The IVBP that i need to solve is the follow: \begin{equation} \begin{cases} u_t=au_{xx} & x>0,t>0,a\in\mathbb{R}^+\\ u(x,0)=B_0e^{-kx}\cos(kx) & x\geq 0,B_0\in\mathbb{R}^+,k\in\mathbb{R}^+\\ u(0,t)=B_0\cos(\omega t) & t\geq 0,B_0\in\mathbb{R}^+,\omega\in\mathbb{R}^+ \end{cases} \end{equation}
How can i solve this using separation of variables method?