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**Unformatted text preview: **10.5: SEPARATION OF VARIABLES AND HEAT CONDUCTION IN A ROD KIAM HEONG KWA The variation of temperature u ( x,t ) in a bar whose axis lies along the x-axis is governed by the heat conduction equation (1) α 2 u xx = u t , < x < L, t > , where the thermal diffusivity α 2 is a parameter which depends only on the material from which the bar is made. Here we assume that the cross-sectional dimension of the bar is sufficiently small such that the temperature u ( x,t ) is constant on any given cross section. We also assume that the initial temperature distribution (2) u ( x, 0) = f ( x ) , ≤ x ≤ L, is specified, where f ( x ) and its derivative f ( x ) are piecewise continu- ous. Next, we also stipulates the homogeneous boundary conditions (3) u (0 ,t ) = 0 , u ( L,t ) = 0 , t > . This means that the two ends of the bar are held at fixed zero temper- atures. It is worth noting that if u n ( x,t ), n = 1 , 2 , ··· ,N , are solutions of (1), then so is any linear combination ∑ N n =1 c n u n ( x,t ) = c 1 u 1 ( x,t ) + c 2 u 2 ( x,t )+ ··· + c N u N ( x,t ) of these solutions. Furthermore, if u n ( x,t ), n = 1 , 2...

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