Luxxt x s dUx sdx Luxxx tx s d2Ux8dx2 Use the Laplace Tran

L{u_x(x,t)} (x, s) = dU(x, s)/dx L{u_xx(x, t)}(x, s) = d^2U(x,8)/dx^2 Use the Laplace Transform Method to solve the following pde:

Solution

Utt = C^2Uxx+sin( x/l)sin(at)

we will split this for better understanding

taking Utt= C^2Uxx

Laplace transform of this is

S^2 U - sU(x,0) - Ut(x,0) = s^2U = c^2Uxx

and U(0,t) =  e^(-st) U(0,t)dt =  e^(-st)mtdt = M(s)

so U(x,s) = Be^(-sa/c) = M(s)e^ -sa/l

By the shift formula

u(x,t) = H(t-x/c)m(t-x/c) = 0 if t-x/c<0

= m(t-x/c) if t-x/c >=0

Uxx= C^2Uxx

U(x,t) = F(x-ct) + G(x+ct)

F(x) = f(x)/2 - 1/2l g(y)dy +C1

G(x) = f(x)/2 + 1/2l  g(y)dy +C2

U(x,t) = 1/2(f(x-ct) + f(x+ct)) + 1/2lg(y)dy

taking second term sin(pix/l)sinat

taking laplace transform

we know that laplace transform of sinat = a/S^2 +a^2

so that

sin (pix) a/S^2 +a^2 + pi x /S^2 +pi^2

hence on substituting the values we get

U(x,t) = l^2/(al)^2 - (c pi)^2 [ al / pi c sin ( pi c x/l) - sin at ] sin ( pi x / l)