Green's matrix explained

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 x 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 x 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 ^{t} _{a} 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 ^{t} _{a} 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 ^{b} _{a} G _{0(t,s)g(s)ds.}

External links

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