Can we concentrate on the general solution with coefficients that fit the equation, like the diff equation? 

Thank you very much,

I have changed some of the initial condition 

Ea := 0.00762014687*t + a*t^2 + b*t^3 + c*t^4 + d*t^5;
E1 := diff(Ea, t);
E2 := subs(t = 435, Ea);
E3 := subs(t = 528, Ea);
E4 := subs(t = 2860, Ea);

Thus the plot Ea looks like in Aph1_mm.mw

Do you think the plot "Ea" is accurate I mean most over 95% of the plot with equation 

E1 = 5.012764943*10^(-24)*Ea/(exp(Ea/(4.100527530*10^(-21))) - 1);

a is a bit lower than Optimization:-NLPSolve('J(10)(A)', A=-10..10);

[0.000140134440210983, [A = -0.000605241506177023]]

and my choice is a:=-0.000000144670357887361 to fit the plot with what I think it should look like.

@mmcdara 

I want to calculus the Time Quanta of the electron in the 1 S orbital of the Hydrogen atom thus I must have the exact value of the wave function of the electron!

You can see details at http://michaelvio.orgfree.com/T&SQ.pdf

Please advice!

electronFlux1.mw

electroSum.mw

I try to write the electron 1 S Hydrogen atom equation This is my site: http://michaelvio.orgfree.com/

Please contact me on my email at michaelvio@yahoo.com or michaelvio@gmail.com to give you all the details.

a=1/137 c= light speed E= Rythgerg energy rb is RadiusBohr and R is a distant constant that results from the Theory that is on the site.

Excellent solution!electronFlux1.mw

I want to consult with you on the calculus that I made ant for Time Quanta of the electron, please!

In the last file, I put the calculus of electron and I want to calculate the Time Quanta of the electron that must be rb/c the time between two consecutive o wave functions of the Hydrogen electron 1 S orbital with the boundary condition the velocity is c/137 all the time at radius R wave function =0 and 

in origin, the derivate is R with the norm condition int(P(r,o),r=0..R)=1

@Thomas Richard 

The attached file is elecSVL4Ps.mw

Maple Worksheet - Error

Failed to load the worksheet /maplenet/convert/elecSVL4Ps.mw .
 

Download elecSVL4Ps.mw

Please advise 

I look forward to your's opinion

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(rb, 0) = 0, f(R, t) = 0, D[1](f)(0, 0) = R, D[2](f)(rb,0)=a*c;
sys := [eq1, iv1]:
Sol := pdsolve(sys, HINT=`*`);
pt := pdetest(Sol, sys);

These are the correct initial value, it works with the first 3 conditions but when I introduced the fourth iv:D[2](f)(rb,0)=a*c the compilation time exceed very much. To solve it I used Change of Var :r=rb*rho; 

@Thomas Richard 

Please help me find a solution that must not contain the _c[1] it must result from the 2 conditions.

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

introducing the iv1 f(rb,0) didn't work The compiling time over 15000sec and still not complete!

infolevel[pdsolve] := 5: # optional
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, f(rb,0)=0;
sys := [eq1, iv1]:
Sol := pdsolve(sys, HINT=`*`);
pt := pdetest(Sol, sys);

Please help!

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

The solution must not contain the _c[1] it must result from the 2 conditions.

Please help!

Excellent answer but something is not ok is entirely my fault the equation is rH''(r)+H'(r)+(rk^2-r^2*b^2/R^2)H(r)=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. The SECOND CONDITION IS TO DERIVATE H'(1/R)=R 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) and if is possible a plot for R=370, k=100 and b=35. Very nice I learn a lot about it! could you add an extra help, please!

Sorry, the equation is rH''(r)+H'(r)+(rk^2-r^2*b^2/R^2)H(r)=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) and if is possible a plot for R=370, k=100 and b=35.

Please advice!.

ok asympt is a good command!

asympt(G, r):

but I'm also interested in a general solution where R= an arbitrary constant, not a value!

@vv 

I'm very interested in an approximate solution but unfortunately is not correct you put R=0 and convert  it into series 

g1:=convert(series(%, R=0), polynom);  

I have my own workout 
 

but I didn't resolve in a recurrent mod with the expression of the general term.
Please advice!

Download SloNew2.mw

 Does Maple 2019 or 2017 come with significant improvement in solving this kind of problem? I did my self the first calculation with Airy it's very easy but I want the solution with series powers with package slode...

Thanks a lot! very useful...could I have on Email?