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 $n \times n$ matrix function of $t$ , which is continuous for $t \in I, a \leq t \leq b$ , where $I$ is some interval.

Now let $x^{1} (t), \dots, x^{n} (t)$ 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) = [x^{1} (t), \dots, x^{n} (t)] .

Now $X (t)$ is an $n \times n$ 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,

\begin{matrix} x^{'} & = X^{'} y + X y^{'} \\ = A X y + X y^{'} \\ = A x + X y^{'} . \end{matrix}

This implies $X y^{'} = g$ or $y = c + \int_{a}^{t} X^{- 1} (s) g (s) d s$ where $c$ is an arbitrary constant vector.

Now the general solution is $x = X (t) c + X (t) \int_{a}^{t} X^{- 1} (s) g (s) d s .$

The first term is the homogeneous solution and the second term is the particular solution.

Now define the Green's matrix $G_{0} (t, s) = {\begin{matrix} 0 & t \leq s \leq b \\ X (t) X^{- 1} (s) & a \leq s < t . \end{matrix}$

The particular solution can now be written $x_{p} (t) = \int_{a}^{b} G_{0} (t, s) g (s) d s .$

External links

An example Archived 2006-03-28 at the Wayback Machine of solving an inhomogeneous system of linear ODEs and finding a Green's matrix from www.exampleproblems.com.

References

↑ Villanueva, R. J.; Jodar, L. (1992). "Generalized Green′s functions for higher order boundary value matrix differential systems". International Journal of Mathematics and Mathematical Sciences. 15 (3): 217938. doi:10.1155/S0161171292000681. ISSN 1687-0425.

[1] Villanueva, R. J.; Jodar, L. (1992). "Generalized Green′s functions for higher order boundary value matrix differential systems". International Journal of Mathematics and Mathematical Sciences. 15 (3): 217938. doi:10.1155/S0161171292000681. ISSN 1687-0425.