Green's matrix

In mathematics, and in particular ordinary differential equations, a Green's matrix helps to determine a particular solution to a first-order inhomogeneous linear system of ODEs. The concept is named after George Green.

For instance, consider $$x'=A(t)x+g(t)\,$$ where $$x\,$$ is a vector and $$A(t)\,$$ is an $$n\times n\,$$ matrix function of $$t\,$$, which is continuous for $$t\isin I, a\le t\le b\,$$, where $$I\,$$ is some interval.

Now let $$x^1(t),\ldots,x^n(t)\,$$ be $$n\,$$ linearly independent solutions to the homogeneous equation $$x'=A(t)x\,$$ and arrange them in columns to form a fundamental matrix:


 * $$X(t) = \left[ x^1(t),\ldots,x^n(t) \right].\,$$

Now $$X(t)\,$$ is an $$n\times n\,$$ matrix solution of $$X'=AX\,$$.

This fundamental matrix will provide the homogeneous solution, and if added to a particular solution will give the general solution to the inhomogeneous equation.

Let $$x = Xy\,$$ be the general solution. Now,



\begin{align} x' & =X'y+Xy' \\ & = AXy+Xy' \\ & = Ax + Xy'. \end{align} $$

This implies $$Xy'=g\,$$ or $$y = c+\int_a^t X^{-1}(s)g(s)\,ds\,$$ where $$c\,$$ is an arbitrary constant vector.

Now the general solution is $$x=X(t)c+X(t)\int_a^t X^{-1}(s)g(s)\,ds.\,$$

The first term is the homogeneous solution and the second term is the particular solution.

Now define the Green's matrix $$G_0(t,s)= \begin{cases} 0 & t\le s\le b \\ X(t)X^{-1}(s) & a\le s < t. \end{cases}\,$$

The particular solution can now be written $$x_p(t) = \int_a^b G_0(t,s)g(s)\,ds.\,$$