7.8: Summary

We have seen throughout the chapter that Green’s functions are the solutions of a differential equation representing the effect of a point impulse on either source terms, or initial and boundary conditions. The Green’s function is obtained from transform methods or as an eigenfunction expansion. In the text we have occasionally rewritten solutions of differential equations in term’s of Green’s functions. We will first provide a few of these examples and then present a compilation of Green’s Functions for generic partial differential equations.

For example, in section 7.4 we wrote the solution of the one dimensional heat equation as

\[u(x, t)=\int_{0}^{L} G(x, \xi ; t, 0) f(\xi) d \xi\nonumber \]

where

\[G(x, \xi ; t, 0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} e^{\lambda_{n} k t},\nonumber \]

and the solution of the wave equation as

\[u(x, t)=\int_{0}^{L} G_{c}(x, \xi, t, 0) f(\xi) d \xi+\int_{0}^{L} G_{s}(x, \xi, t, 0) g(\xi) d \xi,\nonumber \]

where

\[\begin{aligned} &G_{c}(x, \xi, t, 0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \cos \frac{n \pi c t}{L}, \\ &G_{s}(x, \xi, t, 0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \frac{\sin \frac{n \pi c t}{L}}{n \pi c / L} . \end{aligned} \nonumber \]

We note that setting \(t=0\) in \(G_{c}(x, \xi ; t, 0)\), we obtain

\[\mathrm{G}_{c}(x, \xi, 0,0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} .\nonumber \]

This is the Fourier sine series representation of the Dirac delta function, \(\delta(x-\xi)\). Similarly, if we differentiate \(G_{s}(x, \xi, t, 0)\) with repsect to \(t\) and set \(t=0\), we once again obtain the Fourier sine series representation of the Dirac delta function.

It is also possible to find closed form expression for Green’s functions, which we had done for the heat equation on the infinite interval,

\[u(x, t)=\int_{-\infty}^{\infty} G(x, t ; \xi, 0) f(\xi) d \xi,\nonumber \]

where

\[G(x, t ; \xi, 0)=\frac{e^{-(x-\xi)^{2} / 4 t}}{\sqrt{4 \pi t}}\nonumber \]

and for Poisson’s equation,

\[\phi(\mathbf{r})=\int_{V} G\left(\mathbf{r}, \mathbf{r}^{\prime}\right) f\left(\mathbf{r}^{\prime}\right) d^{3} r^{\prime},\nonumber \]

where the three dimensional Green’s function is given by

\[G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\frac{1}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|} .\nonumber \]

We can construct Green’s functions for other problems which we have seen in the book. For example, the solution of the two dimensional wave equation on a rectangular membrane was found in Equation (6.1.26) as

\[u(x, y, t)=\sum_{n=1}^{\infty} \sum_{m=1}^{\infty}\left(a_{n m} \cos \omega_{n m} t+b_{n m} \sin \omega_{n m} t\right) \sin \frac{n \pi x}{L} \sin \frac{m \pi y}{H}, \text { (7.153) }\label{eq:1} \]

where

\[ a_{n m}=\frac{4}{L H} \int_{0}^{H} \int_{0}^{L} f(x, y) \sin \frac{n \pi x}{L} \sin \frac{m \pi y}{H} d x d y,\label{eq:2} \]

\[b_{n m}=\frac{4}{\omega_{n m} L H} \int_{0}^{H} \int_{0}^{L} g(x, y) \sin \frac{n \pi x}{L} \sin \frac{m \pi y}{H} d x d y,\label{eq:3} \]

where the angular frequencies are given by

\[\omega_{n m}=c \sqrt{\left(\frac{n \pi}{L}\right)^{2}+\left(\frac{m \pi}{H}\right)^{2}} .\label{eq:4} \]

Rearranging the solution, we have

\[u(x, y, t)=\int_{0}^{H} \int_{0}^{L}\left[G_{c}(x, y ; \xi, \eta ; t, 0) f(\xi, \eta)+G_{s}(x, y ; \xi, \eta ; t, 0) g(\xi, \eta)\right] d \xi d \eta,\nonumber \]

where

\[G_{c}(x, y ; \xi, \eta ; t, 0)=\frac{4}{L H} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \tilde{\xi}}{L} \sin \frac{m \pi y}{H} \sin \frac{m \pi \eta}{H} \cos \omega_{n m} t\nonumber \]

and

\[G_{s}(x, y ; \xi, \eta ; t, 0)=\frac{4}{L H} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \sin \frac{m \pi y}{H} \sin \frac{m \pi \eta}{H} \frac{\sin \omega_{n m} t}{\omega_{n m}} .\nonumber \]

Once again, we note that setting \(t=0\) in \(G_{c}(x, \xi ; t, 0)\) and setting \(t=0\) in \(\frac{\partial G_{c}(x, \zeta ; t, 0)}{\partial t}\), we obtain a Fourier series representation of the Dirac delta function in two dimensions,

\[\delta(x-\xi) \delta(y-\eta)=\frac{4}{L H} \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \sin \frac{m \pi y}{H} \sin \frac{m \pi \eta}{H} .\nonumber \]

Another example was the solution of the two dimensional Laplace equation on a disk given by Equation (6.3.28). We found that

\[u(r, \theta)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right) r^{n} .\label{eq:5} \]

\[a_{n}=\frac{1}{\pi a^{n}} \int_{-\pi}^{\pi} f(\theta) \cos n \theta d \theta, \quad n=0,1, \ldots,\label{eq:6} \]

\[b_{n}=\frac{1}{\pi a^{n}} \int_{-\pi}^{\pi} f(\theta) \sin n \theta d \theta \quad n=1,2 \ldots\label{eq:7} \]

We saw that this solution can be written as

\[u(r, \theta)=\int_{-\pi}^{\pi} G(\theta, \phi ; r, a) f(\phi) d \phi,\nonumber \]

where the Green’s function could be summed giving the Poisson kernel

\[G(\theta, \phi ; r, a)=\frac{1}{2 \pi} \frac{a^{2}-r^{2}}{a^{2}+r^{2}-2 a r \cos (\theta-\phi)} .\nonumber \]

We had also investigated the nonhomogeneous heat equation in section 9.11.4,

\[\begin{array}{r} u_{t}-k u_{x x}=h(x, t), \quad 0 \leq x \leq L, \quad t>0 . \\ u(0, t)=0, \quad u(L, t)=0, \quad t>0, \\ u(x, 0)=f(x), \quad 0 \leq x \leq . \end{array}\label{eq:8} \]

We found that the solution of the heat equation is given by

\[u(x, t)=\int_{0}^{L} f(\xi) G(x, \xi ; t, 0) d \xi+\int_{0}^{t} \int_{0}^{L} h(\xi, \tau) G(x, \xi ; t, \tau) d \xi^{\tau} d \tau,\nonumber \]

where

\[G(x, \xi ; t, \tau)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \tilde{\xi}}{L} e^{-\omega_{n}^{2}(t-\tau)}\nonumber \]

Note that setting \(t=\tau\), we again get a Fourier sine series representation of the Dirac delta function.

In general, Green’s functions based on eigenfunction expansions over eigenfunctions of Sturm-Liouville eigenvalue problems are a common way to construct Green’s functions. For example, surface and initial value Green’s functions are constructed in terms of a modification of delta function representations modified by factors which make the Green’s function a solution of the given differential equations and a factor taking into account the boundary or initial condition plus a restoration of the delta function when applied to the condition. Examples with an indication of these factors are shown below.

1. \[g\left(x, y, z ; x^{\prime}, y^{\prime}, c\right)=\sum_{\ell, n} \underbrace{\frac{2}{a} \sin \frac{\ell \pi x}{a} \sin \frac{\ell \pi x^{\prime}}{a} \frac{2}{b} \sin \frac{n \pi y}{b} \sin \frac{n \pi y^{\prime}}{b}}_{\delta \text {-function }}[\underbrace{\sinh \gamma_{\ell n} z}_{\text {D.E. }} / \underbrace{\sinh \gamma_{\ell n} c}_{\text {restore } \delta}] .\nonumber \]
2. \[g\left(r, \phi, \theta ; a, \phi^{\prime}, \theta^{\prime}\right)=\sum_{\ell, m} \underbrace{Y_{\ell}^{m *}\left(\psi^{\prime} \theta^{\prime}\right) Y_{\ell}^{m *}(\psi \theta)}_{\delta \text {-function }}[\underbrace{r^{\ell}}_{\text {D.E. }} / \underbrace{a^{\ell}}_{\text {restore } \delta}] .\nonumber \]
3. \[g\left(x, t ; x^{\prime}, t_{0}\right)=\sum_{n} \underbrace{\frac{2}{L} \sin \frac{n \pi x}{L} \sin \frac{n \pi x^{\prime}}{L}}_{\delta-\text { function }}[\underbrace{e^{-a^{2} k_{n}^{2} t}}_{\text {D.E. }} / \underbrace{e^{-a^{2} k_{n}^{2} t_{0}}}_{\text {restore } \delta}] .\nonumber \]

We can extend this analysis to a more general theory of Green’s functions. This theory is based upon Green’s Theorems, or identities.

1. This is easily proven starting with the identity

   \[\nabla \cdot(\varphi \nabla \chi)=\nabla \varphi \cdot \nabla \chi+\varphi \nabla^{2} \chi,\nonumber \]

2. This is proven by interchanging \(\varphi\) and \(\chi\) in the first theorem and subtracting the two versions of the theorem.

The next step is to let \(\varphi=u\) and \(\chi=G\). Then,

\[\int_{V}\left(u \nabla^{2} G-G \nabla^{2} u\right) d V=\oint_{S}(u \nabla G-G \nabla u) \cdot \hat{\mathbf{n}} d S .\nonumber \]

As we had seen earlier for Poisson’s equation, inserting the differential equation yields

\[u(x, y)=\int_{V} G f d V+\oint_{S}(u \nabla G-G \nabla u) \cdot \hat{\mathbf{n}} d S .\nonumber \]

If we have the Green’s function, we only need to know the source term and boundary conditions in order to obtain the solution to a given problem.

In the next sections we provide a summary of these ideas as applied to some generic partial differential equations.\(^{1}\)

Note

This is an adaptation of notes from J. Franklin’s course on mathematical physics.

Laplace’s Equation: \(\nabla^{2} \psi=0\).

1. Boundary Conditions
   1. Dirichlet \(-\psi\) is given on the surface.
   2. Neumann \(-\hat{\mathbf{n}} \cdot \nabla \psi=\frac{\partial \psi}{\partial n}\) is given on the surface.

      Note

      Boundary conditions can be Dirichlet on part of the surface and Neumann on part. If they are Neumann on the whole surface, then the Divergence Theorem requires the constraint

      \[\int \frac{\partial \psi}{\partial n} d S=0\nonumber \]

2. Solution by Surface Green's Function, \(g(\mathbf{\vec{r}},\mathbf{\vec{r}}')\).
   2. Neumann conditions

      \[\begin{gathered} \nabla^{2} g_{N}\left(\overrightarrow{\mathbf{r}}, \overrightarrow{\mathbf{r}}^{\prime}\right)=0, \\ \frac{\partial g_{N}}{\partial n}\left(\overrightarrow{\mathbf{r}}_{s}, \overrightarrow{\mathbf{r}}_{s}^{\prime}\right)=\delta^{(2)}\left(\overrightarrow{\mathbf{r}}_{s}-\overrightarrow{\mathbf{r}}_{s}^{\prime}\right), \\ \psi(\overrightarrow{\mathbf{r}})=\int g_{N}\left(\overrightarrow{\mathbf{r}}, \overrightarrow{\mathbf{r}}_{s}^{\prime}\right) \frac{\partial \psi}{\partial n}\left(\overrightarrow{\mathbf{r}}_{s}^{\prime}\right) d S^{\prime} . \end{gathered}\nonumber \]

      Note

      Use of \(g\) is readily generalized to any number of dimensions.

Homogeneous Time Dependent Equations

1. Typical Equations
   1. ​​​​​Diffusion/Heat Equation \(\nabla^{2} \Psi=\frac{1}{a^{2}} \frac{\partial}{\partial t} \Psi\).
   2. Schrödinger Equation \(-\nabla^{2} \Psi+U \Psi=i \frac{\partial}{\partial t} \Psi\).
   3. Wave Equation \(\nabla^{2} \Psi=\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} \Psi\).
   4. General form: \(\mathcal{D} \Psi=\mathcal{T} \Psi\).
2. Initial Value Green’s Function, \(g\left(\overrightarrow{\mathbf{r}}, \overrightarrow{\mathbf{r}}^{\prime} ; t, t^{\prime}\right)\).
   1. Homogeneous Boundary Conditions
      1. \[\Psi(\overrightarrow{\mathbf{r}}, t)=\int g\left(\overrightarrow{\mathbf{r}}, \overrightarrow{\mathbf{r}}^{\prime} ; t, t_{0}\right) \Psi\left(\mathbf{r}^{\prime}, t_{0}\right) d^{3} \mathbf{r}^{\prime},\nonumber \]

         where

         \(g\left(\mathbf{r}_{s}\right)\) satisfies homogeneous boundary conditions.

      2. The first two properties in (a) above hold, but

         \[\begin{aligned} &g_{c}\left(\mathbf{r}, \mathbf{r}^{\prime} ; t_{0}, t_{0}\right)=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) \\ &\dot{g}_{s}\left(\mathbf{r}, \mathbf{r}^{\prime} ; t_{0}, t_{0}\right)=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) \end{aligned} \nonumber \]

         Note

         For the diffusion and Schrödinger equations the initial condition is Dirichlet in time. For the wave equation the initial condition is Cauchy, where \(\Psi\) and \(\Psi\) are given.

   2. Inhomogeneous, Time Independent (steady) Boundary Conditions
      1. Solve Laplace’s equation, \(\nabla^{2} \psi_{s}=0\), for inhomogeneous B.C.’s
      3. Then \(\Psi(\mathbf{r}, t)=\Psi_{t}(\mathbf{r}, t)+\psi_{s}(\mathbf{r})\).

         Note

         \(\Psi_{t}\) is the transient part and \(\psi_{s}\) is the steady state part.

3. Time Dependent Boundary Conditions with Homogeneous Initial Conditions
   1. or

      \[\Psi(\mathbf{r}, t)=\int_{t_{0}}^{\infty} \frac{\partial h_{N}}{\partial n}\left(\mathbf{r}, \mathbf{r}_{s}^{\prime} ; t, t^{\prime}\right) \Psi\left(\mathbf{r}_{s}^{\prime}, t^{\prime}\right) d t^{\prime}\nonumber \]

   3. Note

      For inhomogeneous I.C.,

      \[\Psi=\int g \Psi\left(\mathbf{r}^{\prime}, t_{0}\right)+\int d t^{\prime} h_{D} \Psi\left(\mathbf{r}_{s}^{\prime}, t^{\prime}\right) d^{3} \mathbf{r}^{\prime} .\nonumber \]

Inhomogeneous Steady State Equation

1. \[\nabla^{2} \psi(\mathbf{r}, t)=f(\mathbf{r}), \quad \psi\left(\mathbf{r}_{s}\right) \quad \text { or } \quad \frac{\partial \psi}{\partial n}\left(\mathbf{r}_{s}\right) \quad \text { given. }\nonumber \]
   1. where \(\nabla^{\prime}\) denotes differentiation with respect to \(r^{\prime}\).
   2. Properties of \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right):\)
      1. \(\nabla^{\prime 2} G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right)\).
      2. \(\left.G\right|_{s}=0\) or \(\left.\frac{\partial G}{\partial n^{\prime}}\right|_{s}=0\).
   3. If there are pure Neumann conditions and \(S\) is finite and \(\int f d^{3} \mathbf{r} \neq\) 0 by symmetry, then \(\left.\vec{n}^{\prime} \cdot \nabla^{\prime} G\right|_{s} \neq 0\) and the Green’s function method is much more complicated to solve.
   4. or

      \[G_{N}\left(\mathbf{r}, \mathbf{r}_{s}^{\prime}\right)=-g_{N}\left(\mathbf{r}, \mathbf{r}_{s}^{\prime}\right) .\nonumber \]

   5. \(G\) satisfies a reciprocity property, \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=G\left(\mathbf{r}^{\prime}, \mathbf{r}\right)\) for either Dirichlet or Neumann boundary conditions.
   6. \(G\left(\mathbf{r}, \mathbf{r}^{\prime}\right)\) can be considered as a potential at \(\mathbf{r}\) due to a point charge \(q=-1 / 4 \pi\) at \(\mathbf{r}^{\prime}\), with all surfaces being grounded conductors.

Inhomogeneous, Time Dependent Equations

1. Diffusion/Heat Flow \(\nabla^{2} \Psi-\frac{1}{a^{2}} \dot{\Psi}=f(\mathbf{r}, t)\).
   1. \[\begin{align} \left[\nabla^{2}-\frac{1}{a^{2}} \frac{\partial}{\partial t}\right] G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t^{\prime}\right) &=\left[\nabla^{\prime 2}+\frac{1}{a^{2}} \frac{\partial}{\partial t^{\prime}}\right] G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t^{\prime}\right)\nonumber \\ &=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) \delta\left(t-t^{\prime}\right) .\label{eq:10} \end{align} \]
   2. Green’s Theorem in 4 dimensions \((\mathbf{r}, t)\) yields

      \[\begin{aligned} \Psi(\mathbf{r}, t) &=\iint_{t_{0}}^{\infty} G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t^{\prime}\right) f\left(\mathbf{r}^{\prime}, t^{\prime}\right) d t^{\prime} d^{3} \mathbf{r}^{\prime}-\frac{1}{a^{2}} \int G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t_{0}\right) \Psi\left(\mathbf{r}^{\prime}, t_{0}\right) d^{3} \mathbf{r}^{\prime} \\ &+\int_{t_{0}}^{\infty} \int\left[\Psi\left(\mathbf{r}_{s^{\prime}}^{\prime}, t\right) \nabla^{\prime} G_{D}\left(\mathbf{r}, \mathbf{r}_{s^{\prime}}^{\prime} t, t^{\prime}\right)-G_{N}\left(\mathbf{r}, \mathbf{r}_{s}^{\prime} ; t, t^{\prime}\right) \nabla^{\prime} \Psi\left(\mathbf{r}_{s}^{\prime}, t^{\prime}\right)\right] \cdot \overrightarrow{d S}^{\prime} d t^{\prime} . \end{aligned} \nonumber \]

   3. Either \(G_{D}\left(\mathbf{r}_{s}^{\prime}\right)=0\) or \(G_{N}\left(\mathbf{r}_{s}^{\prime}\right)=0\) on \(S\) at any point \(\mathbf{r}_{s}^{\prime}\).
   4. \(\hat{\mathbf{n}}^{\prime} \cdot \nabla^{\prime} G_{D}\left(\mathbf{r}_{s}^{\prime}\right)=h_{D}\left(\mathbf{r}_{s}^{\prime}\right), G_{N}\left(\mathbf{r}_{s}^{\prime}\right)=-h_{N}\left(\mathbf{r}_{s}^{\prime}\right)\), and \(-\frac{1}{a^{2}} G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t_{0}\right)=\) \(g\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t_{0}\right)\).
2. Wave Equation \(\nabla^{2} \Psi-\frac{1}{c^{2}} \frac{\partial^{2} \Psi}{\partial^{2} t}=f(\mathbf{r}, t)\).
   1. \[\begin{align} \left[\nabla^{2}-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\right] G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t^{\prime}\right) &=\left[\nabla^{\prime 2}-\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\right] G\left(\mathbf{r}, \mathbf{r}^{\prime} ; t, t^{\prime}\right)\nonumber \\ &=\delta\left(\mathbf{r}-\mathbf{r}^{\prime}\right) \delta\left(t-t^{\prime}\right) .\label{eq:11} \end{align} \]
   3. Cauchy initial conditions are given: \(\Psi\left(t_{0}\right)\) and \(\Psi{\Psi}\left(t_{0}\right)\).
   4. The wave and diffusion equations satisfy a causality condition \(G\left(t, t^{\prime}\right)=0, \quad t^{\prime}>t .\)​​​​​​​