# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 1 Introduction
Finally, we examine the boundary term, showing how the numerical equivalent of the integral term becomes a localized contribution, which is easy to incorporate into the overall time-evolving scheme. Being an exploratory study, we limit ourselves in this initial work to simple spatial domains--one-dimensional--in order to avoid overshadowing the behavior and performance of the method with the complexity of grid design. To conclude this introduction, we present a brief historical review of the use of weak solutions as numerical methodologies to provide some context and rationale for the construction of the method presented in this work. Some Historical Background on the Use of Integral Solutions as Numerical SchemesThe use of weak or integral schemes to solve PDEs has a long history, running parallel to the development of _classical_ numerical schemes. Notably, the foundational works of [PERSON] and [PERSON] [1; 2; 3] introduced efficient numerical methods based on the Green function, incorporating theoretical advances like Path-sum techniques and stochastic differential equations, although constrained by the computational resources of their time. Following these developments, [PERSON] and [PERSON] [4; 5] and others [6; 7; 8] applied similar methodologies, advancing Lagrangian particle methods. [PERSON] [9; 10] further linked the numerical Green function to probability distributions, while the Distributed Approximating Function theory (DAF) [11] provided a formal mathematical basis for propagating initial conditions. Among the many adaptations of this theory towards the use of weak solutions, we point out the work by [PERSON] and [PERSON] [12], who developed and introduced the Discrete Singular Convolution (DSC) method, applying weak solutions to Navier-Stokes and fluid-like PDEs [13; 14]. Furthermore, with the wave of more sophisticated numerical techniques, some of these works had interesting follow-ups with different discretization techniques, such as wavelets [15], integral kernels as substitutes of the Dirac delta [16], Fast transforms [17], Chebyshev discretization [18], Monte-Carlo simulations [19] and so on. Among these, three prominent families of numerical methods are worth highlighting due to their relevance to our work. The first of them being the Exponential Integral methods (ExpRK). These methods separate the PDE into quasilinear and nonlinear terms, using an exponential operator as the propagating element--similar to the Green function [20; 21; 22; 23]. They are widely used for stiff problems, allowing us to implement arbitrary nonlinear terms with high levels of accuracy. The next family of numerical methods are the Lagrangian or particle methods. These methods are highly versatile and often connected to weak solutions of PDEs. Recent advances, particularly in Smoothed Particle Hydrodynamics (SPH), have improved energy conservation, boundary conditions, and time integration [24; 25; 26]. The last family of methods are the Lattice Boltzmann (LB) methods. These methods provide a mesoscopic representation of dynamics and ability to simulate complex systems with high accuracy. This approach intertwines with the interpretation of the Green function as the probability distribution of the quasilinear operator, in the same manner that Fokker-Planck and Langevin equations are different scale representations of the problem. Therefore, while they are limited by the careful selection of parameters like relaxation times and time steps to ensure stability and accuracy, often validated through agreement with analytical solutions, they allow us to directly handle complex nonlinearities. Despite challenges in stability, particularly for coarse or under-resolved meshes, LB methods remain powerful tools for simulating nonlinear dynamics and exploring parameter spaces across a wide array of physical systems [27; 28; 29]. Modern refinements of these original methods include high-order integral algorithms for nonlinear problems (often applied to plasma physics) [30; 31; 32; 33].