# Exploratory Study of a Green Function Based Solver for Nonlinear Partial Differential Equations
## 2 Mathematical Foundations of the Method  
The aim of the method is to solve balance-like partial differential Equations (PDEs) that can be cast without losing generality as:  
\[\left(\frac{\partial}{\partial t}-\mathcal{L}_{\vec{x},t}\right)u(\vec{x},t)= \mathfrak{A}_{\vec{x},t}u(\vec{x},t)=\Phi(\vec{x},t), \tag{1}\]  
where \(u\) is the unknown function and \(\vec{x}\) is the \(n\)-dimensional space under study. The time variable, \(t\), is left separated from the dimensionality of the problem as the method is defined for time-evolving solutions. Hence, the operator \(\mathcal{L}_{\vec{x},t}\) will include the spatial derivatives of the PDE and, when grouping spatial and temporal derivatives, we will refer to the more general operator, \(\mathfrak{A}_{\vec{x},t}\).  
A general, quasilinear (we use the quasilinear definition presented in [42]) differential spatial operator \(\mathcal{M}_{\vec{x},t}\) can be defined as follows:  
\[\mathcal{M}_{\vec{x},t}=\sum_{i=1}^{\alpha_{2}}\vec{a}_{i}(\vec{x},t)\cdot \frac{\partial^{i}}{\partial\vec{x}^{i}}, \tag{2}\]where \(o_{2}\) is the highest order of the spatial derivatives and \(\vec{a}_{i}\) will be called _transport_ coefficients. It is important to recall that their dimension will vary together with the order of the derivative to keep the general PDE as a scalar, i.e., the first-order derivative will require a vector transport coefficient, the second-order a matrix, the third a third-order tensor, and so on. Same thing goes for the product between the coefficient and the set of derivatives--it will be the standard inner product with the corresponding order of the derivative that is applied at. The highest order of the temporal derivative, \(o_{1}\), is set equal to \(1\) for this work, therefore it is omitted through the derivation.  
Finally, the function \(\Phi\) groups the rest of the terms of the equation that do not fit with the ones already described and will be called, for simplicity, non-homogeneous or nonlinear term (although it might be linear over some of the variables under study). This term can group not only arbitrary terms but also quasilinear terms that might fit into the definition of the operator (2).  
In the following, we define the spatial operator of the PDE as the highest order derivative term, of order \(o_{2}\), and the rest will be grouped into the \(\Phi\) term. This choice is justified not only for simplicity but also to reach a mathematical procedure and solutions of the problem in a systematic and consistent manner. Hence, the mathematical shape of the spatial operator of the PDE, \(\mathcal{L}_{\vec{x},t}\), can be defined as follows,  
\[\mathcal{L}_{\vec{x},t}=\vec{a}_{o_{2}}(\vec{x},t)\cdot\frac{\partial^{o_{2}} }{\partial\vec{x}^{o_{2}}}. \tag{3}\]  
Together with Equation (1), a set of initial and boundary conditions will close the mathematical formulation of the problem object of this work. We define a mathematical control volume of study, \(\mathcal{V}\), and its corresponding boundary surface, \(\mathcal{S}\). This control volume is divisible in a differential sense, being \(d\mathcal{V}\) its element of volume or differential volume and, in a same manner, \(dS\) the surface element or differential surface. The problem of interest will be an initial value problem (IVP), meaning that the boundary conditions in time are always set at the initial time frame, \(t_{0}\).  
The mathematical shape of the boundary conditions of the problem will vary depending on the spatial order of the PDE. For each \(o_{2}\), and for different volumes of study, different allowed boundary conditions will arise, each one with their own peculiarities and mathematical shape. Thus, we leave their particular expression aside for the moment and they will be introduced afterwards for each specific problem.