Are i and j indices or just names?
If they are indices, say i = 1 .. n , j = 1 .. n, then you could clarify the question by writing out all the equations when n = 2. Can i be equal to j?
If i and j are just names and you only have one equation then you need to know what Tj(t) is (and Hi,Uij,Ci of course).

@KatePirs In one of your by now numerous postings of this question it has been pointed out (I think by Carl Love) that your x-range might as well be considerably smaller. Anyway, try this:

IBC2 := {ui(-10, t) = 0, ui(10, t) = 0, ui(x, 0) = 0, ur(-10, t) = 0, ur(10, t) = 0, ur(x, 0) = 1/cosh(x)};
sol := pdsolve({l1, l2}, IBC2, funcs, numeric, time = t);
sol:-plot([ui,ur],t=10);
sol:-animate([ui,ur],t=500,frames=100);
sol:-plot(ur,t=0,x=-10..10,numpoints=500);
plot(1/cosh(x),x=-10..10);


The similarity is striking.

@KatePirs In one of your by now numerous postings of this question it has been pointed out (I think by Carl Love) that your x-range might as well be considerably smaller. Anyway, try this:

IBC2 := {ui(-10, t) = 0, ui(10, t) = 0, ui(x, 0) = 0, ur(-10, t) = 0, ur(10, t) = 0, ur(x, 0) = 1/cosh(x)};
sol := pdsolve({l1, l2}, IBC2, funcs, numeric, time = t);
sol:-plot([ui,ur],t=10);
sol:-animate([ui,ur],t=500,frames=100);
sol:-plot(ur,t=0,x=-10..10,numpoints=500);
plot(1/cosh(x),x=-10..10);


The similarity is striking.

@helix  Something like this:

with(plots):
p1:=animate(arrow,[<cos(tend),sin(tend),0.9*tend>,<-cos(tend),-sin(tend),0>,color=red],tend=0..50,axes=boxed,frames=150);
alpha:=1:
p2:=animate(textplot3d,[[[-1.8,0,alpha*tend,piecewise(tend<=12,"",tend<=25,"position","")]]],tend=0..50,axes=boxed,frames=150);
display(p1,p2);

@helix  Something like this:

with(plots):
p1:=animate(arrow,[<cos(tend),sin(tend),0.9*tend>,<-cos(tend),-sin(tend),0>,color=red],tend=0..50,axes=boxed,frames=150);
alpha:=1:
p2:=animate(textplot3d,[[[-1.8,0,alpha*tend,piecewise(tend<=12,"",tend<=25,"position","")]]],tend=0..50,axes=boxed,frames=150);
display(p1,p2);

There are various ways. But we shall need the details, i.e. all the lines of code.

@KatePirs
I don't get that error message, but after supplying values for a and m I get complaints about initial/boundary conditions.
You have boundary conditions, but no initial conditions!

The help page for pdsolve,numeric says:

"The pdsolve(PDEsys,conditions,numeric,vars,options) command returns a module that can be used to compute numerical solutions for time-based PDE systems over a fixed finite 1-space interval."


That must mean that initial conditions must be given also.

@KatePirs
I don't get that error message, but after supplying values for a and m I get complaints about initial/boundary conditions.
You have boundary conditions, but no initial conditions!

The help page for pdsolve,numeric says:

"The pdsolve(PDEsys,conditions,numeric,vars,options) command returns a module that can be used to compute numerical solutions for time-based PDE systems over a fixed finite 1-space interval."


That must mean that initial conditions must be given also.

@Axel Vogt Actually the worksheet was saved with the data. If you execute the assignment to stiff you will see that it is a symmetric tridiagonal matrix with diagonal elements = 2 and the other -1, as would have been produced by:
LinearAlgebra:-BandMatrix([-1,2,-1], 1, 21);

It would be more interesting and useful if you gave us the 21x21 matrix. Maybe upload a worksheet.

Works for me too in 16.02.

Works for me too in 16.02.

Why repost?

http://www.mapleprimes.com/questions/145196-How-To-Compare-Or-Perform-Phase-Portrait--Diffthetat

For help with the syntax you could look at my answer to your question about the damped driven pendulum.

http://www.mapleprimes.com/questions/145198-The-Damped-Driven-Pendulum

I don't get anything I can see.

Could you give us the equations in Maple syntax in text form we can copy and paste or as an uploaded worksheet?