I want to solve the diffusion equation in porous media with some external forcing at the origin. The regular diffusion equation- ∂h∂t=D∇2h\frac{\partial h}{\partial t}=D\nabla^2h But I want to solve for pore pressure, so I am using p=ρghp=\rho g h where h is the pore pressure and h is the hydraulic head and D is a constant. Also, I am simplifying this for 1-D, so this equation becomes - ∂p∂t=D∂2p∂2x\frac{\partial p}{\partial t}=D\frac{\partial^2 p}{\partial^2 x} As initial conditions, p=0 at t=0 ∀\forall x . x ∈\in {-25,5}. At the origin (x=0), there is direct injection from a well, and p changes with time according to a function, let's say it is a simple straight line, p=mtp=mt . How do I solve a problem like this where p is not only changing due to the diffusion but also because of the external forcing? How do I implement this into the diffusion equation? I don't really want the actual solution but rather the approach to solving this problem also if there is anything wrong with my logic. The last time I took an ODE or PDE course was in my first year of undergrad so I am very rusty.