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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. 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.\,}

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.\,}

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