Question: Getting undefined terms in output of pdsolve

Getting undefined terms in pdsolve output. Am I expecting too much of pdsolve to be able to solve such complex systems? If so, is there another way I could go about trying to solve such a system with approximations rather than analytically.

Here is the system which produces the undefined terms when solved using pdsolve:

Get the following for w[3](y,z):

Code below:

restart;

with(PDEtools); umain := 1-exp(-y)+sum(A^i*u[i](y, z), i = 1 .. 5); vmain := -1+sum(A^i*v[i](y, z), i = 1 .. 5); wmain := sum(A^i*w[i](y, z), i = 1 .. 5); pde[main] := diff(vmain, y)+diff(wmain, z); pde[main[2]] := vmain*(diff(wmain, y))+wmain*(diff(wmain, z))-(diff(wmain, y$2));

sys[1] := [A*coeff(pde[main[2]], A) = 0, w[1](0, z) = sin(z), w[1](infinity, z) = 0]; pdsolve(sys[1]); w[1] := proc (y, z) options operator, arrow, function_assign; sin(z)*exp(-y) end proc; pdex[1] := diff(w[1](y, z), z)+diff(v[1](y, z), y) = 0; pdexbound := v[1](0, z) = 0; sysx[1] := [pdex[1], pdexbound]; pdsolve(sysx[1])

v[1] := proc (y, z) options operator, arrow, function_assign; cos(z)*(exp(-y)-1) end proc

for i from 2 to 3 do pde[i] := coeff(pde[main[2]], A^i) = 0; sys[i] := [pde[i], w[i](0, z) = 0, w[i](infinity, z) = 0]; pdsolve(sys[i]); assign(op(0, lhs(pdsolve(sys[i]))) = unapply(rhs(pdsolve(sys[i])), op(lhs(pdsolve(sys[i]))))); sysx[i] := [diff(w[i](y, z), z)+diff(v[i](y, z), y) = 0, v[i](0, z) = 0]; pdsolve(sysx[i]); assign(op(0, lhs(pdsolve(sysx[i]))) = unapply(rhs(pdsolve(sysx[i])), op(lhs(pdsolve(sysx[i]))))) end do