This looks like a one-dimensional advection equation with a source term. The numerical solution can be found here

https://math.stackexchange.com/questions/317696/advection-equation-with-fx-cdot-ux-t-source-term

http://www.henley.ac.uk/web/FILES/maths/J_Hudson.pdf

https://earth.usc.edu/~becker/teaching/557/problem_sets/problem_set_fd_advection.pdf

You can also use FEM tools to model it. And if you want a fancy way to do it, you can use Green's function to obtain the solution.

This looks like a one-dimensional advection equation with a source term. The numerical solution can be found here

https://math.stackexchange.com/questions/317696/advection-equation-with-fx-cdot-ux-t-source-term

http://www.henley.ac.uk/web/FILES/maths/J_Hudson.pdf

https://earth.usc.edu/~becker/teaching/557/problem_sets/problem_set_fd_advection.pdf

You can also use FEM tools to model it. And if you want a fancy way to do it, you can use Green's function to obtain the solution.

Yes, this is a linear 1D advection equation (for generally fixed x,y values) with variable coefficient w (that is, however, constant in z) and a source term. That can be written as Da/Dt=f, expressing the fact that while the quantity a travels along the path-line dz/dt=w it is no longer constant due to the action of f. You can integrate f along the path-line to solve for a or you can use a very simple numerical discretization

Dear Utkarsh, the PDE, as first order linear equation, can be solved formally using the method of characteristics (which is not the Method of Lines): w is the tangent to characteristic of the equation (see prof. Denaro answer). Method of characteristic is easily explained e.g. on Ian Sneddon, Elements of Partial Differential Equation.  Gianluca