Find the solution to ut uxx u u0 t u1 t 0 ux 0 3 sinpix

Find the solution to u_t = u_xx + u, u(0, t) = u(1, t) = 0, u(x, 0) = 3 sin(pix) + sin(2pix).

Solution

Let u_t = u_xx+u =k where k is a constant

Consider u_t =k

U(t) = kt +C

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Next consider u_xx+u = k

Auxialary equation has solution as i and -i

Hence solution is

U(x) = A cos x + B sin x

Hence general solution for U is

U(x,t) = (kt +C)(A cos x + B sin x)

Now substitute initial conditions:

u(1,t) = (kt+c) (-A) = 0

u(0,t) = (kt+C)(A)=0

It follows that A =0

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u(x,0) = (+C)(Bsin x) = 3 sin pix + sin 2pix

BC sin 3x= (3 sin pix+sin 2pix)

Hence solution is

U(X,t) = kBt sin x +(3 sin pix+sin 2pix)(sinx)/sin3x