Dear @tomleslie, you are right. The Maple numeric method is more accurate than my method.

What are the simple methods for finding maximum relative error, and max. absolute error at the interval?

Thanks, @vv, 

This is relative error, right?

How to find max. absolute error at the interval?

Dear @Carl Love, thanks for your feedback.

1- You are right, I corrected the post.

2- I removed local variable K.

3- Yes, it must be "*". I corrected it, too.

4- phii is not an external procedure. I corrected it as you say. (  phi[n, m+s-2*j](t) )

 I think the multiplication is commutative and also the matrix  is a symmetric matrix. 

Dear @tomleslie, thanks for your interest.

I think I am a misunderstanding. So, I edited the question.

After finding matrices by hand, we can write maple code for the matrices like you (Thanks again). But I want to find the matrices M,C,K etc. from the directly from the following equation by Maple. 

@Kitonum 

Thank you.

 I edited the question according to corrections you say.

@Rouben Rostamian  

Thank you very much.

You are right, it is not u^4. ( that' s u^+)

I edited the question

@tomleslie 

Dear @tomleslie, 

The exact solution in the code is right, but I made a typo while writing the exact solution in the question. I edited the question.

I don' t know whether you know Legendre wavelets method for solving ODE's. But If you glance at the theory of Legendre wavelets, you will see it' is a very simple but powerful method.

**My OBJECTIVE in the question:**
I want to solve the ODE by Legendre wavelets method and in order to achieve it, I wrote a Maple code. But, something is wrong. I didn' t understand what is the problem. Could you help me, please?
 

@Joe Riel thanks. That' s what I want.

@tomleslie 
Lastly; are the following codes right?
 

Let y(x,t):=sin(Pi*x)*dudt(t),

in order to find 3D graph of y(x,t)

  plot3d( sin(Pi*x)*dudt(t),x=0..2,
        t=0..1,
        title= typeset(diff(u(t),t), " vs ", t)
      );
  

and in order to find y(x,t) for some values (x,t)

A:=unapply( sin(x)*dudt(t), (x,t));
A(1,2);

@tomleslie 

Thank you so much. It is what I want to find.

@Rouben Rostamian  

Thank you very much.

How to find derivative of u(t) after finding numerically as follows?

restart:
ode:=diff(u(t),t,t)+diff(u(t),t)-6*u(t)=0:
ICs:=u(0)=1,D(u)(0)=0:
dsolve({ode,ICs},numeric);

Dear @tomleslie,

I found elements of Matrix 
NOW, How to create  Matrix A:=  by the elements in the_code.mw

J:=proc(j1,j2,j3,m1,m2,m3)
local i,f:
i:=max(0,j2-j3-m1,j1-j3+m2):
f:=min(j1+j2-j3,j1-m1,j2+m2):
if m1+m2+m3<>0 then:
0
elif j3>j1+j2 then:
# printf("Error. Does not satisfy the triangle relation");
0
elif j3<abs(j1-j2) then:
# printf("Error. Does not satisfy the triangle relation");
0
elif abs(m1)>j1 or abs(m2)>j2 or abs(m3)>j3 then:
0
else:
simplify(((-1)^(j1-j2-m3))*((((j1+j2-j3)!*(j1-j2+j3)!*(-j1+j2+j3)!*(j1+m1)!*(j1-m1)!*(j2+m2)!*(j2-m2)!*(j3+m3)!*(j3-m3)!)/(j1+j2+j3+1)!))^(1/2)*sum(((-1)^t)/((j1+j2-j3-t)!*(j1-m1-t)!*(j2+m2-t)!*(j3-j2+m1+t)!*(j3-j1-m2+t)!*t!),t=i..f));
end if;
end proc:

###### That is matrix which we want to find ###############
#psi_1aa.psi_1bb:=add(J(aa,bb,c,0,0,0)*J(aa,bb,c,0,0,0)*sqrt(2*(2*aa+1)*(2*bb+1)*(2*c+1))*psi(1,c),c=abs(aa-bb)..(aa+bb));
#psi_2aa.psi_2bb:=add(J(aa,bb,c,0,0,0)*J(aa,bb,c,0,0,0)*sqrt(2*(2*aa+1)*(2*bb+1)*(2*c+1))*psi(2,c),c=abs(aa-bb)..(aa+bb));
#Let A:= Psi.Transpose(Psi). So, it is matrix which we want to find.


aa:=0:bb:=0: 
A(1,1):=add(J(aa,bb,c,0,0,0)*J(aa,bb,c,0,0,0)*sqrt(2*(2*aa+1)*(2*bb+1)*(2*c+1))*psi(1,c),c=abs(aa-bb)..(aa+bb)); 
# it is element of A in first row and column or psi_10.psi_10
                               
aa:=1:bb:=0: 
A(2,1):=add(J(aa,bb,c,0,0,0)*J(aa,bb,c,0,0,0)*sqrt(2*(2*aa+1)*(2*bb+1)*(2*c+1))*psi(c),c=abs(aa-bb)..(aa+bb)); 
# it is element of A in second row and column or psi_11.psi_10 
#We can proceed so so for aa=0..M-1 and bb=0..M-1

#Similarly; Let's find psi_20.psi_20
A(M+1,1):=add(J(aa,bb,c,0,0,0)*J(aa,bb,c,0,0,0)*sqrt(2*(2*aa+1)*(2*bb+1)*(2*c+1))*psi(2,c),c=abs(aa-bb)..(aa+bb));
 # it is element of A in M+1 row and column or psi_20.psi_10

@tomleslie 

Thank you for your reply and feedback.

I edited the question and wrote  psi[n,m](t) for any n,m

Dear @acer ,

I understood now, thanks.

I asked it in here. But anyone doesn' t reply. So I asked as a new question. Sorry.