The eigenfunctions and eigenvalues aresinn+ 1/2Lπxn+ 1/2Lπ2,n= 1,2, . . .Thus, we can expandv(x, t) =∞n=1vn(t) sinn+ 1/2LπxAtt= 0 we have−x−1 =v(x,0) =∞n=1vn(0) sinn+ 1/2Lπxsovn(0) =−2LL0(x+ 1) sinn+ 1/2LπxdxSubstitute the expansion is the PDE and equate coeﬃcients˙vn(t) +kn+ 1/2Lπ2vn(t) = 0vn(0) =−2LL0(x+ 1) sinn+ 1/2LπxdxThe solution is thenvn(t) =v
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