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 , , −− ++ =∆ =∆ ≈ xx Ti l Til Ti l Txi x tl t x and the time derivative is given by ( ) ( ) ,1 , , +− t Til Tx ix t lt t So the PDE is discretized as ( ) ( ) ( ) ( ) 2 2 , 1 , 2 , 1 , ,1 12 , 1 , 1 , , / κ − − + + = ∆∆ +=− + − + + =∆∆   T il T i l l tx cTil cTi l Ti l c t x Time x=L x t i- 1, i, i+ 1 l+ 1, l 2 2 ∂∂ = TT
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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. Likewise, there is no right-hand side point for x = L ( i=N x ). For these points, the boundary condition should be used.
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This note was uploaded on 08/27/2010 for the course ME ME4906 taught by Professor Dr.g.p.zheng during the Spring '10 term at NYU Poly.

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3669201626 - Lecture Notes ME4906, Mechanical Engineering,...

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