basically m not able to direct the dsolve output to this command..
any suggestions
thanks

I have to plug that expression for y(r) in two simultaneous integral equations and solve them using NRapson. I need to keep changing everything till Nrapson starts converging.
Thanks again for all the help
MS

First we solve DE for initial conditions and then get values of y(.1) and D(y)(.1). Then we solve DE for those conditions. It's basically combination of Robert's and your advice.
ade:= diff(y(r),r,r)+2*diff(y(r),r)/r+486*polylog(3/2,-exp(1-1/43*y(r))) = 0;
ys:= dsolve({de, y(0) = 0, D(y)(0) = 0}, y(r), series, order=10);
yn := convert(evalf(ys), polynom);yn1 := eval(yn, r = .1); dyn1 := eval(diff(yn, r));
dyn2 := eval(dyn1, r = .1);S := dsolve({de, y(.1) = 1.266393114, (D(y))(.1) = 25.18200436}, y(r), numeric);with(plots); odeplot(S, [r, y(r)], r = 0.1e-1 .. 200);
# I get a plot from this step, but I need an expression for y(r). A crude method is to do S(1);S(2) and so on and copy paste those values for interpolation.
Would you mind helping me with a generic code for that step? And asymptotic potential is the solution for the DE at r-> infinity.
Thank you
MS

I got y(r) values from the formula at few points n did an Interpolation to get a solution. But I am not able to write a generic code for doing this. So, i just copy pasted points (a stupid laborious process).
1) Is there any generic way to get an expression for y(r) from the plot above?
2) How can i calculate asymptotic potential for the diff equation at r->infinity as i need to subtract it from the potential expression in step(1) to get a complete potential that satisfies BC's ?
Thanks
MS

Thanks a million. I have 2 questions now.
1) How can i get an expression for y(r) from the plot above?
2) How can i calculate asymptotic potential for the diff equation at r-> as i need to subtract it from the expression in step(1)?
Thanks again for your help.
MS

Thanks for this. I have two questions.
You have changed your initial conditions at 1 rather than at 0. I don't think we can do that as we don't know the value of D(y)(1).
I just tried this simple thing:
de := diff(y(r), r, r)+2*(diff(y(r), r))/r+9*(16*43)*Pi^2*sqrt(Pi/(2*(1/43)^3))*polylog(3/2, -exp(1/43*(43-y(r))))/(16*Pi^2*sqrt(43)^3) = 0; dsolve[interactive]({de, y(0) = 0, (D(y))(0) = 0})
and used truncated series method in 'solve symbolically' option.
I am getting a solution like this:
y(r) = -(3/4)*sqrt(86)*sqrt(Pi)*polylog(3/2, -exp(1))*sqrt(43)*r^2-(1161/80)*Pi*polylog(1/2, -exp(1))*polylog(3/2, -exp(1))*r^4+O(r^6)
I am not sure if that is correct though. Would you mind having a look.
Thanks again
MS

yeah, i asked my prof about it... He told me to first solve this DE at initial conditions y(0)=0 and D(y)(0)=0 .
Then calculate the asymptotic potential y(r->∞) and subtract this from the entire potential function.
So, the immediate problem reduces to soving this nonlinear DE at IC's y(0)=0 and y'(0)=0.
Thanks for this insight Robert. Any suggestions now on solving this DE for initial conditions.
Thanks again
MS

I see what you are saying sir. Its polylog(3/2, -exp(1/43*(43-y(r))).
Notice if polylog is given by Ln(a,z) then z is -exp(...) and not exp(..).
I guess thats why you are getting complex roots and I hope this makes things clear.
Again any advice is appreciated.
Thanks
MS

By "without approx" my prof meant not to approximate anything for polylog as i was taking few polynomial terms in place of polylog.
And here is the worksheet. You have to run this once and then delete this line ''' for i from 0 to no-1 do y[i+1] := (dr+i*dr)^2*(-2.731/20^2) end do''' from sheet and then u can set a for loop. This above line is for initial values.

View
I guess, i need to evaluate the polylog function using evaluate command and insert this as a subroutine in Rk integration...If you have any clue about doing this, please let me know.
Again, thank you and appreciate your patience
MS

I will send you the worksheet Patrick whenever I know how to post it. I can also email it to you.
But still I am hanging in mid air.
In my prof's words regarding solving this nonlinear DE: "Solve without
approximations. They key ingredient is to use a subroutine to evaluate the
polylogs and then integrate the
differential equation with help of a Runge-Kutta like scheme. You can
find both in programs like Mathematica, Maple, Matlab etc. "
I will really appreciate if I can get some help to figure a way out.
Thanks
MS

Yeah i know, i cannot change it to a linear equation...
How can I post the maple worksheet with the code here......?

1) I have already tried substituting few polynomial terms in place of polylog but still the DE remains non-linear.
2) OK regarding Solution of the DE: if u approximate polylog by few polynomial terms in y(r), then substitute that polynomial in y(r)(polylog) as another variable say a(r), one can write the DE in terms of a(r) which is still non-linear. That nonlinear DE in a(r) can be solved by using 'equilibrium iteration method or equilibrium Newton's method' numerically. Then the solutions values can be interpolated to get an expression for a(r) and for y(r). 'Numerical methods for eng and scientists: by Hoffman'.
I have to use this value of y(r) in some other expressions and so i need accurate rather than approximate values. Thats why I am emphasizing on using a subroutine to evaluate polylog and then integrating the differential equation with the help of a Runge-Kutta like scheme (according to my prof).
Hope this helps and thank you
MS

Even if we transform the BVP into a set of two eqns, still y(r) is inside polylog ang it creates trouble.
Just in case you need proper values of constants, C=486, C1 = 1/43, C2=43....
I have tried all the basic known methods in maple. The correct way is to a use a subroutine to solve polylog and then integrate the differential equation with help of a Runge-Kutta like scheme (according to my prof).
Thank you for your time,
MS

Hello sir,
here y(r) is a potential and y(r -> infinity)=0. this has a solution as our professor has solved this problem. this is a part of summer homework problem set.
Our Prof emailed a hint saying to use a subroutine to solve polylog and then integrate the differential equation with help of a Runge-Kutta like scheme.
Would you mind throwing some light on how to make a subroutine for this case.
Really appreciate your help
MS

here C, C2 and C3 are just numbers.