Asymptotic solution of a system of ODEs

Question

I have the following system of Ordinary Differential Equations (ODEs) together with initial values

ode = {-4 * (1 + u * a[u]) * f'[u] == (a[u]^3 + 1/4 * a[u] + (3 a[u]^2 + 1/4) * u * a'[u]) + 2 (a[u] + u * a'[u]) * f[u], 
       a'[u] == 1/2 * u * f'[u], 
       a[0] == -1/2, f[0] == 0}

which can be easily solved numerically:

{ra, rf} = NDSolveValue[ode, {a, f}, {u, 0, 20}]

and asymptotically for $u\rightarrow0$.

AsymptoticDSolveValue[odetph, {a[u], f[u]}, {u, 0, 3}]

(* {-(1/2) + u^2/64 + u^3/128, u/16 + (3 u^2)/128 + (3 u^3)/512} *)

But I need to know an asymptotic expansion of a[u] as $u\rightarrow\infty$. Even the coefficient of the lowest order term $\sim 1/u$ would be very nice to know. Is there a trick to obtain such a solution with MA?

Answer 1

A simple approach is to introduce the ansatz $$ f(u)=\sum_{k\ge0}\frac{f_k}{u^k},\qquad a(u)=\sum_{k\ge0}\frac{a_k}{u^k} $$ which you plug into the ODE and solve for the coefficients. I find \begin{aligned} f(u)&=\frac{3}{8}-\frac{1}{u^2}+\frac{3}{2 u^4}-\frac{7}{2 u^8}+\frac{33}{2 u^{12}}+\cdots\\ a(u)&=-\frac{1}{u}+\frac{1}{u^3}-\frac{2}{u^7}+\frac{9}{u^{11}}+\cdots \end{aligned}

Check:

ode[[1 ;; 2]] /. {f -> (3/8 - 1/#^2 + 3/(2 #^4) - 7/(2 #^8) + 33/(2 #^12) &), a -> (-(1/#) + 1/#^3 - 2/#^7 + 9/#^11 &)}
Series[%, {u, ∞, 12}] // Simplify

(* {True, True} *)

You can of course add more terms, with symbolic coefficients, and find their correct value by solving the equation order by order. I'm sure you know what I mean but feel free to ask, of course.

The numeric minus the asymptotic solutions differ by

which is so close to 0 that the numerics overwhelm the error. Might need to increase accuracy to get a meaningful plot...