Consider the PDE $$u_t=Du_{xx}+x+t$$ where $t>0, x \in (0,a)$ and $u(x,0)=0, u_x(0,t)=u_x(a,t)=0$.
I'm told that a particular solution is given by $$p(x,t)=xt+\frac{t^2}{2}$$ and I'm asked to use this in order to find the solution.
I've tried defining $v(x,t)=u(x,t)-p(x,t)$ where $p(x,t)$ is the particular solution above. Then $v(x,t)$ satisfies $v_t=Dv_{xx}$ but doesn't satisfy the boundary conditions so this doesn't seem to be a very useful approach.
How should I do this?