@Carl Love you are right and I agree with you that "There's no indication of what's "sufficiently close" or how to converge to correct initial values. " Its a hit and trial based algorithm and much boring.

I face another issue here that when we choose a particular value for B, then for different values of infinity we get different solutions. It seems that infinity play a role of parameter here. 

But I think that logically and mathematically these steps are correct. 

Step 5 is about the concept of shooting method that selects such type of initial guess which shoots the other boundary condition (Shooting Method). Eq (2.19) in step 5 is that another boundary condition.

@Carl Love I derived one solution through MATLAB, but it's difficult to find the other three solutions.

Thank you so much for your time.

@Carl Love 

Soory sir, i have not received an email from you. 

my email 

arif.qau@hotmail.com

aukhan@math.qau.edu.pk

@Preben Alsholm 

Here is the same case. N represents infinity.

@Preben Alsholm Really Sorry sir, but I need solutions for the full range of parameters which I have already mentioned at the start of my question i.e (0 <= gamma <= 10,      0 <= rho & nu <= 200).

Really sorry for my mistake and I am thankful to you to assist. 

@Carl Love oh sorry it was a misunderstanding. The solution by Preben Alsholm did not give me results when I choose  abserr<1.5e-1, and parameters values {gamma = 1, rho, nu = {2,20,100,200}}.

If we choose abserr=1.5e-1, then for gamma = 1, rho=nu = 20 it gives me f''(0)=-399.999968119932, while the the published result is f''(0)=-4.1947.

Here I share some published results for particular values of parameters

gamma=1, rho=nu = {2,20,100,200}, then f''(0)={-0.3515;-4.1947;-11.3575;-16.7974}

@Carl Love I am trying to review the said manuscript. Results are already made and published by the author of this manuscript. I am trying to generate these results. because without that, I am not able to move forward. 

Sorry, I don't know that this is a Preben's problem or results of this problem known as Preben's solution. Because I hear about Preben's Problem or Preben's solution for the 1st time from you.

I am trying to attach that manuscript here but found attachment failed. 

If you would give me your email address, I will send this manuscript to you via email.

Thanks

@Carl Love I'll be very thankful to you for this favor.

@Carl Love Thanks for your reply.

Sorry @Carl Love I did not know about Preben's solution. Actually, this is a published paper which i m trying to review it. the author used Shooting technique (in c++) to solve this problem. I tried to solve it by using Mathematica, MATLAB and know Maple but failed to get the correct result.

I attached this paper, Please check section 3.2.

https://aip.scitation.org/doi/10.1063/1.5025721

@Preben Alsholm Thanks for your reply.

I check it and it does not give me the required results.

Is there any other routine rather than "midrich", which handles this problem?

@Carl Love yes i want this result. Thank you. 

@Preben Alsholm Its work by simply increasing the initmesh point. Thanks alot.

@Preben Alsholm thanks alot. But there is some typo mistake in the 3rd boundary condition as i mention below. If we set this boundary condition then the solution is not convergent.

D(F)(N) not equal to 0, but, D(F)(N)=1

@tomleslie if it is possible that what you say, then it may possible that 1st we find the solution for small value of lambda (e.g lambda=-0.3) and use this solution as initial guess to find the solution for large value of lambda (for better results). By making use of for loop or other else. (solution exis for lambda=-0.0,...,-0.4). 

@Kitonum actually I have to find the value of A. A is basically an eigenvalue. And it is find with the help of that one extra condition.