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 $$f(w,x)=\frac{1}{2 \sqrt{\pi}} \int_0^\infty \frac{e^{-\frac{\ln(t)^2}{4}}}{t} e^{x t^{\ln\left(w^{1/2}\right)}}dt$$ 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.

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.

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 $$f(w,x)=\frac{1}{2 \sqrt{\pi}} \int_0^\infty \frac{e^{-\frac{\ln(t)^2}{4}}}{t} e^{x t^{\ln\left(w^{1/2}\right)}}dt$$ 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

• Is the series solution unique? Is it the only smooth solution?
• 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
• 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

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 $$f(w,x)=\frac{1}{2 \sqrt{\pi}} \int_0^\infty \frac{e^{-\frac{\ln(t)^2}{4}}}{t} e^{x t^{\ln\left(w^{1/2}\right)}}dt$$ 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.