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SPlease be clear and show all work!

Problem 4 (Hand In): Solve the inhomogeneous heat conduction problem with heat loss, a time dependent source, and subject to time dependent boundary conditions: \[ \begin{aligned} u_{t} &=u_{x x}+e^{-2 t} \sin (x), 00 \\ u(0, t) &=1, \text { and } \frac{\partial u(\pi / 2, t)}{\partial x}=t \\ u(x, 0) &=\sin (3 x)+1 \end{aligned} \]

Answer:- Given Equation is , ut=uxx+e?2t.sin?(x) We will solve