@Thomas Richard Thanks for your answer. Sorry there were a little mistak in above PDEs the corrected PDEs are

PDE1 := diff(u(y, t), t)+diff(u(y, t), t, t) = diff(u(y, t), y, y)-u(y, t);
PDE2 := v(y, t)+diff(v(y, t), t) = diff(u(y, t), y);
ICandBC := {u(0, t) = 0, u(3, t) = 0, u(y, 0) = 0, v(y, 0) = 0, (D[2](u))(y, 0) = 0};
pds := pdsolve({PDE1, PDE2}, ICandBC, numeric)

Again I got some error given bellow

Error, (in pdsolve/numeric/par_hyp) input system is too far from a 'standard' form (see ?pdsolve,numeric for more detail)

@tomleslie dear In the attachment I wrote the same code as given in above. But in the attached file I used summation form instead of computing integrals which definitely reduces the computational time. My question is can we write our required matrix F in “proc” which reduces more computational time? If yes please help me. I am waiting for your kind response.

nonlinear.mw

@tomleslie that one have some error I just come to know. The above one is new I need help about it

@Preben Alsholm thanks a lot for couple of positive response. Your last response is fruitful for me. But it still take a lot of time in evolution. Can you reduce it computation time. Your positive response is highly appreciable. 

@Preben Alsholm please see the the following file

hELP.mw

@Preben Alsholm thank you for your response. I also have set of numeric values for the above integration. 

x = i/(1.*M1), y = j/(1.*M2) for i=1..M1 and j=1..M2

Now how to evalute it? I tried but got error.

@Preben Alsholm thank you for your answer. Can we integrat it via some numerical method and how?

@Preben Alsholm thanks for your response. I made a little mistake in my asked question. The function U should be the following one

U := q^6*sin(p);

instead of U := y^6*sin(x);

Now please guide me. Thanks

@tomleslie Thanks for your response. In the above code, integration part take alot of timw when NN=4. I need to run it for higher values like NN=7,8,9. Please Let me know how we can reduce computational time here?

@Christian Wolinski thanks for your positive response. I have another problem regarding above querry suppose,

Sol := {u[1, 1, 1, 1] = 0.33004814594903018204e-1, u[1, 1, 2, 1] = 0.67470784550800487342e-1, u[1, 2, 1, 1] = 0.67470784550800487340e-1, u[1, 2, 2, 1] = .13881687114056674193, u[2, 1, 1, 1] = 0.63417136832119584739e-1, u[2, 1, 2, 1] = .13209553008466760265, u[2, 2, 1, 1] = .13209553008466760265, u[2, 2, 2, 1] = .27601195211329433114}

is a set of points if apply your command like

for i while i <= nops(Sol) do (proc (x, y) assign(x, y) end proc)(op(op(i, Sol))) end do

fail to see the value of u[2, 2, 2, 1]; 

File is also attached for more justification

Help.mw

@Christian Wolinski thanks

@acer thanks for your positive response!

@Carl Love really sorry to disturbe you again. Actually I did a little mistake. Order of block matrix should be M1M2×M1M2 which have null and F1 matrices of order M2×M2.

Like M1=2 and M2 = 2

Matrix(2, 2, {(1, 1) = O, (1, 2) = O, (2, 1) = F1[2, 1], (2, 2) = O})

where null and F1 matrices of order 2 by 2

and for M1=2 and M2= 3

Matrix(2, 2, {(1, 1) = O, (1, 2) = O, (2, 1) = F1[2, 1], (2, 2) = O})

where null and F1 matrices of order 3 by 3

for M1=3 and M2=2 then we have block matrix

Matrix(3, 3, {(1, 1) = O, (1, 2) = O, (1, 3) = O, (2, 1) = F1[2, 1], (2, 2) = O, (2, 3) = O, (3, 1) = O, (3, 2) = F1[3, 2], (3, 3) = O})

where null and F1 matrices of order 2 by 2

@Carl Love thanks I got it :)
Now I want to define a block matrix F like

restart; with(LinearAlgebra); with(linalg);

M1 := 3; M2 := 3;

OM2 := Matrix(1 .. M2, 1 .. M2, shape = zero);

IM2 := IdentityMatrix(M2)
nu := 1;

printlevel := 3;

for r1 from 2 while r1 <= M1 do

for s1 while s1 <= r1-1 do

if type(r1+s1, odd) then

F1[r1, s1] := (2*(2*s1-2+2*nu))*IM2

end if

end do end do;

F is M1 by M2 block matrix have entries F1[r1, s1] others entries are null matrix (OM2)

It did it for M1=M2=3 and define F manully like

F := blockmatrix(M1, M2, [OM2, OM2, OM2, F1[2, 1], OM2, OM2, OM2, F1[3, 2], OM2])
But I want to know general way how to define F if M1 and M2 large like 10.

@Carl Love thanks for you kind response. May be meaning is same but I used the block matrix termonology. Anyhow, how to define it for any value of M.