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 ${\displaystyle x'=A(t)x+g(t)\,}$ where ${\displaystyle x\,}$ is a vector and ${\displaystyle A(t)\,}$ is an ${\displaystyle n\times n\,}$ matrix function of ${\displaystyle t\,}$, which is continuous for ${\displaystyle t\in I,a\leq t\leq b\,}$, where ${\displaystyle I\,}$ is some interval.

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

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

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

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 ${\displaystyle x=Xy\,}$ be the general solution. Now,

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

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

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

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

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

The particular solution can now be written ${\displaystyle x_{p}(t)=\int _{a}^{b}G_{0}(t,s)g(s)\,ds.\,}$