How can I solve Einstein’s equation and calculus of the value of the K constant in Einstein's equation and the value of the tensor stress energy that fits in this equation?

QTBend.docxSqBend.mw

elctroWav.mw have the equation 1 :

eq1 := 2*m*(E + e0^2/(rb*rho))*g(rho, t)*(-rb*rho + R)/(h^2*R) + R*(diff(g(rho, t), rho, rho)*(-rb*rho + R)/R - 2*diff(g(rho, t), rho)*rb/R)/(rb^3*rho) - diff(g(rho, t), t, t)*(-rb*rho + R)/(a^2*c^2*R) = 0;

with iv1:

iv1 := D[1](g)(0, 0) - g(0, 0)*rb/R = rb*R, g(1, 0)*(R - rb)/R = 0, D[2](g)(rb, 0) = a*c;

As in maple document electroWav.mw

The issue is that there is an unsolved part with RootOf how can I rezolve it?

-R*h^2*RootOf(AiryBi(-(R*_Z^3*h^2 + 2*e0^2*m*rb^2)/(_Z^2*R*h^2)))^3

Please advice...

How do I solve completely the diferential equation and also speed up the compilation of  the time is over 3000sec: 

eq1 := 2*m*(E + 8*Pi*epsilon/r)*f(r, t)/h^2 + R*diff(f(r, t), r $ 2)/r - diff(f(r, t), t $ 2)/(a^2*c^2) = 0;

iv1 := f(r, 0) = 0, f(R, t) = 0, D[1](f)(0, 0) = R;

Sol := pdsolve([eq1, iv1]);

Where f(r,t) is the function of variable r and t  in spherical coordinate and m, E, h, R, rb, a, and c are constants.

I also want to find the exact value of f(r,t) with the condition f(rb, 0) = 0; and diff(f(rb,t),t)=a*c for the value t=0 and if is possible the pulsation of the sinusoidal solution of f(r,t). [the solution is a combination of AiryAi ; AiryBi and sinusoidal sin(a*c*sqrt(-2*E*m - _c[1])*t/h)]. I didn't find the value of _c[1] for the 2 additional condition above.

The issue is the period of time between 2 consecutive zero of the f(r,t)=0

tks

Int1 := int(exp(-z*(R^2*k^2 - b^2*z)/(R*b))/(z*HeunB(0, k^2*R^2/(b*sqrt(R*b)), R^3*k^4/(4*b^3), 0, -sqrt(R*b)*z/R)^2), z = R .. r);

into cylindrical coords with z axis simetry and radius r;

where R, k, b are constants >0;

And HeunB is Maple funtion

I apreciate the exact calculus but maybe an aproximation is ok but or a plot.

Please advise! 
         

I have to solve the equation rH''(r)+H'(r)+(rk^2-r^2*b^2/R^2)=0 where k, b, and R are real constant positive number, with condition H(R)=0 and H(1/R)=R to be solved into series of power. I know from the literature that xy''+y'+xy=0, can't be solved in terms of elementary function(see G.Nagy-ODE-November 29, 2017) that's why I'm interested in an approximate solution based on series, or any results as long as it satisfied the too condition H(R)=0 and H(1/R)=R of the real function H(r). 

Please advice!.