@Carl Love 

Here is an Anm table of first values for n and m (we can observe that some Amn values are equal to zero) :

@Carl Love 

Thank you again for your time :)

I dig your explanations. It's clear for me.

Nevertheless, I tried evalf(A) with restart several times
I always get 2.956602889.
So I decided to compute the same problem with MATHEMATICA.
And I get 288/\[Pi]^4, that is...2.9566.

So I am a little bit confused now.

My MATHEMATICA code below :
 

ClearAll[x, y,t]

L = 2;
H = 3;
c = 0.3;
f1[x_] := Piecewise[{{x, 0 <= x <= L/2}, {L - x, L/2 < x <= L}}];
f2[y_] := Piecewise[{{y, 0 <= y <= H/2}, {H - y, H/2 < y <= H}}];

Plot[{f1[x], f2[x]}, {x, 0, 3}, 
 PlotLabels -> Placed[Automatic, Above]]

A[n_, m_] := 
 4/(L H) Integrate[
   3 f1[x]*f2[y] Sin[(n \[Pi])/L x] Sin[(m \[Pi])/H y], {x, 0, L}, {y,
     0, H}]

A[1, 1]

I tried with other n and m values, and Maple & Mathematica give the same result.
 

@Carl Love 

well done!
I didn't know this formulation.
I just begin wih integrals by Maple.

But I think my problem is the one you told me in your previous answer. It is a number and not a function.
And I have some difficulties to compute this number.
See my last post.

And thank you very much for your help. 

Now, it is very late in France : 1 o'clock in the morning and I've got the blues. It's time for me to go to sleep, dreaming about my integrals :). So, I will let you for a while. My apologies.

@Carl Love

So the above expression is not a function but a value depending on x and y (piecewise). Isn't it?
Let's call this expression A.
A:=(2*Int(3*piecewise(x < 1, x, 1 < x, 2 - x)*piecewise(y < 3/2, y, 3/2 < y, 3 - y)*sin(Pi*x/2)*sin(Pi*y/3), [y = 0 .. 3, x = 0 .. 2]))/3
If I write evalf(A) I get 2.956602889
And when I take each combination (there are 4), I get 72/Pi^2 = 7.295125220 for each of them.

If I well understand the Maple notation, I get these 4 expressions that I have evaluated.

(2*Int(3*x*y*sin(Pi*x/2)*sin(Pi*y/3), [y = 0 .. 3, x = 0 .. 2]))/3 gives 72/Pi^2

(2*Int(3*(2 - x)*y*sin(Pi*x/2)*sin(Pi*y/3), [y = 0 .. 3, x = 0 .. 2]))/3 idem

(2*Int(3*x*(3 - y)*sin(Pi*x/2)*sin(Pi*y/3), [y = 0 .. 3, x = 0 .. 2]))/3 idem

(2*Int(3*(2 - x)*(3 - y)*sin(Pi*x/2)*sin(Pi*y/3), [y = 0 .. 3, x = 0 .. 2]))/3 idem

Where is my mistake?

@Carl Love 

oups!!!
You are right. It's my mistake.

@tomleslie 

Thank you for your answer.
Have a nice day.

And can MAPLE 2021 do it? 
Thanks in advance for yopur answer.

@Rouben Rostamian  
Thank you, you are right, I've made a trivial mistake :(

@P2prod 
My new code below :

restart;
with(plots);
with(PDEtools);
a := 1;
b := 1;
rho := 8166;
h := 10^(-3);
Dp := 19.2615;
WavEq := rho*h*diff(w(x, y, t), t $ 2)/Dp + diff(w(x, y, t), x $ 4) + 2*diff(w(x, y, t), x $ 2, y $ 2) + diff(w(x, y, t), y $ 4) = 0;
solalone := pdsolve(WavEq);

ic := w(x, y, 0) = sin(7*x)*cos(4*y), D[1](w)(x, y, 0) = 0;
bc := w(0, y, t) = 0, w(a, y, t) = 0, w(x, 0, t) = 0, w(x, b, t) = 0, D[1 $ 2](w)(0, y, t) = 0, D[1 $ 2](w)(a, y, t) = 0, D[2 $ 2](w)(x, 0, t) = 0, D[2 $ 2](w)(x, b, t) = 0;

solicbc := pdsolve([WavEq, ic, bc])

Maple gives a solution for equation without ic and bc. 
For equation with initial conditions and boundary conditions, the provided solution is ().

I am a beginner in Maple and PDE with bi-harmonic, so  I am not sure to do wright. Be indulgent.

@Rouben Rostamian  

Thank you very much for your answer. The code you send me is perfectly running.

And I am really sorry, I mixed u and w. The unknown is w(x,y,t). Sorry again.
And you are right, my boundary conditions are insufficient.
In fact, I was inspired by another MaplePrimes post telling us "In Maple 2020.2, one can analytically solve the heat equation in 2 spatial dimensions, at least for "simple" boundary conditions". So I tried my wave equation by imitation with "simple" BC. Bad idea.

I will retry with the good boundary conditions involving derivatives, but I am pretty sure it will not work with my 2020.1 version.
I just need to buy the new one :)

Thanks again.
Pascal77

remark : equation and BCs come from this study : https://www.intechopen.com/books/advanced-engineering-testing/effect-of-various-edge-conditions-on-free-vibration-characteristics-of-isotropic-square-and-rectangu