SOLVE FIRST ORDER 2D PDEs

Hello everybody,

I have a numerical model for a thermal spray process which is described by a relationship of the type:

where x,y and z are the three space dimensions and t is the time. The phenomenon is described by the following type of First Order 2D PDE:

I can obtain both Initial Conditions and Boundary Conditions but they will be different each time I apply the model.

I tried to understand if pdepe was feasible but I have too many variables for it. I also tried with Partial Differential Equation Toolbox but my equation does not match the Divergence Form. I think this problem could be solved with the characteristics method but changing IC and BC makes it unfeasible. Is there any Matlab function or numerical method to solve this equation numerically?

[darova]

Can you show these?

Answer 1

[Daniele Vanerio]

I was finally able to numerically solve the equation. It is explained in this article together with its application:

https://doi.org/10.1016/j.jmapro.2021.05.013

Answer 2

[Daniele Vanerio]

Thank you for your answer.

g is

where excluding x,y and z all the other values are constants.

f is an interpolating function and corresponds to this

but an easier form could be found if necessary. In the case all the other values are constants too.

[darova]

It's too complicated, don't know how to solve. I thought FDM method can be applied, but it's not