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 × n matrix function of t , which is continuous for t ∈ I , a ≤ t ≤ b , where I is some interval.

Now let x 1 ( t ) , … , 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:

Now X ( t ) is an n × n matrix solution of X ′ = A X .

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 = X y be the general solution. Now,

This implies X y ′ = g or y = c + ∫ a t X − 1 ( s ) g ( s ) d s where c is an arbitrary constant vector.

Now the general solution is x = X ( t ) c + X ( t ) ∫ a t X − 1 ( s ) g ( s ) d s .

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 ) = { 0 t ≤ s ≤ b X ( t ) X − 1 ( s ) a ≤ s < t .

The particular solution can now be written x p ( t ) = ∫ a b G 0 ( t , s ) g ( s ) d s .