Construct DifferentialMatrices and Kernel for LevinRule for this integral and ODE set

Question

I've made a lot of progress on my problem the last few days thanks to all the help I've received on here. I think I'm upto the final step of greatly improving the performance of NIntegrate[..] on my setup.

I have an integral that looks like

$$\int^{x_f}_{x_i} \mathrm{d}x\, \exp{(ix)} \exp{(i\omega t(x))} \phi^*(x)$$

(* refers to complex conjugation).The $\exp{(ix)}$ is slowly oscillatory compared with everything else. The $x_i,x_f$ are negative; typical values might be $-115,-75$, the closer you get to $-4/3$ the more problematic things get, $x>x_h=-4/3$. Similarly the bigger $\omega$ gets the more oscillatory things are.

The function $t(x)$ is

$$t(x)=3x_h (x/x_h)^{1/3}+x-3x_h\text{arctanh}{\left((x_h/x)^{1/3}\right)}$$

and satisfies

$$\frac{dt}{dx}=\frac{1}{1-\left(\frac{x_h}{x}\right)^{2/3}}$$

and

$$\phi''(r)+\frac{2(r-1)}{r(r-2)} \phi'(r)+\left(\frac{\omega^2 r^2}{(r-2)^2}-\frac{2}{r(r-2)}\right)\phi(r)=0$$

where $r=2(x/x_h)^{2/3}$. So this can be transformed into an ODE in x.

Is it possible using this information to construct the "DifferentialMatrices" and "Kernel" of the LevinRule?

If this turns out not to be possible, is it possible to tell Mathematica to use LevinRule only in some finite range of integration where one of the above factors becomes non-oscillatory and I can lump it in the amplitude.

Answer 1

How about this?

The $\phi$ ODE in terms of $x$ looks like

$$\frac{d^2\phi}{dx^2}+\left(\frac{5-3(x_h/x)^{2/3}}{3(x-(x/x_h)^{1/3}x_h)}\right)\frac{d\phi}{dx}+\left[\frac{(x/x_h)^{2/3}\omega^2}{((x/x_h)^{2/3}-1)^2}-\frac{4}{9}\frac{2}{(x/x_h)^{4/3}x_h^2((x/x_h)^{2/3}-1)}\right]\phi=0$$

Let's write this as

$$\frac{d^2\phi}{dx^2}+f(x)\frac{d\phi}{dx}+g(x)\phi=0$$

which defines $f(x),g(x)$.

Now to construct Kernel and Differential Matrices

Make the choice:

$$w_1=\exp(i\omega t(x))\phi^*$$ $$w_2=w_1'=i\omega t' w_1+\exp{(i\omega t)}(\phi^{*})'$$

The second equation gives us something that will be useful later: $\exp(i\omega t) (\phi^*)'=w_2-i\omega t' w_1$

Then we automatically have components $A_{11}=0$, $A_{12}=1$ of the differential matrices the tricky part is getting the other two.

Now take the derivative of $w_2$ which is $w_1''$, using the useful relation I mentioned and the ODE in $\phi$, this can be shown to be

$$w_2=w_1[-i\omega t'(2i\omega t'-f)+i\omega t''-g-\omega^2 (t')^2]+w_2(2i\omega t'-f)$$

This gives us

$$A_{12}=[-i\omega t'(2i\omega t'-f)+i\omega t''-g-\omega^2 (t')^2]$$ $$A_{22}=(2i\omega t'-f)$$

Now all that is left is to simply sub into this $f(x),g(x),t'(x),t''(x)$ that we know from above.

Then the rules are specified as "Kernel->{w_1,w_2}", "DifferentialMatrices"->{{A_{11},A_{12}},{A_{21},A_{22}}.

EDIT:

Strangely this shows no time improvement (it actually takes a bit longer than default) although gives answer in agreement.