Applied linear operators and spectral methods/Greens functions 2

Green's functions for linear differential operators
Consider the one-dimensional heat equation given by

- \cfrac{d^2 u}{d x^2} = f(x) \qquad x \in (0,1),u(0) = 0, u(1) = 0 ~. $$ This equation has Green's function $$g(x,y)$$ which satisfies

- \frac{\partial^2 g}{\partial x^2} = \delta(x-y) \qquad x,y \in (0,1) $$ The Green's function is given by

g(x,y) = \begin{cases} (1-y)~x & 0 \le x < y \\ (1-x)~y & y < x \le 1 \end{cases} $$ or,

g(x,y) = (1-y)~x + (y-x)~H(x-y) ~. $$ Therefore,

g_x(x,y) = (1-y) - H(x-y) + (y-x)\frac{\partial }{\partial x}[H(x-y)] = (1-y) - H(x-y) + (y-x)\delta(x-y) $$ Now,

\left\langle x~\delta(x), \phi \right\rangle = \left\langle \delta(x), x~\phi \right\rangle = (x~\phi)(0) = 0 $$ Hence,

g_x(x,y) = (1-y) - H(x-y) $$ And,

g_{xx}(x,y) = - \delta(x-y) $$ Note that the second derivative of $$g$$ is a delta function.

We can use this observation to arrive at a more general description of the Green's function for a particular differential equation. That is, for a $$n$$th order linear differential operator we would want the $$n$$th derivative of $$g(x,y)$$ to be like a delta function. Thus the $$(n-1)$$th derivative of $$g(x,y)$$ should be like a Heaviside function and all lower derivatives should be continuous.

In particular, consider the operator $$\mathcal{L}$$ acting on $$g$$ such that

\mathcal{L}~g = \delta(x-y) $$ with

\mathcal{L} = a_n(x)~\cfrac{d^n}{dx^n} + a_{n-1}(x)~\cfrac{d^{(n-1)}}{dx^{(n-1)}} + \dots a_0(x) $$ where $$a_i(x) \in C^{\infty}$$. If we integrate this equation across the point $$x = y$$ from $$x = y^{-1}$$ to $$x = y^{+}$$ we get

\left.a_n(x) \cfrac{d^{n-1}g}{dx^{n-1}} \right|_{x = y^{-}}^{x = y^{+}} = \int_{y^{-1}}^{y^{+}} \delta(x)~\text{d}x = 1 $$ This condition is called a  Jump condition.

This suggests that the Green's function $$g(x,y)$$ satisfying

\mathcal{L}_x g(x,y) = \delta(x-y) $$ in the sense of distributions has the properties that


 * 1) $$\mathcal{L}_x g(x,y) = 0$$ for all $$x \ne y$$.
 * 2) $$\cfrac{d^k g(x,y)}{d x^k}$$ is continuous at $$x = y$$ for $$k = 0, 1, \dots, n-2$$.
 * 3) $$ \left.\cfrac{d^{n-1}g(x,y)}{dx^{n-1}} \right|_{x = y^{-}}^{x = y^{+}} = \cfrac{1}{a_n(y)}$$.
 * 4) $$g(x,y)$$ must satisfy all appropriate  homogeneous boundary conditions.

If the Green's function exists then

u(x) = \int_a^b g(x,y)~f(y)~\text{d}y ~. $$

Example
Let us consider a second order differentiable linear operator, i.e., $$n = 2$$ on $$[a,b]$$ with  separated boundary conditions,

\begin{align} \alpha_1~u(a) + \beta_1~u'(a) & = 0 \qquad \implies \mathcal{B}_1[u(a)] = 0 \\ \alpha_2~u(b) + \beta_2~u'(b) & = 0 \qquad \implies \mathcal{B}_2[u(b)] = 0 \end{align} $$ where $$\mathcal{B}_1$$ and $$\mathcal{B}_2$$ are  boundary operators.

The differential equation

\mathcal{L}_x g(x,y) = 0 $$ and its required continuity conditions are satisfied is

g(x,y) = \begin{cases} A_1~u_1(x)~u_2(y) & a \le y < x \\ A_2~u_2(x)~u_1(y) & y < x \le b          \end{cases} $$ The first two terms above are to satisfy the boundary conditions while the third terms gives us continuity.

The jump condition is that

\left.\cfrac{dg(x,y)}{dx} \right|_{x = y^{-}}^{x = y^{+}} = \cfrac{1}{a_2(y)} = A [u_2'(y)~u_1(y) - u_1'(y)~u_2(y)] = A~W(y) $$ where $$W(y)$$ is the  Wronskian.

Therefore,

A = \cfrac{1}{a_2(y)~W(y)}, W(y) \ne 0 $$

For the heat equation

\mathcal{L} u = - u'' $$ we can take

u_1(x) = x,u_2(x) = (1-x),\text{and} W(y) = -1 $$ That gives us $$A = 1$$ and we recover the same $$g(x,y)$$ as before.

In the general case, the solution is

u(x) = \int_a^x \cfrac{u_2(x)~u_1(y)~f(y)}{a_2(y)~W(y)}~\text{d}y + \int_x^b \cfrac{u_1(x)~u_2(y)~f(y)}{a_2(y)~W(y)}~\text{d}y $$