Does the Green's Function for an IVP always converge while integrating?

Question

I'm having some trouble solving an ODE using the Green's function method. The problem I'm working in is the simple harmonic oscillator equation $$ L[y(t)]=f(t)$$ $$ L = \frac{d^2}{dt^2}+\omega^2$$ $$y(0)=1,y'(0)=0$$ Now I'll construct my Green's function. I have two linear independent solutions for the homogeneous version ( $f(t)=0$ ) of my problem: $y_1(t)=\cos(\omega t)$ and $y_2(t)=\sin(\omega t)$.

I know that the Green's function satisfies $L[G(t,\xi)]=\delta(t-\xi)$. For $t\neq\xi$ we have that $L[G(t,\xi)]=0$.

My function should be $$ G(t,\xi)=\left\{\begin{matrix} A\cos(\omega t) + B \sin(\omega t),& t<\xi \\ C\cos(\omega t) + D \sin(\omega t),& \xi<t \end{matrix}\right. $$

Applying the initial conditions $G(0,\xi)=1,G'(0,\xi)=0$ we find that $A=1$ and $B=0$. $$ G(t,\xi)=\left\{\begin{matrix} \cos(\omega t),& t<\xi \\ C\cos(\omega t) + D \sin(\omega t),& \xi<t \end{matrix}\right. $$

Using the continuity of the function at $t=\xi$ and the discontinuity of the derivative at $t=\xi$ we should have $$C \cos(\omega \xi)+D \sin(\omega \xi)-\cos(\omega \xi)=0 \\ -C\sin(\omega \xi)+D\cos(\omega \xi)+\sin(\omega\xi)=\frac{1}{\omega}$$

Solving this system we find that $C=1-\dfrac{\sin(\omega\xi)}{\omega}, \;D=\dfrac{\cos(\omega\xi)}{\omega}$.

Finally, the Green's function should be $$ G(t,\xi)=\left\{\begin{matrix} \cos(\omega t),& t<\xi \\ \dfrac{1}{\omega}\left[\omega-\sin(\omega\xi)\right]\cos(\omega t) + \dfrac{1}{\omega}\cos(\omega\xi)\sin(\omega t),& \xi<t \end{matrix}\right. $$

Now I should be able to find a solution for the ODE using the Green's function because $y(t)=\int_0^\infty f(\xi)\,G(t,\xi)\,d\xi$

I'm having trouble with this last step. Some of the functions I should try are $f(t)=e^{-t}$ and $f(t)=\cos(t)$. For the case where the "forcing function" is a cosine, the integral does not converge. I tried the following \begin{align*} y(t) &=\int_0^\infty f(\xi)\,G(t,\xi)\,d\xi\\ &=\int_0^t \cos(\xi)\left[ \frac{1}{\omega}\left[\omega-\sin(\omega\xi)\right]\cos(\omega t) + \frac{1}{\omega}\cos(\omega\xi)\sin(\omega t)\right]d\xi\\ &\quad+\int_t^\infty \cos(\xi)\,\cos(\omega t)\,d\xi \end{align*}

The first integral is not hard to compute and it give us an answer. But the second integral does not converge. I thought that the Green's function should give us a solution for this problem because this ODE has a solution that we can easily get with other methods.

So, my question is: what is going wrong with my construction? Is my Green's function wrong or am I taking the integral in the wrong way with the wrong limits? Thanks in advance!

Answer 1

The idea is to write $$ y(t)=y_0(t)+\int_0^tG(t,ξ)f(ξ)\,dξ, $$ where

• $y_0$ satisfies the equation with homogeneous rhs and the given initial conditions, and
• $G_ξ(t)=G(t,ξ)$ satisfies $L[G_ξ]=δ_ξ$ with homogeneous IC. This has as a first consequence that $G_ξ(t)=0$ for $t<ξ$, only the delta pulse changes this to a non-zero solution.

Thus $$ y_0(t)=\cos(ωt),\\ G_ξ(t)=\Theta(t-ξ)\frac{\sin(ω(t-ξ))}{ω}, $$ where $Θ$ is the Heaviside or unit-jump/ramp function.