I am currently working on a small research project as a part of my degree. The project is centered around Continuous-time Markov Chains, and based on this paper :
https://hal.archives-ouvertes.fr/hal-03102188/
Currently, I should verify if a proposed analytical solution to the following equation, a version of the Eikonal equation, agrees with the numerical one, by numerically solving the following on Matlab, and comparing it with the analytical solution:
$$e^{d\phi(x,y)\over dy}- (y+1)e^{-d\phi(x,y)\over dx}-1=0$$ I have found the following initial and boundary conditions: $$ \phi(0,0)=0$$ $${d\phi(0,0)\over dy}=log(1+e^{d\phi(0,0)\over dx})$$
I am not super well versed in PDE's, and much less so in Matlab. I have the following questions:
- Would you say the conditions I have found contain enough information? I am not sure if I have everything I need to solve this?
- The Matlab guide for solving PDE's wants the coefficients in a very specific form. How would I go about reformulating this so that Matlab will work for me? How would I address the exponential outer functions of the partial derivatives?