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I have a differential equation, which I want to expand as a series at a finite $r$ value - the horizon, and at $\infty$. The functions involved are definitely cumbersome, and I have recreated my code below:

f[r_] := (r/rs)^(2 e) (((1 - (rs/r)^(1 + e)) (2 + e + e (rs/r)^(1 + e))^2)/(16 (1 + e)^4 (1 + d^2L^2 g0^2 rs^2/(16 r^4)))) ((2 + e)^2 -e^2 (rs/r)^(1 +e));
g[r_] := (((r/rs)^(1 + e) - 1) ((r/rs)^(1 +e) (2 + e)^2 - e^2))/((1 + d^2 L^2 g0^2 rs^2/(16 r^4))((e + (r/rs)^(1 + e) (2 + e))^2)) // Simplify ;
h[r_] := r^2 (1 + d^2 L^2 g0^2 rs^2/(16 r^4));
k[r_] := Sqrt[g[r]/f[r]] h[r] // FullSimplify;
\[CapitalDelta]2[r_] := g[r] h[r] + a^2 // FullSimplify;

Here, d,g0,e,rs are all constants. I want to solve the resulting differential equations, for different values of these parameters. Consider the following sample values.

e = 0.01;
d = 0;
rs = 2 M;
L = 0;
g0 = 0;

The actual equation is defined as follows:

$Assumptions = {M > 0, a > 0, r > 0, a < M, w > 0};
rplus = M + Sqrt[M^2 - a^2];
rminus = M - Sqrt[M^2 - a^2];
f2[r_] := \[CapitalDelta]2[r]/(k[r] + a^2) // Simplify;
Gaos[r_] := (k'[r] \[CapitalDelta]2[r])/(2 (k[r] + a^2)^2) // FullSimplify
KKaos = (k[r] + a^2) \[Omega] - a m // Simplify;

Vaos = (KKaos^2 - \[CapitalDelta]2[r] \[Lambda])/(k[r] + a^2)^2 - Gaos[r]^2 - f2[r] Gaos'[r] // Simplify;

eqnaos = f2[r]^2 R''[r] + f2'[r] f2[r] R'[r] + Vaos R[r] ;

For one thing, I am unable to even Simplify the potential Vaos or the equation eqnaos. After running the Simplify command for a long time, I get an error that asks me to increase the TimeConstraint. But when I do this, I either get the same error again, or the kernel just restarts.

The actual series expansion I want to perform is given below:

R[r_] := (r - rplus)^AA h2[r];
eqn1aos = eqnaos/(r - rplus)^AA;
ruleHaosold = {AA -> (I (a m - 2 M (M + Sqrt[-a^2 + M^2]) w))/(2 Sqrt[-a^2 + M^2])};
ORDH = 3;
ssaos = Series[eqn1aos //. ruleHaosold, {r, rplus, ORDH}];

The last series expansion keeps running, and I get no output and no error. I have tried running this overnight as well.

If I set e = 0 the entire set of functions and the equation simplifies greatly (and is a known simpler case of the problem I'm trying to solve) and the series expansion and the rest of my code works perfectly.

I also have a similar series expansion at $\infty$ which also suffers from the same problem where it just keeps running indefinitely. How do I fix this?

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1 Answer 1

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Don't define rplus at the beginning, but after Series taken. Otherwise it is too complex for Series.

$Assumptions = {M > 0, a > 0, r > 0, a < M, w > 0};

(*   rplus=M+Sqrt[M^2-a^2];
     rminus=M-Sqrt[M^2-a^2];   *)

f2[r_] = \[CapitalDelta]2[r]/(k[r] + a^2) // Simplify;
Gaos[r_] = (k'[r] \[CapitalDelta]2[r])/(2 (k[r] + a^2)^2) // 
Together // Simplify;
KKaos = (k[r] + a^2) \[Omega] - a m // Together // Simplify;
Vaos = (KKaos^2 - \[CapitalDelta]2[r] \[Lambda])/(k[r] + a^2)^2 - 
 Gaos[r]^2 - f2[r] Gaos'[r] // Together // Simplify;
eqnaos = f2[r]^2 R''[r] + f2'[r] f2[r] R'[r] + Vaos R[r] //  Together //
Simplify;


R[r_] = (r - rplus)^AA h2[r]
eqn1aos = eqnaos/(r - rplus)^AA // Together // Simplify
ruleHaosold = {AA -> (I (a m - 
      2 M (M + Sqrt[-a^2 + M^2]) w))/(2 Sqrt[-a^2 + M^2])} // 
Together // Simplify
ORDH = 3;

ssaos = Series[eqn1aos //. ruleHaosold, {r, rplus, ORDH}]

Normal[ssaos] /. rplus -> M + Sqrt[M^2 - a^2]
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  • $\begingroup$ What does Together do and why have you used it with Simplify? Does it make the simplification easier? Because rplus and rminus are anyway not used anywhere in the equation and potential, but Simplify still doesn't work there. $\endgroup$
    – newtothis
    Jan 30, 2022 at 7:06
  • $\begingroup$ I tried running Vaos with Together and Simplify like you suggested. After about 10 mins, evaluation is automatically aborted and the kernel just restarts. $\endgroup$
    – newtothis
    Jan 30, 2022 at 7:26
  • $\begingroup$ @newtothis , Together and Simplify are not absolutly neccessarry, but make things much faster. Try it without them ! The main point is, not to define rplus at the beginning, as i already wrote ! $\endgroup$
    – Akku14
    Jan 30, 2022 at 7:34
  • $\begingroup$ When I tried running it overnight, I did not have Together and Simplify, and I did not have rplus, rminus defined before the series. It still did not return anything in the morning. $\endgroup$
    – newtothis
    Jan 30, 2022 at 7:43
  • $\begingroup$ Besides M and w, you also use m and [Omega], but you do not mention it in Assumptions. $\endgroup$
    – Akku14
    Jan 30, 2022 at 8:09

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