# 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
#### 2.3.2 Second-Order Operator: Diffusion Equation  
This operator corresponds to the well-known Heat transfer equation, that smooths down initial conditions through the spatial domain. From the perspective of the integral scheme, we start by characterizing the general open domain solution, Equation (22) for \(o_{2}=2\). \[ \begin{split}\mathcal{G}_{2}\big{(}\vec{x},t\,|\,\vec{x}^{\prime},t^{\prime}\big{)}&=\frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}} \mathrm{Exp}\Big{[}\mathrm{i}\vec{\omega}\cdot\big{(}\vec{x}-\vec{x}^{\prime }\big{)}-\vec{\bar{a}}_{2}:\vec{\omega}^{T}\vec{\omega}\big{(}t-t^{\prime} \big{)}\Big{]}d\Omega\\ &=\frac{1}{\sqrt{\big{(}\det\vec{\bar{a}}_{2}\big{)}\big{[}4\pi(t -t^{\prime})\big{]}^{3}}}\mathrm{Exp}\bigg{[}-\frac{1}{4(t-t^{\prime})}\big{(} \vec{x}-\vec{x}^{\prime}\big{)}\vec{\bar{a}}_{2}^{-1}\big{(}\vec{x}-\vec{x}^{ \prime}\big{)}^{T}\bigg{]},\\ \vec{\bar{a}}_{2}&=a_{2}\vec{\bar{I}}\longrightarrow \frac{1}{\sqrt{\big{[}4a_{2}\pi(t-t^{\prime})\big{]}^{3}}}\mathrm{Exp}\bigg{[}- \frac{(\vec{x}-\vec{x}^{\prime})^{2}}{4a_{2}(t-t^{\prime})}\bigg{]}.\end{split}  \tag{26}\]  
So the open domain Green function (ODGF) of the second-order operator is a normalized Gaussian function, centered in zero under the variable \((\vec{x}-\vec{x}^{\prime})\) and a standard deviation, \(\sigma_{2}\), of  
\[\sigma_{2}=\sqrt{a_{2}(t-t^{\prime})}. \tag{27}\]  
It can be easily verified that the initial condition for the Green function is satisfied by this mathematical shape: When the final time frame tends to the initial one, the Dirac delta is recovered. The support of the Gaussian tends to zero concentrating its area at position \((\vec{x}-\vec{x}^{\prime})\), defining the limit towards a zero standard deviation Gaussian as a usual representation of the Dirac delta [50]. Contrary to the first-order Green function, the Gaussian function is more numerically friendly to implement. Indeed as it is presented in expression (26) the solution of the open domain integral scheme (4) can be seen as a convolution operation of the extended initial condition with a smoothing function. \[u(\vec{x},t_{0}+\tau)=\{\mathcal{G}_{2}(\tau)*[u(t_{0})+\tau\Phi(\mathrm{t})] \}(\vec{x}). \tag{28}\]  
Thus, any non-essential discontinuity [51] present on the non-homogeneous terms or at the initial condition will not be transmitted into the next time frame, smoothed down by the Gaussian [52]. Going back into the short-time approximation, the Taylor series expansion of the problem, with a non-constant transport coefficient will result into:  
\[ \begin{split}&\mathcal{G}\big{(}\vec{x},\tau\,|\,\vec{x}^{\prime},0\big{)}=\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}+\tau a_{2,ij}\
abla_{i }\
abla_{j}\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}+\frac{1}{2}\tau^{2}a _{2,ji}a_{2,kl}\
abla_{i}\
abla_{j}\
abla_{k}\
abla_{l}\delta\big{(}\vec{x}- \vec{x}^{\prime}\big{)}\\ &+\frac{1}{2}\tau^{2}\big{\{}\big{(}a_{2,ij,t}+a_{2,jj}\
abla_{k} \
abla_{l}a_{2,kl}\big{)}\
abla_{i}\
abla_{j}+2a_{2,jj}\
abla_{k}a_{2,kl} \
abla_{l}\
abla_{i}\
abla_{j}\big{\}}\delta\big{(}\vec{x}-\vec{x}^{\prime} \big{)}+\mathcal{O}\Big{(}\tau^{3}\Big{)}\\ &=\mathrm{Exp}\Big{[}\tau\vec{\bar{a}}_{2}(\vec{x},0):\vec{\
abla }\vec{\
abla}\Big{]}\delta\big{(}\vec{x}-\vec{x}^{\prime}\big{)}+\frac{1}{2} \tau^{2}\psi_{2}+\mathcal{O}\Big{(}\tau^{3}\Big{)},\end{split}  \tag{29}\]  
where we have used [PERSON]'s notation for the first terms for simplicity. To clarify this solution even further, we present the 1D Taylor expansion too:\[