@Carl Love Would it be possible in Maple to make unassign not evaluate its arguments, so that uevaluation quotes were not necessary?
If you restrict unassign to a fixed number of arguments it is easy enough by just using the existing unassign:
p:=proc(x::evaln) unassign(x) end proc;
a:=7:
p(a);
a;
An attempt to extend this to a sequence of arguments using the seq-modifier:
q:=proc(x::seq(evaln)) unassign(x) end proc;
Error, `evaln` cannot be used within `seq` modifier for parameter `x` in procedure q

@acer My guess is that there is no (significant) difference between a Danish keybord and a Norwegian.
Ctrl+Alt+=  works in Maple 17.02 on a Danish keyboard.

@wzelik

Using that the matrix A is symmetric and has an LU-decomposition with no switching of rows, i.e. A = L.U, A can also be written as A = L.D1.L^T, where D1 is the diagonal matrix having the diagonal of U as its diagonal.
A will be negative definite iff all diagonal elements of U are negative. Since all your constants are positive all the elements but the last one U[n,n] are always negative. 
The last element U[n,n] is
((n-1)*hext*hint*`δx`+hext*lambda-hint*lambda)/(`δx`*((n-1)*hint*`δx`+lambda)*rho*cp);

Since `δx` = L/(n-1) this can be written

(L*hext*hint+hext*lambda-hint*lambda)*(n-1)/(L*(L*hint+lambda)*rho*cp)

and this is negative iff

L*hext*hint+hext*lambda-hint*lambda < 0.

Notice that this inequality does not contain n, thus it doesn't help to increase n.
The inequality can be rewritten as
L < lambda*(1/hext-1/hint)
For your values of the constants L, lambda, hext, and hint this inequality is not satisfied.

This part
(diff(v(r,t),r)-v(r,t)/r+beta*diff(diff(v(r,t),r),t)
- 1/r * beta*diff(v(r,t),t)
-Re*(1+lambda)*exp(1-p(r,t))*diff(v(r,t),t)-(1+lambda)*exp(1-p(r,t))*g2
starts with a parenthesis '(' , but its partner ')' is missing.
Furthermore, Re has a multiplication sign following it. It should just be Re(....).
Your IBC has sint, it should be sin(t).

@wzelik I get an error 404. File or directory not found.

@Carl Love My guess is that the intended meaning was plotting z as a function of x which is accomplished as Adri van der Meer is describing or as I commented to the asker's question.
But only the asker can tell us, what he meant. Clearly these different interpretations of the intended meaning are quite different.

It is not clear to me what you want to do: What does "after indicate to extract u(t) etc. " mean?

Change the last line to
odeplot(sol,[x(t),z(t)],0..10,numpoints=200,color=blue,legend="z",scaling=constrained);

@bunmipopoola From the picture you posted it appears that the maximal time considered is 3.1 ms, i.e. 3.1*10^(-3) s. Why do you use .0000031? Is the unit of time not seconds?
Are you sure that you have the correct equation eq1? Is the sign simply wrong?
It follows from eq1 that D(F)(t) > 0 for all time t. Thus F is increasing. If F is the amount of fuel, then that is clearly wrong, since you are not gathering wood for your wood burning stove.

@bunmipopoola Did you try all my lines?
Try copying the whole thing from my answer and paste it into a fresh worksheet. Then execute.
I just tried that very thing myself in Maple 17 (but the version shouldn't matter much).
No problem.

@bunmipopoola Did you try all my lines?
Try copying the whole thing from my answer and paste it into a fresh worksheet. Then execute.
I just tried that very thing myself in Maple 17 (but the version shouldn't matter much).
No problem.

@Gaia Now the issue with +- seems  resolved: It meant -.
You need eqt to be a polynomial not an equation. Simply remove =0.
You won't have any luck with algcurves[parametrize] though.

@traiannp If the system really is inconsistent, then that means there just is no solution!

@traiannp If the system really is inconsistent, then that means there just is no solution!

In Maple 17
solve(sys, [c,d,e,f,g,h,i,j,k,l,m,n,p]);

returns [], i.e. it didn't find any solution.