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I am solving Regge-Wheeler equations for electromagnetic perturbations of a Schwarzschild black hole. See this paper for Regge-Wheeler equation.

The differential equation that must be solved is of this type:

$$\dfrac{d^2A}{dt^2} - \left( 1-\dfrac{2M}{r} \right)\dfrac{d}{dr}\left[ \left( 1-\dfrac{2M}{r} \right) \dfrac{dA}{dr} \right] + \left(1-\dfrac{2M}{r} \right) \dfrac{l(l+1)}{r^2} A = 0$$

The idea is to solve first the static problem:

$$- \left( 1-\dfrac{2M}{r} \right)\dfrac{d}{dr}\left[ \left( 1-\dfrac{2M}{r} \right) \dfrac{dS}{dr} \right] + \left(1-\dfrac{2M}{r} \right) \dfrac{l(l+1)}{r^2} S = 0$$

with the boundary condition that $S(2M)=0, S(\infty)=0$ and to solve the dynamic problem with the initial conditions that $\dfrac{dA}{dt}(0,r)=0, A(0,r)=S(r)$

I can solve the static problem with ParametrcicNDSolveValue, but I cannot solve the dynamic problem because I don't know how to impose the second initial condition on $A(t,R)$.

I have compactified the infinite interval in $r$ with the change of variable $r\rightarrow \dfrac{t+1}{1-t}$, so my naive code looks like this:

sol = ParametricNDSolveValue[{-(1 - (2 M)/((t + 1)/(1 - t))) 1/
  2 (t - 1)^3 D[S[t], {t, 2}] - (1 - (2 M)/((t + 1)/(1 - t))) (
  2 M)/((t + 1)/(1 - t))^2 1/
  2 D[S[t], t] + (1 - (2 M)/((t + 1)/(1 - t))) (
  l (l + 1))/((t + 1)/(1 - t))^2 S[t] == 0, S[0.4] == 0, S[0.99] == 0}, S, {t, 0.4, 0.99}, l]

din = ParametricNDSolveValue[{D[
  A[T, t], {T, 2}] - (1 - (2 M)/((t + 1)/(1 - t))) 1/
  2 (t - 1)^3 D[A[T, t], {t, 2}] - (1 - (2 M)/((t + 1)/(1 - t))) (
  2 M)/((t + 1)/(1 - t))^2 1/
  2 D[A[T, t], t] + (1 - (2 M)/((t + 1)/(1 - t))) (
  l (l + 1))/((t + 1)/(1 - t))^2 A[T, t] == 0,  Derivative[1, 0][A][0, t] == 0, A[0, t] == sol[l][t]}, A, {T, 0, 10}, l]

but Mathemartica complains that

No functions were specified for output from NDSolveValue

Where am I wrong? The value I used for the mass is $M=1$, $l$ is any integer larger than 2.

I use Mathematica 11.3 if needed.

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  • $\begingroup$ Please check you're code, the differential equations didn't evaluate. $\endgroup$ Oct 11, 2020 at 9:34
  • $\begingroup$ @UlrichNeumann Corrected, sorry for the errors. $\endgroup$
    – mattiav27
    Oct 11, 2020 at 9:48
  • $\begingroup$ In the 2nd ode there is a part D[A[T, r], {T, 2}] which isn't transformed and should be expressed by t $\endgroup$ Oct 11, 2020 at 9:54
  • $\begingroup$ @UlrichNeumann Corrected that too, sorry $\endgroup$
    – mattiav27
    Oct 11, 2020 at 9:57
  • $\begingroup$ The second ode in A[T,t] requires two arguments. Add , {t, 0.4, 0.99} in the second NDSolve ! $\endgroup$ Oct 11, 2020 at 10:13

1 Answer 1

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Try

din = ParametricNDSolveValue[{D[A[T, t], {T, 2}] - (1 - (2 M)/((t + 1)/(1 - t))) 1/2 (t - 1)^3 D[A[T, t], {t,2}] - (1 - (2 M)/((t + 1)/(1 - t))) (2 M)/((t + 1)/(1 -t))^2 1/2 D[A[T, t],t] + (1 - (2 M)/((t + 1)/(1 - t))) (l (l +1))/((t + 1)/(1 - t))^2 A[T, t] == 0,Derivative[1, 0][A][0, t] == 0, A[0, t] == sol[l][t]}
,A, {t, 0.4, 0.99}, {T, 0, 10}, l]

which is evaluated without error (Mathematica v12, Windows10)

Plot3D[din[3][T, t], {t, 0.4, 0.99}, {T, 0, 10}]

enter image description here

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