"The young theoretical physicists of a generation or two earlier subscribed to the belief that: If you haven't done something important by age 30, you never will. Obviously, they were unfamiliar with the history of George Green, the miller of Nottingham…" - Julian Schwinger$^1$
HomeGreen's functions are routinely used to solve inhomogeneous ordinary and partial differential equations.$^{2}$ Green's function approaches are also used to solve homogeneous differential equations with inhomogeneous boundary conditions. Here we will discuss some of the properties of the Green's functions and highlight their importance in a reaction-diffusion equation context.
Some background on Green's functions
First we consider an example of Green's functions with spatial variables only. Suppose we are to solve an inhomogeneous Sturm-Lioville problem $$ \hat{L}y(x)=-\frac{d}{dx}(p(x)\frac{d}{dx}y(x))+q(x)y(x)=f(x) $$ subject to the homogeneous boundary conditions $$ \alpha_1y(a)+\alpha_2\frac{dy}{dx}\Bigr|_{x=a}=0 \\ \beta_1y(b)+\beta_2\frac{dy}{dx}\Bigr|_{x=b}=0 \\ x\in[a,b] $$ This problem is transposed into a problem to solve for a function $G(x|x')$ such that $$ \hat{L}G(x|x')=\delta(x-x') $$ where the function $G(x|x')$ obeys the same homogeneous boundary conditions as above. Clearly, the inhomogeneous term has been replaced with a point source term. The idea behind solving for $G(x|x')$ is that the general solution to the inhomogeneous problem ($y(x)$) obeys $$ y(x)=\int^b_a f(x')G(x|x')dx' $$ $G(x|x')$ is called a Green's functions or sometimes even referred to as the fundamental solution to the operator $\hat{L}$. In our example case the analytical expressions for the Green's function of the Sturm-Liouville problem are well known. Hence solving the inhomogeneous problem involves evaluating the above integral. The physical interpretation of the above equation is very interesting as well. The Green's function represents the response of the system at $x$ because of a point impulse at $x'$. Thus, the solution to the inhomogeneous problem (or the total effect at a point $x$) is a superposition of responses to point impulses (at $x'$) of strength $f(x')$. Many operators frequently used in physics and engineering have analytical expressions for the associated Green's function.
Green's functions for partial differential equations are functions of the spatial ($x$) as well as time ($t$) variables. We illustrate this by considering the solution of the 1-D diffusion problem with an arbitrary initial condition. $$ \frac{\partial \rho(x,t)}{\partial t}=\alpha^2 \frac{\partial^2 \rho(x,t)}{\partial x^2} $$ subject to the initial condition $\rho(x,0)=\phi(x)$. Fourier transforming both sides with respect to the spatial variable yields $$ \frac{\partial \tilde{\rho}(k,t)}{\partial t}=-\alpha^2k^2\tilde{\rho}(k,t) $$ The above equation is an ODE and can be easily solved to give $$ \tilde{\rho}(k,t)=\tilde{\rho}(k,0)e^{-\alpha^2k^2t} $$ Now we first consider a case when the initial condition is a point impulse $\rho(x,0)=\delta(x-x')$. The fourier transform of the impulse initial condition is $\tilde{\rho}(k,0)=e^{-i k x'}$. Thus, the inverse fourier transform of equation 7 yields $$ \rho(x,t)=\frac{1}{2\pi} \int_{-\infty}^{\infty} dk \ e^{ikx}e^{-ikx'}e^{-\alpha^2k^2t}=\frac{1}{\sqrt{4\pi\alpha^2t}}e^{-(x-x')^2/4\alpha^2t} $$ where completion of squares is used to convert the integral into a Gaussian integral. Once again starting from equation 7 if we were to solve for an arbitrary initial condition instead of the impulse we must fourier invert both sides of equation 7. First we note that the individual fourier inversions of the quantities on the RHS yields $F^{-1}[\tilde{\rho}(k,0)]=\phi(x)$ and $F^{-1}[e^{-\alpha^2k^2t}]=\frac{1}{\sqrt{4\pi\alpha^2t}}e^{x^2/4\alpha^2t}$. Additionally, we know that the fourier transform of a convolution integral is a product of fourier transforms of the individual functions. Thus, the inverse fourier transform of equation 7 results in a convolution integral $$ \rho(x,t)=\int_{-\infty}^{\infty}dx' \ \phi(x') \frac{1}{\sqrt{4\pi\alpha^2t}} e^{-(x-x')^2/4\alpha^2t} $$ Note that the solution to the impulse response (equation 8) appears inside the integral for the complete solution (equation 9). Thus, the function $\frac{1}{\sqrt{4\pi\alpha^2t}}e^{(x-x')^2/4\alpha^2t}$ is referred to as the impulse-response function or Green's function for the 1-D diffusion equation. Thus, formally the complete solution is $$ \rho(x,t)=\int_{-\infty}^{\infty}dx' \ \phi(x') G(x,t|x') \\ G(x,t|x')=\frac{1}{\sqrt{4\pi\alpha^2t}}e^{(x-x')^2/4\alpha^2t} $$ The Green's function ($G(x,t|x')$) here physically represents the response of the system at a point $x$ and at time $t$ due to an impulse at at a point $x'$ and time $t=0$. Thus, the complete response at a point $x$ at time $t$ is a continuous superposition of the intial $t=0$ impulses of strength $\phi(x')$.
The inhomogeneous reaction-diffusion equation
Here we summarize how the Green's function solution to the general inhomogeneous reaction-diffusion equation relates to the complete solution of the problem. We came across a similar treatment in the book by Evans et. al.$^3$ (see Chapter 5). The key observation is note that the procedure described there may be applied to the reaction-diffusion problem as well. We consider the governing equation $$ \frac{\partial u(\vec{r},t)}{\partial t}=\nabla^2u(\vec{r},t)-u(\vec{r},t)+f(\vec{r},t) $$ where the first term on the RHS is the diffusion term, the second term is the kinetic consumption term (first order) and the last term is the inhomogeneous term. Note that we have ignored the constants like the diffusion coefficient ($D$), rate constant ($k$) for simplicity as it doesn't affect the analysis that follows. Rearranging the above equation yields $$ \Big( \nabla^2-1-\frac{\partial}{\partial t} \Big)u(\vec{r},t)=-f(\vec{r},t) $$ The Green's function as shown previously is obtained by replacing the inhomogeneous term with a point source in spatial and time variables $$ \Big( \nabla^2-1-\frac{\partial}{\partial t} \Big)G(\vec{r},t|\vec{r}',0)=-\delta^3(\vec{r}-\vec{r}')\delta(t-0) $$ We now Laplace transform both sides of equations 12 and 13 with respect to $t$ to yield $$ \nabla^2\tilde{u}-\tilde{u}-s\tilde{u}+u(\vec{r},0^-)=-\tilde{f} \\ \nabla^2\tilde{G}-\tilde{G}-s\tilde{G}+G(\vec{r},0^-|\vec{r}')=-\delta^3(\vec{r}-\vec{r}') $$ where the Laplace transform limit started slightly to the left of $0$ (denoted as $0^-$) instead of $0$. This is to ensure that the Dirac delta function in $t$ is comfortably integrated over. The term $u(\vec{r},0^-)$ is nothing but the initial condition $u(\vec{r},0)$. The term $G(\vec{r},0^-|\vec{r}')$, however, is zero. This is because of the causality property of Green's functions. In other words, Green's functions are responses of the system to impulse reponses applied at time $t=0$. The response of the system before the impulse is applied is declared to be zero. We now multiply the top equation in 14 by $\tilde{G}$ and the bottom equation by $\tilde{u}$ and subtract the two resulting relations to yield $$ \tilde{G}\nabla^2\tilde{u}-\tilde{u}\nabla^2\tilde{G}+u(\vec{r},0)\tilde{G} =-\tilde{f}\tilde{G}+\delta^3(\vec{r}-\vec{r}')\tilde{u} $$ Both sides of the equations can now be integrated over the volume $V$ $$ \int_V(\tilde{G}\nabla^2\tilde{u}-\tilde{u}\nabla^2\tilde{G})\ d^3\vec{r}'+\int_Vu(\vec{r},0)\tilde{G}\ d^3\vec{r}' =-\int_V\tilde{f}\tilde{G}\ d^3\vec{r}'+\tilde{u}(\vec{r},s) $$ We may now use Green's second identity to convert the volume integral term into a surface integral term as given by $$ \int_V(\tilde{G}\nabla^2\tilde{u}-\tilde{u}\nabla^2\tilde{G})\ d^3\vec{r}' =\oint_S(\tilde{G}\nabla\tilde{u}-\tilde{u}\nabla\tilde{G}).\hat{n}\ d^2\vec{r}' $$ Using the above equation in equation 16 and rearranging the terms yields $$ \tilde{u}(\vec{r},s)=\int_V\tilde{f}(\vec{r}',s)\tilde{G}(\vec{r},s|\vec{r}')d^3\vec{r}' + \int_Vu(\vec{r}',0)\tilde{G}(\vec{r},s|\vec{r}')\ d^3\vec{r}' + \oint_S(\tilde{G}\nabla\tilde{u}-\tilde{u}\nabla\tilde{G}).\hat{n}\ d^2\vec{r}' $$ The inverse Laplace transform of the LHS is our complete solution of interest! We additionally note that the first and last terms of the RHS are products of Laplace transforms. Hence, the inverse Laplace of these terms will yield a convolution integral in time. Accordingly, $$ u(\vec{r},t)=\int_0^t\int_Vf(\vec{r}',t')G(\vec{r},t-t'|\vec{r}')\ d^3\vec{r}' dt' +\int_Vu(\vec{r}',0)G(\vec{r},t|\vec{r}')d^3\vec{r}' +\int_0^t\oint_S \Big[G(\vec{r},t|\vec{r}')\nabla u(\vec{r},t-t')-u(\vec{r}',t)\nabla G(\vec{r},t-t'|\vec{r}') \Big].\hat{n}\ d^2\vec{r}'dt' $$ which is a general solution to the inhomogeneous reaction-diffusion problem that can be used after solving for the Green's function. We can now derive some physical insight with the help of the above equation. The complete solution $u(\vec{r},t)$ is a sum of three terms. The first term is the response of the system to the inhomogeneous term. Note that integral over all volume represents the superposition of impulses applied in the domain of the problem and the Green's function has a time argument of $t-t'$ which only accounts for the time elapased since the impulse. The integral over time accounts for the superposition of impulses in time. The second term propagates the response of the system arising from the initial condition. The third term is the response of the system because of the boundary conditions. Although it may appear as if $u(\vec{r},t)$ appears on both sides of the above equation, the RHS term is evaluated at the surface of the domain $V$. This implies that the terms inside the integral are determined from the boundary conditions of the problem. Additionally, for homogeneous boundary conditions it can be shown that the third term is zero. This may be easily understood for a one dimensional reaction-diffusion problem with homogeneous boundary conditions.
1. Biography edited by Ng Y. Julian Schwinger: The Physicist, the Teacher, and the Man. World Scientific, 1996.
2. Farlow S. Partial Differential Equations for Scientists and Engineers. Dover, 1993.
3. Evans, G., Blackledge, J., and Yardley, P. Analytic Methods for Partial Differential Equations. Springer, 1999.