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# We must of course study the behavior of 225 however

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Unformatted text preview: ivity is positive. In order to construct nite-di erence approximations of (2.3.1): ( ) introduce a uniform grid of spacing =1 , 0, and on the strip (0 1) ( 0) (Figure 2.3.1) ( ) evaluate (2.3.1a) at the mesh point ( ) and ( ) replace the partial derivatives by forward time (2.1.8) and centered space (2.1.9) di erences to obtain i x ii =J J> t j t> xn t iii 2.2. A Heat Equation 17 t (j,n) n 2 1 0 1 2 J j x Figure 2.3.1: Computational grid used for the nite di erence solution of the heat conduction problem (2.3.1). +1 ; un n uj t t +1 ; 2uj + uj ;1 2 x n ; 2 ( tt)n+ = j j uj u n n 2 ; 12 ( x )n+ ] uxxxx j (2.3.2a) where ;1 1, 0 1. (With second spatial derivatives in (2.3.1) there seems little point in using forward or backward spatial derivatives and we have not done so.) Neglecting the discretization error terms, we get the nite di erence scheme < < < < +1 ; U n n Uj j t or, solving for +1 ; 2Uj + Uj ;1 2 x n = Uj n n (2.3.2b) +1 , n Uj +1 n Uj = ;1 + (1 ; 2r)Uj n rUj n + n rUj +1 (2.3.3a) whe...
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