# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 2 Mathematical Foundations of the Method
### Integral Solution of the Problem in Terms of the Green Function
For the purposes of this work, which aims to explore numerical implementation, only the final expressions used are introduced. So the only term that appears in expression (4) that is not in an explicit form is the last one, that includes the boundary contributions. As it will be dependent of the operator \(\mathcal{L}\), we need to state its order first, \(o_{2}\), in order to fully develop the term. To do so, we first look at the definition of the concomitant function, \(\vec{p}\) that the generalized Green theorem introduces,  
\[\vec{\
abla}\cdot\vec{P}(u,vs.) =u\mathcal{L}vs.-vs.\widetilde{\mathcal{L}}u\] \[\vec{\
abla}\cdot\widetilde{\vec{P}}(u,vs.) =u\widetilde{\mathcal{L}}vs.-vs.\mathcal{L}u\]  
where the tilde quantities refer to the mathematical problem defined by the adjoint operator and \(u\) and \(v\) stand for general test functions. Introducing the condition that the Green function's flux must be zero at the boundaries--or, from the perspective of its boundary conditions, they must be the ones of the PDE but in a homogeneous expression--we can recover the possible boundary conditions for a given derivative order, \(o_{2}\). For this initial exploration we will only look into the first two orders, \(o_{2}=\{1,2\}\), that coincide, respectively, with the advective problem [44] and the diffusive one. * First-order operator From expression (5) we can obtain the generalized flux, \(\vec{P}_{1}(u,v)\) for the advective problem, and particularizing it for the boundary term of the integral scheme (4), we recover, \[\widetilde{\vec{P}}_{1}(u,\mathcal{G}_{1})=-\vec{a}_{1}u\mathcal{G}_{1} \longrightarrow\iint\vec{a}_{1}u\mathcal{G}_{1}\cdot d\vec{S}^{\prime}dt.\] (6) Therefore, this operator will only allow for Dirichlet boundary conditions, imposed over the unknown function, \(u\), and the Green function will necessarily be a generalized Dirac delta at the boundaries in the same way it was at the initial condition. Second-order operator On the other hand, this operator allows for Dirichlet, Neumann and Mixed boundary conditions as its generalized flux is, \[\widetilde{\widetilde{P}}_{2}(u,\mathcal{G}_{2})=u\vec{\
abla}\cdot\left( \vec{\bar{q}}_{2}\mathcal{G}_{2}\right)-\mathcal{G}_{2}\vec{\bar{q}}_{2}\cdot \vec{\
abla}u.\] (7) Therefore, depending on which of the three possible kinds of boundary condition we want to impose over the unknown function, we will need the Green function to _cancel_ the rest. For example, if we want to impose a Dirichlet boundary condition over \(u\) the term with its derivative, i.e., the Neumann one, need to be canceled, therefore forcing the Green function to have also a Dirichlet boundary condition. Furthermore, the same logical procedure can be applied for Neumann boundaries, from which we can derive the already introduced condition: the Green function must satisfy the homogeneous version of the unknown function in order not to contribute with spurious information at the boundaries. Consequently, the full explicitly defined integral scheme that defines the solution of the partial differential equation can be defined, for the First and Second operators as, * First-order operator \[ \begin{split} u(\vec{x},t)&=\int u\left(\vec{x}^{ \prime},t_{0}\right)\mathcal{G}_{1}\left(\vec{x},t\mid\vec{x}^{\prime},t_{0} \right)d\mathcal{V}^{\prime}\\ &+\iint\Phi\left(\vec{x}^{\prime},t^{\prime}\right)\mathcal{G}_{ 1}\left(\vec{x},t\mid\vec{x}^{\prime},t^{\prime}\right)d\mathcal{V}^{\prime} dt^{\prime}\\ &+\iint\left[\vec{a}_{1}u\left(\vec{x}^{\prime},t^{\prime} \right)\mathcal{G}_{1}\left(\vec{x},t\mid\vec{x}^{\prime},t^{\prime}\right) \right]\cdot d\vec{S}^{\prime}dt^{\prime}.\end{split} \] (8)
* Second-order operator \[