I'm interested in finding the (unique?) solution to the set of delay differential equations $$f_w(w,x) = xf(w,w^2x)+w^3x^2f(w,w^4x), $$ $$f_x(w,x) = wf(w,w^2x)$$ With the initial condition $f(1,x) = e^x$ and $(w,x) \in \mathbb{C}^2$

For $|w| \leq 1$, I've found the series $f(w,x) = \sum_{n=0}^\infty \frac{w^{n^2} x^n}{n!}$ satisfies the equations, but it doesn't converge for $|w|>1$.

For $|w|>1$, I have considered the integral $$f(w,x)=\frac{1}{2 \sqrt{\pi}} \int_0^\infty \frac{e^{-\frac{\ln(t)^2}{4}}}{t} e^{x t^{\sqrt{\ln\left(w\right)}} }dt$$ which seems numerically to satisfy the differential equation, though it converges so badly at most places that its fairly useless.

A few questions I have regarding this set of differential equations

Question 01: Is the series solution unique? Is it the only smooth solution?

Question 02: How can we numerically approximate this delay differential equation? As far as I know, Mathematica's NDSolve doesn't support delay differential equations with non-constant delays, and it seems other computation programs similarly don't have support for this type of differential equation.

Question 03: Are there methods to analytically approximate a solution? I'm considering that perhaps there is some way to resum the divergent series that gives an asymtotic solution.


1 Answer 1


Its looks like one solution to this is the function $$f(w,x) = 1+\frac{1}{2\pi i} \int_{\frac{1}{2} - i \infty}^{\frac{1}{2} + i \infty} w^{t^2} (-x)^t \Gamma(-t) dt $$ Which is obtained by applying the residue theorem and assuming the other parts of the contour cancel out somehow. This solutions agrees with the approximations provided by the previous function, though this one converges well enough to be able to numerically confirm that it satisfies the solution to the delay differential equations. It has a branch cut at $\theta = -\pi$, so for certain angles of $w$ and $x$ the differential equation needs to evaluate the function at different branches to satisfy the differential equations. We can obtain all of the branches as $$f_n(w,x) = 1+\frac{1}{2\pi i} \int_{\frac{1}{2} - i \infty}^{\frac{1}{2} + i \infty} w^{t^2} e^{ \left(\ln(-x)+2\pi i n\right)t} \Gamma(-t) dt $$

What is left is prove uniqueness. This solution is interesting to me because the function $f(w,x)$ when $|w|<1$ is a complex function with a natural boundary, and so if we can prove uniqueness of the solution for $f(w,x), |w|>1$ it may provide an interesting way to 'analytically continue' the inner function beyond its natural boundary.


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