# 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  
Having defined the short time approximation, we are ready to obtain a solution for the Green function knowing that it must satisfy the original partial differential equation without the non-homogeneous terms and satisfying its homogeneous boundary conditions. We start by approximating the Green function at time \(t\) by a Taylor series, recovering,  
\[ \begin{split}\mathcal{G}\big{(}\vec{x},t\,|\,\vec{x}^{\prime},t^ {\prime}\big{)}&=\mathcal{G}\big{(}\vec{x},t^{\prime}\,|\,\vec{x} ^{\prime},t^{\prime}\big{)}+\big{(}t-t^{\prime}\big{)}\mathcal{G}_{t}\big{(} \vec{x},t^{\prime}\,|\,\vec{x}^{\prime},t^{\prime}\big{)}+\\ &+\frac{1}{2}\big{(}t-t^{\prime}\big{)}^{2}\mathcal{G}_{tt}\big{(} \vec{x},t^{\prime}\,|\,\vec{x}^{\prime},t^{\prime}\big{)}+(\dots)+\frac{1}{n!} \big{(}t-t^{\prime}\big{)}^{n}\mathcal{G}_{(n)}\big{(}\vec{x},t^{\prime}\,|\, \vec{x}^{\prime},t^{\prime}\big{)},\end{split}  \tag{15}\]  
where the subscript \(\mathcal{G}_{t}\) indicates time derivative. In general, for a quasi linear operator as defined in expression (3) the transport coefficients \(a_{0_{2}}\) will depend on space, \(\vec{x}\), and time, \(t\), so the expression of the time derivatives of the Green function higher than \(1\) will be the result of the successive application of the operator \(\mathcal{L}_{\vec{x},t}\) and the complexity on the derivatives and number of terms will increase exponentially,  
\[\mathcal{G}_{t}=\mathcal{L}_{\vec{x},t}\mathcal{G},\,\,\,\mathcal{G}_{tt}= \big{(}\mathcal{L}_{\vec{x},t}\mathcal{G}\big{)}_{t}=\big{(}\mathcal{L}_{\vec {x},t}\big{)}_{t}\mathcal{G}+\mathcal{L}_{\vec{x},t}\mathcal{G}_{t}=\big{(} \mathcal{L}_{\vec{x},t}\big{)}_{t}\mathcal{G}+\mathcal{L}_{\vec{x},t}^{2} \mathcal{G}. \tag{16}\]  
Indeed, following the general Leibniz rule [48] we recover a general expression for the \(n\)-th derivative  
\[\mathcal{G}_{(n)}=\sum_{k=0}^{n-1}\binom{n-1}{k}\big{(}\mathcal{L}_{\vec{x},t }\big{)}_{(n-1-k)}\mathcal{G}_{(k)}, \tag{17}\]  
that will be applied recursively until the \(n\)-th-order is lowered to \(1\). If the coefficients inside \(\mathcal{L}_{\vec{x},t}\) do not depend on \((\vec{x},t)\), then the expansion simplifies considerably, being able to relate each time derivative to the successive application of the spatial operator:  
\[\mathcal{G}_{t}=\mathcal{L}\mathcal{G},\,\,\,\mathcal{G}_{tt}=\mathcal{L}^{2 }\mathcal{G},\,\,\,(\dots),\,\,\,\mathcal{G}_{(n)}=\mathcal{L}^{n}\mathcal{G}. \tag{18}\]  
In this situation, if we substitute conditions (18) and the fact that the initial condition of the Green function will be a Dirac delta into the Taylor series expansion, Equation (15), we recover:  
\[\mathcal{G}\big{(}\vec{x},t\,|\,\vec{x}^{\prime},t^{\prime}\big{)}=\overbrace{ \left[1+\big{(}t-t^{\prime}\big{)}\mathcal{L}+\frac{1}{2}\big{(}t-t^{\prime} \big{)}^{2}\mathcal{L}^{2}+(...)+\frac{1}{n!}\big{(}t-t^{\prime}\big{)}^{n} \mathcal{L}^{n}\right]}\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}. \tag{19}\]  
We stress that this result is only valid for operators whose transport coefficients do not depend on \((\vec{x},t)\), i.e., transport coefficients that are constant in space and time. In general, when the coefficients do present such dependencies, although we will still recover the exponential term, due to the Leibniz rule a number of extra terms or _corrections_ to the solution will appear. Although these terms will modify the shape of the Green function, they will only begin to appear when the time derivative is applied over the operator itself, that is, starting at the second-order in the Taylor expansion. Consequently, they will be a second-order contribution in time and, the lower the time step is, the lower the influence of these terms on the final solution.