The first error is that you define the procedure named GenerateStencil, but in PoissonSolve you call Stencil, not GenerateStencil.

There are other errors, but that's a start.

It is -m, but modulo 2*Pi, depending on the branch of the logarithm.

simplify(evalc(I*ln(cos(m)+I*sin(m)))) assuming -Pi < m, m <= Pi;

                                                           -m

Is the assumption -Pi < m <= Pi valid in your case?

First, end each of the EQs with a semicolon. Then assign the initial conditions to a variable, say ics, and end with a semicolon. Then

dsolve({EQ||(1..5), ics}, {q||(1..5)(T)}, method= laplace, convert_to_exact= false):
allvalues(%):  #obtain approximate roots of polynomials.
evalc(%):  #convert exp(a*I*T) to sin cos forms.
simplify(%, zero);  #Remove spurious imaginary parts.

a:= proc(n) option remember; a(0)+a(n-1) end proc:  a(0):= 1:
a(4);

                                                  5

Your phi46N in the matrix is what is known as an "escaped local" variable. (This is the problem that you suspected.) So, it is different from the variable phi46N which you enter at the top level. To refer to this variable, you need to extract it from the context in which it occurs. So one form of a correct solve command is

solve({seq(eq[i], i= 1..6)}, convert(phi4N, set));

Constructing the eq[i] in a loop is actually superfluous, so you could do

solve(convert(phiN =~ phi4N, set), convert(phi4N, set));

Set Digits to a higher value (and higher than 15), redo the computation, and subtract the new results for the eigenvalues from the original. You can do the same with the eigenvectors except that they made be reversed in sign.

When the number variables being solved for is less than the number of equations you should generally use eliminate instead of solve.

eliminate(sys, drivers);

Note that eliminate always returns pairs of sets: The first set is the elimination equations; the second set is the original equations with the appropriate variables eliminated via the substitutions from the first set.

In the pdsolve command, use the options timestep and spacestep. For example,

pds := pdsolve(pde, IBC, numeric, timestep=.01, spacestep= 1);

I cannot vouch for the accuracy of the plots that you get with this, but you will get the plots instead of error messages. I tried smaller values, but the left sides of the plots still have that sharp drop. But note that the boundary condition u(t,-50) = 0 necessitates that the left sides go to 0, so maybe those plots are accurate.

Here's how to make the input the differential equation itself. The procedure preprocessODE takes the ODE (given in the standard Maple form accepted by dsolve) and returns the function f that is passed to eul. I made a slight modification to eul so that it calls preprocessODE. (I am not sure if this will work in Maple 5; the error command syntax might be different. Let me know.)

preprocessODE:= proc(eq::`=`)
local d, y, x, f;
    d:= indets(eq, specfunc(anything, diff));
    if nops(d) = 0 then error "No derivative found." end if;
    if nops(d) > 1 then error "More than 1 derivative found." end if;
    d:= d[];
    y:= op([1,0], d);  x:= op([1,1], d);
    f:= {solve(eq, d)};
    if nops(f) <> 1 then error "Cannot isolate derivative." end if;
    unapply(subs(y(x)= y, f[]), [x,y])
end proc:
    
eul:=proc(ODE::`=`,h,x0,y0,xn)
  local f,no_points,x1,y1,i;
  f:= preprocessODE(ODE);
  no_points:=round(evalf((xn-x0)/h));
  x1:=x0;
  y1:=y0;
 
  for i from 1 to no_points do
      y1:=y1+evalf(h*f(x1,y1));
      x1:=x1+h
  end do;
  y1
end proc:
       

Example of use:

eul(diff(y(x), x) = x^2*y(x)^3, .01, 0, 1, 1);

1.68181553277060

You cannot use indexed variables where the index or the base are the same as the variable of integration. So you need to change lambda[v], lambda[t], and t[c]. I changed them to L__v, L__t, and t__c  (which print as subscripted variables in Maple 17 & 18). Also, you need to assume that the lambdas are positive.

int(
     int(
          L__v*L__t*exp(-L__v*v-L__t*t),
          v = (1/2)*(q[p]+q[p]*t__c*t+2*S[di]*h*t)/(h*t) .. infinity
     ), t = 0 .. infinity
) assuming L__v > 0, L__t > 0;

The command to plot the cylinder (which is a surface, not a curve) is plot3d, not spacecurve. You may keep the rest of the command as is. However, I do recommend that you omit the options numpoints= 4000, scaling= constrained, color= blue, and thickness= 2 at least until you get the actual plot. The axis is a curve, and you may use spacecurve to plot that (and the thickness and color options would be more a appropriate for the axis).

Please attach your file as Maple worksheet instead of a pdf so that I can work with it. I believe that option is working again. (I've seen a few worksheets posted today.)

Here's an example using fsolve and dsolve's option output= listprocedure:

Sol:= dsolve(
     {
          diff(a[1](t),t$2) = -a[1](t), diff(a[2](t),t$2) = -a[2](t),
          a[1](0)=1, a[2](0)=0, D(a[1])(0)=0, D(a[2])(0)=1
     }, numeric, output= listprocedure
):
A[1]:= eval(a[1](t), Sol): #extract procedure for a[1]
A[2]:= eval(a[2](t), Sol):
fsolve(A[1], 1..infinity);
                        10.9955744378927
fsolve(A[2], 1..infinity);
                        3.14159269513977

It is also possible to locate the zeros with the events option to dsolve. I'll leave it to someone else to give an example of that.

Edit: Square brackets after D changed to parentheses.

Is this what you're looking for?

solve({0<x, x<1, x^2+y^2 - 1 > 0}, x, parametric);

If that's not what you want, could you give a specific solved example of what you want?

Your whole procedure could be replaced by the following:

ff:= (N::posint)-> Matrix([[-2 $ 2*N+1], [0 $ 2*N+1] $ 2*N-1, [-3 $ 2*N+1]]);

I don't know if correcting this will solve your problem, but you have an error in your second boundary condition. You have u(y,0) = 0, but it should be u(0,y) = 0.