# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 2 Mathematical Foundations of the Method
### Closed Expressions for the Green Function
#### 2.3.1 First-Order Operator: Advection Equation  
For this problem, expression (22) becomes,  
\[\mathcal{G}_{1}\big{(}\vec{x},t\mid\vec{x}^{\prime},t^{\prime}\big{)}=\frac{1 }{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\text{Exp}\big{\{}\mathbf{i}\,\vec{\omega} \cdot\big{[}\vec{x}-\vec{x}^{\prime}+\vec{a}_{1}\big{(}t-t^{\prime}\big{)} \big{]}\big{\}}d\Omega=\delta\big{[}\vec{x}-\vec{x}^{\prime}+\vec{a}_{1}\big{(} t-t^{\prime}\big{)}\big{]}. \tag{23}\]  
As it can be observed, the Green function collapses into Dirac delta that takes information from point \(\vec{x}^{\prime}\) and _advecs_ it towards point \(\vec{x}\) that is placed at a distance \(\vec{a}_{1}(t-t^{\prime})\), hence the usual name _advection_ equation. If the advection is not constant, we need to go for the Taylor series expansion, that particularized for the first-order operator can be defined as:  
\[\mathcal{G}\big{(}\vec{x},\tau\mid\vec{x}^{\prime},0\big{)} =\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}+\tau\vec{a}_{1}( \vec{x},0)\cdot\vec{\
abla}\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}+\frac{ 1}{2}\tau^{2}\vec{a}_{1}^{T}\vec{a}_{1}(\vec{x},0):\vec{\
abla}\vec{\
abla} \delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)} \tag{24}\] \[=\text{Exp}\Big{[}\tau\vec{a}_{1}(\vec{x},0)\cdot\vec{\
abla} \Big{]}\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}+\frac{1}{2}\tau^{2}\psi_{ 1}+\mathcal{O}\Big{(}\tau^{3}\Big{)}.\]  
Equation (24) is the one-dimensionalits Eigenfunctions do not accept a closed domain form but a semi-open one, in the sense that each surface that limits the volume of study will advect or receive information from its surroundings in an independent manner. Yet, by linear combination of all the _semi-opened_ domains a full closed domain problem can be recovered. The shape of this Green function allows to define, within the short-time approximation, a region of influence of the boundaries, that isolates their evolution from the rest of the domain in the transition from the initial state, \((x^{\prime},t_{0})\) to the short time one in the future, \((x^{\prime},t_{0}+\tau)\). This region will be defined by the advection coefficient, hence separating an inner region and a boundary one, with the magnitude \(\tau\vec{a}_{1}\), as the main driver of the separation between zones. Moreover, as the advection coefficient is a vector with a defined direction its interaction with the surface normal will define whether the information leaves the domain or enters it. That, while both conditions can be understood as Dirichlet boundary conditions, depending on the mathematical set of equations we are solving might imply some hidden relations between the variables being solved, as it happens in the case of Euler equations in fluid dynamics. We can then classify the boundaries associated with the first-order operator into _active_ or _passive_ boundaries. Together with the classification, in Figure 1 we present an example for a boundary situation that combines both active and passive boundaries as the boundary condition is a sine function. A positive advection will imply that the boundary acts as an active source and introduce information inside the domain, therefore an active boundary, and negative advection will mean that the solution at that boundary point will come from a source point inside the domain, hence, passive boundary (see Figure 1). Figure 1: Example of the boundary behavior of the first-order Green function. If a boundary has an advection equal to zero, i.e., a non-permeable wall condition, the situation will be equivalent to a passive boundary. The boundary point will simply accumulate the information being advected towards it and if there is no other additional term, the solution will diverge after all the information has been concentrated at the last point. This occurs, for example, in an impinging polluted flow facing a non-permeable wall, where the fluid advects the inert heavy particles of the pollutant towards the wall concentrating at the impinging point until they occupy a region where the some transversal advection appears, moving them outwards.