11.5.5.2 Mixed problems

Problem: Solve the mixed problem: \[ \left\{\begin{array}{c} u_{t t}=u_{x x}+t^{2} x, \quad t>0 \\ \left.u\right|_{t=0}=x^{2},\left.\quad u_{t}\right|_{t=0}=0 \end{array}\right. \]