Question: Need help in solving nonlinear elliptic pde

Hello, I am having problem in solving the pde in the image using maple. Due to its nonlinear nature it has been solved for small value of v using first order perturbation technique and seperation of variable method into radial and angular part in many papers. I am having trouble in proceeding as Maple complains about Boundary/Initial condition.Please tell me if Maple can provide any help in improving existing solution or providing new solution? I can post the full solution procedure by the method i mentioned if needed.

restart

ode0 := (diff(xi^2*(diff(theta[E](xi), xi)), xi))/xi^2 = -theta[E](xi)^n

(2*xi*(diff(theta[E](xi), xi))+xi^2*(diff(diff(theta[E](xi), xi), xi)))/xi^2 = -theta[E](xi)^n

(1)

bc0 := theta[E](0) = 1, (D(theta[E]))(0) = 0

theta[E](0) = 1, (D(theta[E]))(0) = 0

(2)

base := dsolve({bc0, ode0}, theta[E](xi), series)

theta[E](xi) = series(1-(1/6)*xi^2+((1/120)*n)*xi^4+O(xi^6),xi,6)

(3)

pde1 := (diff(xi^2*(diff(psi(xi, mu), xi)), xi))/xi^2+(diff((-mu^2+1)*(diff(psi(xi, mu), mu)), mu))/xi^2 = -psi(xi, mu)^n+v

(2*xi*(diff(psi(xi, mu), xi))+xi^2*(diff(diff(psi(xi, mu), xi), xi)))/xi^2+(-2*mu*(diff(psi(xi, mu), mu))+(-mu^2+1)*(diff(diff(psi(xi, mu), mu), mu)))/xi^2 = -psi(xi, mu)^n+v

(4)

bc1 := psi(0, mu) = 1, (D[1](psi))(0, mu) = 0, (D[2](psi))(0, mu) = 0, limit(psi(xi, mu), v = 0) = rhs(base)

psi(0, mu) = 1, (D[1](psi))(0, mu) = 0, (D[2](psi))(0, mu) = 0, psi(xi, mu) = series(1-(1/6)*xi^2+((1/120)*n)*xi^4+O(xi^6),xi,6)

(5)

pdsolve(pde1, [bc1], psi(xi, mu))

Error, (in pdsolve/sys) too many arguments; some or all of the following are wrong: [[psi(xi,mu)], [psi(0,mu) = 1, D[1](psi)(0,mu) = 0, D[2](psi)(0,mu) = 0, psi(xi,mu) = series(1-1/6*xi^2+1/120*n*xi^4+O(xi^6),xi,6)]]

 

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Download Nonlinear_Elliptic_PDE_in_Spherical_Coordinate.mw

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