7.4: Green’s Functions for 1D Partial Differential Equations

In Section 7.1 we encountered the initial value green’s function for initial value problems for ordinary differential equations. In that case we were able to express the solution of the differential equation \(L[y]=\) \(f\) in the form \[y(t)=\int G(t, \tau) f(\tau) d \tau,\nonumber \] where the Green’s function \(G(t, \tau)\) was used to handle the nonhomogeneous term in the differential equation. In a similar spirit, we can introduce Green’s functions of different types to handle nonhomogeneous terms, nonhomogeneous boundary conditions, or nonhomogeneous initial conditions. Occasionally, we will stop and rearrange the solutions of different problems and recast the solution and identify the Green’s function for the problem.

In this section we will rewrite the solutions of the heat equation and wave equation on a finite interval to obtain an initial value Green;s function. Assuming homogeneous boundary conditions and a homogeneous differential operator, we can write the solution of the heat equation in the form \[u(x, t)=\int_{0}^{L} G\left(x, \xi ; t, t_{0}\right) f(\xi) d \xi .\nonumber \] where \(u\left(x, t_{0}\right)=f(x)\), and the solution of the wave equation as \[u(x, t)=\int_{0}^{L} G_{c}\left(x, \xi, t, t_{0}\right) f(\xi) d \xi+\int_{0}^{L} G_{s}\left(x, \xi, t, t_{0}\right) g(\xi) d \xi .\nonumber \] where \(u\left(x, t_{0}\right)=f(x)\) and \(u_{t}\left(x, t_{0}\right)=g(x)\). The functions \(G\left(x, \xi ; t, t_{0}\right)\), \(G\left(x, \xi ; t, t_{0}\right)\), and \(G\left(x, \xi ; t, t_{0}\right)\) are initial value Green’s functions and we will need to explore some more methods before we can discuss the properties of these functions. [For example, see Section.]

We will now turn to showing that for the solutions of the one dimensional heat and wave equations with fixed, homogeneous boundary conditions, we can construct the particular Green’s functions.

Heat Equation

In Section 3.5 we obtained the solution to the one dimensional heat equation on a finite interval satisfying homogeneous Dirichlet conditions, \[\begin{align} &u_t=ku_{xx},\quad 0<t,\quad 0\leq x\leq L, \nonumber \\ &u(x, 0)=f(x), \quad 0<x<L, \nonumber \\ &u(0, t)=0, \quad t>0, \nonumber \\ &u(L, t)=0, \quad t>0 .\label{eq:1}\end{align}\] The solution we found was the Fourier sine series \[u(x, t)=\sum_{n=1}^{\infty} b_{n} e^{\lambda_{n} k t} \sin \frac{n \pi x}{L},\nonumber \] where \[\lambda_{n}=-\left(\frac{n \pi}{L}\right)^{2}\nonumber \] and the Fourier sine coefficients are given in terms of the initial temperature distribution, \[b_{n}=\frac{2}{L} \int_{0}^{L} f(x) \sin \frac{n \pi x}{L} d x, \quad n=1,2, \ldots .\nonumber \]

Inserting the coefficients \(b_{n}\) into the solution, we have \[u(x, t)=\sum_{n=1}^{\infty}\left(\frac{2}{L} \int_{0}^{L} f(\xi) \sin \frac{n \pi \xi}{L} d \xi\right) e^{\lambda_{n} k t} \sin \frac{n \pi x}{L} .\nonumber \] Interchanging the sum and integration, we obtain \[u(x, t)=\int_{0}^{L}\left(\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \tilde{\xi}}{L} e^{\lambda_{n} k t}\right) f(\xi) d \xi .\nonumber \]

This solution is of the form \[u(x, t)=\int_{0}^{L} G(x, \xi ; t, 0) f(\xi) d \xi .\nonumber \] Here the function \(G(x, \xi ; t, 0)\) is the initial value Green’s function for the heat equation in the form \[G(x, \xi ; t, 0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} e^{\lambda_{n} k t} .\nonumber \] which involves a sum over eigenfunctions of the spatial eigenvalue problem, \(X_{n}(x)=\sin \frac{n \pi x}{L}\).

Wave Equation

The solution of the one dimensional wave equation (2.1.2), \[\begin{align} u_{t t} &=c^{2} u_{x x}, \quad 0<t, \quad 0 \leq x \leq L,\nonumber \\ u(0, t) &=0, \quad u(L, 0)=0, \quad t>0,\nonumber \\ u(x, 0) &=f(x), \quad u_{t}(x, 0)=g(x), \quad 0<x<L,\label{eq:2} \end{align}\] was found as \[u(x, t)=\sum_{n=1}^{\infty}\left[A_{n} \cos \frac{n \pi c t}{L}+B_{n} \sin \frac{n \pi c t}{L}\right] \sin \frac{n \pi x}{L} .\nonumber \] The Fourier coefficients were determined from the initial conditions, \[\begin{align} &f(x)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi x}{L},\nonumber \\ &g(x)=\sum_{n=1}^{\infty} \frac{n \pi c}{L} B_{n} \sin \frac{n \pi x}{L},\label{eq:3} \end{align}\] as \[\begin{align} A_{n} &=\frac{2}{L} \int_{0}^{L} f(\xi) \sin \frac{n \pi \xi}{L} d \xi,\nonumber \\ B_{n} &=\frac{L}{n \pi c} \frac{2}{L} \int_{0}^{L} f(\xi) \sin \frac{n \pi \xi}{L} d \xi .\label{eq:4} \end{align}\]

Inserting these coefficients into the solution and interchanging integration with summation, we have \[\begin{align} u(x, t)=& \int_{0}^{\infty}\left[\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \cos \frac{n \pi c t}{L}\right] f(\xi) d \xi\nonumber \\ &+\int_{0}^{\infty}\left[\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \frac{\sin \frac{n \pi c t}{L}}{n \pi c / L}\right] g(\xi) d \xi\nonumber \\ =& \int_{0}^{L} G_{c}(x, \xi, t, 0) f(\xi) d \xi+\int_{0}^{L} G_{s}(x, \xi, t, 0) g(\xi) d \xi .\label{eq:5} \end{align}\] In this case, we have defined two Green’s functions, \[\begin{align} &\mathrm{G}_{c}(x, \xi, t, 0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \xi}{L} \cos \frac{n \pi c t}{L}\nonumber \\ &\mathrm{G}_{s}(x, \xi, t, 0)=\frac{2}{L} \sum_{n=1}^{\infty} \sin \frac{n \pi x}{L} \sin \frac{n \pi \tilde{\xi}}{L} \frac{\sin \frac{n \pi c t}{L}}{n \pi c / L}\label{eq:6} \end{align}\] The first, \(G_{c}\), provides the response to the initial profile and the second, \(G_{s}\), to the initial velocity.