I would be thankful to anyone who can present an analytical solution to the following inhomogeneous PDE equation:
$$\frac{\partial{u}}{\partial{t}}= \alpha\frac{\partial^2{u}}{\partial{x^2}}-ku$$
$$u(0,t) = 0$$
$$u(1,t) = M_R$$
$$u(x,0) = x*f(x)$$
where k, $\alpha$ and $M_R$ are constants and k>0.
Set first $u= ve^{-kt}$ so that $ \partial_t u+ku=e^{-kt}(\partial_t v-kv+kv)=e^{-kt}\partial_t v. $ The equation becomes $$ \partial_t v=\alpha\partial_x^2 v, \quad v(0,t)=0, \quad v(1,t)=M_R e^{kt}, \quad v(x,0)=x\ast f(x). $$ You may assume $\alpha =1$ by writing $v(x,t)=w(x, \alpha t)$ and get then $$ \partial_t w=\partial_x^2 w, \quad w(0,t)=0, \quad w(1,t)=M_R e^{kt/\alpha}, \quad w(x,0)=x\ast f(x), $$ ending-up with an Initial-value and boundary problem for the heat equation. You can follow the standard Fourier method by looking for $2$-periodic functions in the variable $x$.
