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
)
{\displaystyle x'=A(t)x+g(t)\,}
where
x
{\displaystyle x\,}
is a vector and
A
(
t
)
{\displaystyle A(t)\,}
is an
n
×
n
{\displaystyle n\times n\,}
matrix function of
t
{\displaystyle t\,}
, which is continuous for
t
∈
I
,
a
≤
t
≤
b
{\displaystyle t\in I,a\leq t\leq b\,}
, where
I
{\displaystyle I\,}
is some interval.
Now let
x
1
(
t
)
,
…
,
x
n
(
t
)
{\displaystyle x^{1}(t),\ldots ,x^{n}(t)\,}
be
n
{\displaystyle n\,}
linearly independent solutions to the homogeneous equation
x
′
=
A
(
t
)
x
{\displaystyle x'=A(t)x\,}
and arrange them in columns to form a fundamental matrix:
X
(
t
)
=
[
x
1
(
t
)
,
…
,
x
n
(
t
)
]
.
{\displaystyle X(t)=\left[x^{1}(t),\ldots ,x^{n}(t)\right].\,}
Now
X
(
t
)
{\displaystyle X(t)\,}
is an
n
×
n
{\displaystyle n\times n\,}
matrix solution of
X
′
=
A
X
{\displaystyle 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
y
{\displaystyle x=Xy\,}
be the general solution. Now,
x
′
=
X
′
y
+
X
y
′
=
A
X
y
+
X
y
′
=
A
x
+
X
y
′
.
{\displaystyle {\begin{aligned}x'&=X'y+Xy'\\&=AXy+Xy'\\&=Ax+Xy'.\end{aligned}}}
This implies
X
y
′
=
g
{\displaystyle Xy'=g\,}
or
y
=
c
+
∫
a
t
X
−
1
(
s
)
g
(
s
)
d
s
{\displaystyle y=c+\int _{a}^{t}X^{-1}(s)g(s)\,ds\,}
where
c
{\displaystyle 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
.
{\displaystyle 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
)
=
{
0
t
≤
s
≤
b
X
(
t
)
X
−
1
(
s
)
a
≤
s
<
t
.
{\displaystyle 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
)
=
∫
a
b
G
0
(
t
,
s
)
g
(
s
)
d
s
.
{\displaystyle x_{p}(t)=\int _{a}^{b}G_{0}(t,s)g(s)\,ds.\,}