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.

For instance, consider $x\text{'}=A(t)x+g(t),$ where $x,$ is a vector and $A(t),$ is an $n\; imes\; n,$ matrix function of $t,$, which is continuous for $tisin\; I,\; ale\; tle\; b,$, where $I,$ is some interval.

Now let $x^1(t),...,x^n(t),$ be $n,$ linearly independent solutions to the homogeneous equation $x\text{'}=A(t)x,$ and arrange them in columns to form a fundamental matrix:

$X(t)\; =\; left\; [\; x^1(t),...,x^n(t)\; ight]\; ,$

Now $X(t),$ is an $n\; imes\; n,$ matrix solution of $X\text{'}=AX,$.

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,

$x\text{'}=X\text{'}y+Xy\text{'},$

$=\; AXy+Xy\text{'},$

$=\; Ax\; +\; Xy\text{'},$

This implies $Xy\text{'}=g,$ 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 .

The particular solution can now be written $x\_p(t)\; =\; int\_a^b\; G\_0(t,s)g(s)ds,$.

External links

* [http://www.exampleproblems.com/wiki/index.php/ODELS4 An example] of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.