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 \( x=Xy\, \) 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 of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
