Partial Differential Equations Manual Solutions Strauss
ve Equation Problem: Solve \( u_{tt} = c^2 u_{xx} \) on \( -\infty < x < \infty \), with initial conditions \( u(x,0) = g(x) \), \( u_t(x,0) = h(x) \). Solution outline: 1. Apply Fourier transform in \( x \): \[ \hat{u}_{tt}(\xi,t) + c^2 \xi^2 \hat{u}(\xi,t) = 0. \] 2. Solve the ODE: \[ \hat{u}(\x