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3669201626 - Lecture Notes ME4906 Mechanical Engineering...

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Lecture Notes ME4906, Mechanical Engineering, PolyU 2008/2009 5.3 Finite Difference for PDE of 1D in Space 5.3.1 Explicit Scheme Consider the problem of 1D unsteady heat transfer defined below, The solution domain is two-dimensional, space x and time t . The length L is divided into N x points and the time advances from t= 0 at an interval of t : t =(1,2,3,…) t . Figure 4. Explicit scheme for time derivative The second space derivative is approximated by ( ) ( ) ( ) ( ) 2 1, 2 , 1, , + + = ∆ = ∆ xx T i l T i l T i l T x i x t l t x and the time derivative is given by ( ) ( ) ( ) , 1 , , + = ∆ = ∆ t T i l T i l T x i x t l t t So the PDE is discretized as ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 , 1 , 1, 2 , 1, , 1 1 2 , 1, 1, , / κ κ + + + = + = + + + = ∆ ∆ T i l T i l T i l T i l T i l t x T i l c T i l c T i l T i l c t x Time x=L x t i- 1, i, i+ 1 l+ 1, l 2 2 κ = T T t x
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1 Notice that, for the points on the boundary, i =1, there is no left-hand side point at i -1.
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