# 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  
We can define the solution of the initial value problem defined by Equation (1) and a set of boundary conditions by means of an integral scheme that separates the contributions of the different terms in play. \[ \begin{split} u(\vec{x},t)&=\int u\big{(}\vec{x}^{ \prime},t_{0}\big{)}\mathcal{G}_{o_{2}}\big{(}\vec{x},t\mid\vec{x}^{\prime},t_{ 0}\big{)}d\mathcal{V}^{\prime}\\ &+\iint\Phi\big{(}\vec{x}^{\prime},t^{\prime}\big{)}\mathcal{G}_{o _{2}}\big{(}\vec{x},t\mid\vec{x}^{\prime},t^{\prime}\big{)}d\mathcal{V}^{ \prime}dt^{\prime}\\ &-\iint\widetilde{\vec{P}}_{\mathcal{L}}\big{[}u\big{(}\vec{x}^{ \prime},t^{\prime}\big{)},\mathcal{G}_{o_{2}}\big{(}\vec{x},t\mid\vec{x}^{ \prime},t^{\prime}\big{)}\big{]}\cdot d\mathcal{S}^{\prime}dt^{\prime}.\end{split}  \tag{4}\]  
where \(\mathcal{G}_{o_{2}}\) is the Green function associated with the operator \(\mathcal{L}\) and \(\widetilde{\vec{P}}_{\mathcal{L}}\) is the generalized flux or _concomitant_ of the problem the shape of which will define the boundary conditions over the surface \(\mathcal{S}\). This expression of the integral scheme is a generalization of what is usually presented as the weak solution of a PDE the details of which can be consulted in [43; 44]. The tilde over the generalized flux refers to the adjoint problem, required as we will introduce to define the generalized Green theorem. We have selected a specific terminology to describe the spatio-temporal dependencies of the Green function, that follows a common physical interpretation of these functions: transition probabilities. As a result, we interpret the Green function as the Kernel that makes the initial information at state \((\vec{x}^{\prime},t^{\prime})\) propagate towards a final state, \((\vec{x},t)\). Therefore, we can identify the three terms that define the time-evolving solution, being propagated from the initial time frame, \(t_{0}\):  
* The first term propagates the initial condition. Starting on time \(t_{0}\), the Green function takes the information at the initial state and _filters_ it through the integral, so the result satisfies the operator \(\mathcal{L}_{\vec{x},t}\) as defined in Equation (3). * The second integral contains all the mathematical terms that are left aside from the operator \(\mathcal{L}_{\vec{x},t}\) and grouped on the nonlinear or _non-homogeneous_ terms. This term is integrated through time and volume, meaning that it includes not only the values of the nonlinear terms at the initial state (like the first term) but also how their variation between states. * The last term takes into account how the boundary conditions of the problem will contribute to the solution. Indeed, it is the surface integral of the adjoint surface flux (or concomitant function) defined by a pair of the unknown function and the Green function. As the shape of the concomitant will depend on the order of the operator, we will introduce it later for the operators of interest in this study. Thus, obtaining the Green function and the expression of the generalized flux will provide a general expression for the solution of the problem. In particular, the Green function must satisfy the homogeneous partial differential equation of the problem, i.e., without the \(\Phi\) term and with a net flux equal to zero at the boundaries. Moreover, we can see that unless it has as initial condition a generalized Dirac delta function, expression (4) will not produce the unknown function for the limit when \(t\) tends towards \(t_{0}\). These two conditions can be understood as the way to guarantee that the Green function does not introduce artificial, spurious information to the problem: neither through the spatial or temporal boundaries. While the fundamental mathematical operative under these assumptions can be extracted from reference [45], an appropriate study with the required details is left for a future extended work.