Question

Determine the laplace transform

i) ∂²u(x,t)/∂t²

ii)∂²u(x,t)/∂x²

Answer 1

i)

$L(u(x,t))=\int^{\infin}_0 e^{-st}u(x,t)dt=U(x,s)$

$L(u_t(x,t))=\int^{\infin}_0 e^{-st}u_t(x,t)dt=e^{-st}u(x,t)|^{\infin}_0+s\int^{\infin}_0 e^{-st}u(x,t)dt=$

$=sU(x,s)-u(x,0)$

$L(u_{tt}(x,t))=s^2U(x,s)-su(x,0)-u_t(x,0)$

ii)

$L(u_x(x,t))=\int^{\infin}_0 e^{-st}u_x(x,t)dt=U_x(x,s)$

$L(u_{xx}(x,t))=\int^{\infin}_0 e^{-st}u_{xx}(x,t)dt=U_{xx}(x,s)$