆˆ (0 , T ] . Then u t ( x 1 , t 1 ) = ku xx ( x 1 , t 1 ) = kv xx ( x 1 , t 1 ) â‰¤ , and so v t ( x 1 , t 1 ) = u t ( x 1 , t 1 )Â± < . This inequality implies that v ( t, x 1 ) > v ( t 1 , x 1 ) for some t âˆˆ (0 , t 1 ) , which contradicts the statement that v achieves its maximum at ( x 1 , t 1 ) . 1...
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 Spring '08
 Howard
 Math, Partial Differential Equations, Boundary, Green's function, 1D Diffusion Equation, MAT51316 Robert PichÂ´

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