@Alfred_F Aim of my work: Seeking improvement over existing solution. What kind of improvement: Currently I am aiming for computational speed improvement and higher order solution. What is the plain: My thought process is to use Chebyshev polynominal in place of Legendre because it has O(NlogN) instead of O(N^2). Stick to spectral method because of its exponential convergence property. Solve for higher order terms which helps describe high speed rotation v. What problem I am facing currently: Can't find solved step by step examples of nonlinear elliptic pde using spectral method in 2D like in my case.I'm new to this method. I following your advice of Galerkin Method and solve using Legendre so i something to check against. Hoping that in large N limit it gets to spectral method as i am using global function. I have copied code using Maple's youtube video  where it uses Galerkin to solve pde. I don't know how to proceed as my A_n  depends on v. And is it even a correct template solution. Hoping for hints.Galerkin_Method.mw

@Alfred_F Thank you for advice, The solution obtained in equation 8 is using Spectral method, which as you mentioned falls under Galerkin method. I am thinking of first replicating the solution using Maple to better understand it. Does Maple know about spectral method, i searched in Maple Help, but couldn't find it. Is there a special package. I wonder does the solution exist if i  use other basis function? Instead of Legendre polynomials as the basis functions.

@nm Thank you for fixing the error, I want to know if Maple has tools of approximate solutions or perturbation method that i can apply. Also how can check how good the solution is if its approximate, does pdtest() work ? I have the solution to the pde1 i.e equation 8 from image, how can i write it in Maple and do pdtest() on truncated series?