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 $t\in I,a\leq t\leq b\,$, 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,

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

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 $G_{0}(t,s)={\begin{cases}0&t\leq s\leq b\\X(t)X^{-1}(s)&a\leq s<t.\end{cases}}\,$

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

External links

• An example Archived 2006-03-28 at the Wayback Machine of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.