Greens Function for PDEs

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In the Eigenfunction Method article we discussed how to find a solution to inhomogeniuous ODE's by the method of Green's function. To generalize this method to find solution to inhomogenious PDEs we make use the general orthogonality property in more dimensions like introduced in the Double Fourier Series article. Denoting the operator of the PDE by $\mathcal{L}$ we write the inhomogenious PDE

\begin{align} \mathcal{L} u(\mathbf{r})=\rho(\mathbf{r}) \tag{1} \end{align}

For a hermitean Operator $\mathcal{L}$, the analogous procedure like for ODEs, gives the Green's function

\begin{align} G(\mathbf{r},\mathbf{r}_0) = \sum_{n=0}^\infty \frac{1}{\lambda_n} u_n(\mathbf{r}) u_n^*(\mathbf{r}_0) \end{align} $G(\mathbf{r},\mathbf{r}_0)$ for homogenious boundary conditions, where $u_n(\mathbf{r})$ are the Eigenfunctions of the homogenious problem. The solution to the inhomogenious problem (1) is \begin{align} u(\mathbf{r})=\int_V G(\mathbf{r},\mathbf{r}_0)\rho(\mathbf{r}_0)dV(\mathbf{r}_0) \tag{2} \end{align} where we integrate $(\mathbf{r}_0)$ over some volume where $\rho(\mathbf{r}_0) \not =0$. Just like for ODE's Green's function is the impulse response to a source at $\mathbf{r}_0$. \begin{align} \mathcal{L}G(\mathbf{r},\mathbf{r}_0) = \delta(\mathbf{r}-\mathbf{r}_0) \end{align}

For Poisson's equation the operator is $\mathcal{L}= \nabla ^2$ and for the Helmholtz's equation $\mathcal{L}= \nabla ^2+\mu^2$. For nonhomogenious boundary conditions the solution to poisson's equation (2) contains additional term, so that the boundary conditions are satisfied (see Poisson's Equation).

Using the convolution theorem the operator of the general linear PDE \begin{align} \mathcal{L}=\sum_{i,j=1}^n a_{ij} \partial_{x_ix_j} + \sum_{i=1}^n b_i \partial_{xi} + c \end{align} we can write the PDE in compact notation as

\begin{align} \mathcal{L}u(\mathbf{x})= \rho(\mathbf{x}_0) \end{align} and the inhomogenious solution to the linear PDE can be written in neat form as \begin{align} u(\mathbf{x})=G(\mathbf{x},\mathbf{x}_0) * \rho(\mathbf{x}_0) \end{align} the convolution of $G(\mathbf{x},\mathbf{x}_0)$ with $\rho(\mathbf{x}_0)$.