Dear @Samir Khan, 

the new features sounds good. Thanks.

I wonder that does Maple 2020 make it easier to add a color bar as follows? or what do you suggest us to add the color bar in plots?

Lastly, how to add the legends to these 3D plots? (As I know legends are only available for 2D plots, but what are the methods for 3D?)

Best regards.

Thank you very much Dear @Kitonum :) 

How to add the legends to the 3D plots? (As I know legends are only available for 2D plots)

 Thank you very much dear @tomleslie for your good answer. 

Lastly, how to add the legends to these 3D plots? (As I know legends are only available for 2D plots)

@acer 

The extension of Matlab files is ".m"

I wonder whether it is possible for Maple, too or not. It may be related to MS-Windows.

Thanks, Dear @Carl Love

I am clear in my third question.

restart:
#Let's define the followings
k:=1:
M:=2:
f:=x*t:  
N:=2^(k-1)*M:   
lambda:=1:
u:=unapply(f,x,t); 
omega:=(n,x)->(1-(2^k*x-2*n+1)^2)^(lambda-0.5); 

#We will define some functions in order to define the matrix psi
Tm:=(t,lambda,m)-> (2*(m-1+lambda)*t*Tm(t,lambda,m-1)-(m-1+2*lambda-1)*Tm(t,lambda,m-2))/m;  
Tm(t,lambda,0):=1:
Tm(t,lambda,1):=2*lambda*t:
Tm2:=(tt,lambda,m)->subs(t=tt,Tm(t,lambda,m)): 
L:=(m,lambda)->piecewise(lambda<>0,(Pi*2^(1-2*lambda)*GAMMA(m+2*lambda))/(m!*(m+lambda)*GAMMA(lambda)^2),lambda=0,2*Pi/m^2,lambda=0 and m=0,Pi);

#Now, we can define the matrix psi
psii:=(n,m,x)->piecewise((n-1)/2^(k-1) <= x and x <= n/2^(k-1), (2^((k)/2)/sqrt(L(m,lambda)))*Tm2(2^(k)*x-2*n+1,lambda,m), 0); 
psi:=(t)->Array([seq(seq(psii(n,m,t),n=0..M-1),m=1...2^(k-1))] ):

# A little edited version of Dear Carl Love's Code 
U:= Matrix(
   N, 
   (i,j)-> 
      int(
         psi(x)(i)*omega(iquo(i-1,M)+1,x)*
            int(u(x,t)*psi(t)(j)*omega(iquo(j-1,M)+1,t), t= 0..1),
         x= 0..1
      )
);

Thanks, @Carl Love,

Firstly; I added your code to my code in the question. But it doesn' t work.  So, I edited something. Can you check is it right, pls.?   edited_answer.mw   

Secondly; I got the following result. u_11=0? or u_11=0.00001 etc.? 

Thirdly; you use omega[iquo(j-1,M)+1](t) instead of omega[n](t). But I think n=iquo(j-1-m,M)+1. Am I false?

@vv, Thanks for your interest.

I checked it, the function omega in the question is right.

Let function omega be any function that is greater than 0. What is your code for solving the question?

@Carl Love 

I think it is not a problem. But If you want, you can select lambda:=3/2 or lambda:=5/2 etc.

Thanks for your interest Dear @Carl Love,

1. Yes, m goes from 0 to M-1, not 1 to M-1.
2. No,  because there is also a function u(x,t) in the inner integral of original post. I edited it. 
3. I edited the post and shared original screenshots from a book. The ns in the two integrals are not different. But I think it can be wrong. If they are ns, I can imagine our code. So, I agree with you. One corresponds to i and one corresponds to j.

 Meanwhile, I use Maple 2019.

Dear @Thomas Richard,

Thank you very much for your valuable contributions. Best regards.

Dear @tomleslie,

Yes, everything is OK now! :)

Many many thanks. I hope everything in your life happens as you desire.

@Kitonum result of the command pdetest is a 1x4 matrix [0,0,0,0] instead of 0. 

What do the elements of the zero matrix stand for?

The first element (first row, first column) of the matrix stands for ... ?

The second element (first row, second column) of the matrix stands for ... ?

etc.

@Rouben Rostamian  thanks for your interest.

I edited and expanded the question.  

@vv many thanks.

What is the purpose of using back quote 'if' ?

What is the difference from using just if?