(a) Showthattheboundaryconditionsu(0,t)=0,ux(l,t)=0leadto
Chapter 5, Problem 5.3.5(choose chapter or problem)
(a) Showthattheboundaryconditionsu(0,t)=0,ux(l,t)=0leadto the eigenfunctions (sin(x/2l), sin(3x/2l), sin(5x/2l),...). (b) If (x) is any function on (0, l), derive the expansion (x)= n=0 Cnsinn+ 1 2x l $ (0 < x < l) by the following method. Extend (x) to the function dened by (x)= (x) for 0 x l and (x)= (2l x) for l x 2l.(This means that you are extending it evenly across x = l.) Writethe Fourier sine series for (x) on the interval (0, 2l) and write the formula for the coefcients. (c) Show that every second coefcient vanishes. (d) Rewrite the formula for Cn as an integral of the original function (x) on the interval (0, l).
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