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 [math]\displaystyle{ x'=A(t)x+g(t)\, }[/math] where [math]\displaystyle{ x\, }[/math] is a vector and [math]\displaystyle{ A(t)\, }[/math] is an [math]\displaystyle{ n\times n\, }[/math] matrix function of [math]\displaystyle{ t\, }[/math], which is continuous for [math]\displaystyle{ t\isin I, a\le t\le b\, }[/math], where [math]\displaystyle{ I\, }[/math] is some interval.

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

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

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

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 [math]\displaystyle{ x = Xy\, }[/math] be the general solution. Now,

[math]\displaystyle{ \begin{align} x' & =X'y+Xy' \\ & = AXy+Xy' \\ & = Ax + Xy'. \end{align} }[/math]

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

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

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

Now define the Green's matrix [math]\displaystyle{ G_0(t,s)= \begin{cases} 0 & t\le s\le b \\ X(t)X^{-1}(s) & a\le s \lt t. \end{cases}\, }[/math]

The particular solution can now be written [math]\displaystyle{ x_p(t) = \int_a^b G_0(t,s)g(s)\,ds.\, }[/math]

External links

• An example of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.