Green's matrix

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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.

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\isin I, a\le t\le 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:

X(t) = \left[ x^1(t),...,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 inhomogenous equation.

Let x = Xy\, be the general solution. Now,

x'=X'y+Xy'\,

= AXy+Xy'\,

= Ax + Xy'\,

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\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_a^b G_0(t,s)g(s)ds\,.

[edit] External links

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