# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 4 Conclusions  
Throughout this work, we have presented the numerical possibilities offered by the weak or integral solution of a nonlinear PDE when transposed into a numerical scheme, revisiting the original methods designed for Fokker-Planck equations and updating them into a more general algorithm. The only requirement for the original shape of the PDE is to have a separable quasilinear operator, which defines the Green function of the problem: a function that satisfies such a quasilinear PDE with homogeneous boundary conditions. The result of this exercise is an integral scheme where the Green function serves as the kernel propagating the different contributions of the solution in a defined state, \((x^{\prime},t_{0})\), towards the next state--understood, within the short-time approximation, as the next time frame. Therefore, we can build an integral time-evolving numerical scheme defined by the difference between time frames, or time step, \(\tau\), that will determine the quality of the numerical solution. Under the short-time approximation, the Green function can be interpreted directly as a transfer function or, from a physical perspective, as a transition probability. Understanding its shape and properties allows one to predict how information evolves in time while satisfying the PDE, and moreover, to define zones of influence between past and future states. This allows for simplifications of the problem, the most straightforward being the separation of the domain of interest into different regions of influence. In this context, we can numerically define boundary and inner regions--zones within the domain that are unaffected by the boundary. As a result, the problem in the inner region is simplified and can be treated as an _open_ domain, whereas the boundary regions are treated as _semi_-open. Under these assumptions, we derive the Green functions of the first- and second-order quasilinear operators in two different but equivalent ways: as discrete Dirac deltas, around which we extrapolate the operator using a Taylor series, and in a closed form, as the solution to its own PDE, i.e., the homogeneous version of the target PDE. Furthermore, at the boundary regions, the second-order operator allows for the application of the method of images, simplifying the process of deriving the semi-open Green function by reflecting it using the boundary as a mirror. With these two expressions established, the next step was to introduce a numerical discretization from which the time-evolving numerical algorithm could emerge. Depending on the shape of this discretization, we may find a solution with poor stability properties. Even when an appropriate discretization yields a low interpolation error, if the resulting discrete matrix does not exhibit suitable behavior or fails to correctly reproduce the time-marching behavior, it may cause indefinite growth over time. To mitigate these issues, we introduced and refined a grid optimization algorithm to ensure the equioscillation condition over a piecewise interpolation, following the methodology of Gauss-Lobato-Chebyshev quadrature. The main conclusion from this is that, outside of the boundary region (defined by the stencil size of the piecewise interpolation, \(2q+1\)), the optimal grid tends to approach a homogeneous discretization. Thus, we can separate the problem both numerically and analytically into inner and boundary problems, assuming that the inner points are defined with a constant grid and a domain of validity equal to half of the grid step. Incorporating this idea into the integral scheme and using a Lagrange-like interpolation, we developed a time-marching interpolation scheme over a stencil of size \(2q+1\) with weights corresponding to the numerical transposition of the Green function. We then conducted an exploratory study to understand the limitations and behavior of these approximations by introducing mathematical complexity into the numerical scheme.