@maple2015 

Is it requisite to write a program for solving NL Eqs?

Tricks for solving (lots of) coupled nonlinear equations numerically

http://mathematica.stackexchange.com/questions/14824/tricks-for-solving-lots-of-coupled-nonlinear-equations-numerically

  Methods for Systems of Nonlinear Equations

@tomleslie 

For m>=4, Maple 18 can't show numerical amounts of FW .

Maple 18 returns F parameyrically instead of FW.

m=2 ---> FW=6.18788052788535979489838829491

m=3 ---> FW=6.11860004299491357506825515146

m=4 ---> FW =

(26.6017792231159512532947857770*b[2]^2+361.474932298954936166628403879*b[4]^2+

294.705660784037569029349281506*b[3]^2+176.452707059743221717141668429*b[2]*b[3]+

179.764515894940777468085569871*b[2]*b[4]+618.168221185375981314638994801*b[3]*b[4])

/(4.29901306323488716726538280258*b[2]^2+54.7320010955844337488514131001*b[4]^2+

48.1304162916072256887555907820*b[3]^2+28.7560435225384216726518943871*b[2]*b[3]+

30.3323036366089904837274461709*b[2]*b[4]+101.906642969089067491841606911*b[3]*b[4])

The exact solution is 6.110

@tomleslie

Thanks a lot

@tomleslie 

Maple 18 shows the bellow value!

7.92004811580310067452441713379

@tomleslie 

Thank you for the attached program

but result of attached file is not coincident with origin code.

I have to improve programming skills for solving time consuming problems in the least amount of time possible.

@Mac Dude thank you

@acer 

Thank you for your great hints

In addition to, I think some articles are good sources

@Markiyan Hirnyk 

Thank you for your comment

@Kitonum 

Therefore it is a bug! Like this:

http://www.mapleprimes.com/questions/204669-How-Does-Maple-Calculate-Floor-Limit

@Kitonum 

Thank you for your comment

I mean domain of the functions f=sqrt(x+2*sqrt(x-1)) and g=sqrt(x-2*sqrt(x-1)) is x>=1, but why domain of function f+g is from -infinity to infinity?

@tomles

new code runs in t=34.328 seconds (Maple18 + RAM 6GB + Core i5 Laptop)

by changing numerical parameters (without restart) because of cache memory it works in t=17.687 seconds.

Separating to smaller parts is an effective method!

Thank you

@tomleslie 

In which J is BesselJ(0,x) and Psi(r) is dimensionless function of w(r) (i.e. W(r)/W_max)

"P" for psi has the minimum amount (B.C.s are Psi(0)=1 and Psi(a)=0)

arbitrary amount of between 0 and 0.5

Thank you for your comments

I need minimize an Integration and I use " Spectral Method" title for my question to clearly show what I need to do.

I use Vriational Calculus and solve Integration analytically by using Euler-Lagrange equation command:

EulerLagrange(f, t, x(t))

But I know want to solve problem numerically with Spectral Method

@Preben Alsholm 

Thank you

@acer  Thank you