I am working on a problem to solve some particle population balances. In the analysis of some experiments I got an equation of this type:
$$\frac{\partial u}{\partial y} - \alpha(x, y) \frac{\partial u}{\partial x} = 0$$
where $\alpha(x, y)$ is a function defined over a collection of data points $(x_i, y_i)$. Need to solve for $u$.
Any ideas how do I even begin to approach such a problem?
I was thinking maybe finding particular solutions treating $\alpha$ as constant for each point in the form
$$\frac{\partial u_i}{\partial y} - \alpha(x_i, y_i) \frac{\partial u_i}{\partial x} = 0$$
which can be easily solved using the method of characteristics. Then using superposition $u = \sum{u_i}$, but that would only give me a discrete representation of $u$ over the data points, and I am not sure if it is entirely correct. Also I only know that $u(x, 0) =0$ so I am not sure if MoC actually works there.
I'd appreciate any help.
With the present wording of the question I cannot give a definitive answer. My comment below is too long to be edited in the comments section. Sorry, I edit it in the answer section, but this is not an answer.
I agree with EditPiAf's comment. I would like add this :
As expected, solving the PDE with the method of characteristics leads to a general solution in which an arbitrary fonction appears. This function has to be determined according to the boundary condition. In the present case the general solution involves $\int \alpha(x,y)dy$ which requires numerical integration since $\alpha(x,y)$ is defined by a set of points. The feasibility of determination of the arbitrary function depends a lot of the shape and distribution of the points. Moreover the condition $u(x,0)=0$ will possibly introduce an Heaviside function. I think that one cannot say more without knowing more precisely about the definition of $\alpha(x,y)$ especially in the vicinity of the boundary $y\simeq 0$.
To go further, could you joint to your question a representative data as an exemple of discret definition of $\alpha(x,y)$.