Using separation of variables to solve the heat equation, ut = kuxx on the interval 0 < x < 1 with boundary conditions ux (0, t ) = 0 and ux (1, t ) = 0, yields the general solution,
∞
u(x,t) = A0 + ?Ane−kλnt cos?nπx? (with λn = n2π2)
n=1DeterminethecoefficientsAn(n=0,1,2,...)whenu(x,0)=f(x)= 0, 1/2≤x<1 .
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