# 3-4 - 170 Finite Element Analysis and Design 4...

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170 Finite Element Analysis and Design 4. One-dimensional heat conduction problem can be expressed by the following differential equation: 2 2 0, 0 dT k Q x L dx where k is thermal conductivity, () Tx temperature, and Q heat source per unit length. Q , the heat generated per unit length, is assumed constant. Two essential boundary conditions are given at both ends: (0) ( ) 0 T T L  . Calculate the approximate temperature using Galerkin method. Compare the approximate solution with the exact one. Hint : Start with assumed solution in the following form: 2 0 1 2 T x c c x c x , and then make it to satisfy two essential boundary conditions. Solution: (a) Exact solution: By integrating the governing equation twice and applying the two boundary conditions, the exact solution becomes 2 ( ) ( ) 2 Q T x Lx x k  (b) Galerkin method: The assumed form of approximate solution should first satisfy the

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3-4 - 170 Finite Element Analysis and Design 4...

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