@janhardo 

I really appreciate you. My mistake was in indices for P matrix. As you mentioned, "Matrix indices in Maple start from 1, so adjust for 0-based indexing", now the problem is clearified to me.
Hope you the best outcome in all life issues.

@Mariusz Iwaniuk 

dear M.I

thanks a lot again for your elegant guide. hope you all the best. 

@Mariusz Iwaniuk 

Dear Mariusz Iwaniuk 

Thanks in advance for your great answer, but I have two question about it and i will be so grateful if you can guide me:

1. In the line "sol := pdsolve([PDE]);" we see 5 constants which should be detemined via BC's and IC's(Boundary and Initial Conditions). But we have only 4 of these conditions. So how we should match them?

2. In the line "SOL := solve.....eval(rhs(sol[1]),t=3)=0,", I have change "eval(rhs(sol[1]),t=3)=0" to eval(rhs(sol[1]),{x=3,t=3)=0". is this true for condition 3? Am I allowded?

Thanks again for any help

@Carl Love 
Dear Carl LOve, 
Hi. In the following code, I can see the general solution but i don't know how apply the initial conditions. also I want to know if there is any analytical solution to the problem using maple?

>restart;
> with(PDEtools):
>E1:=[ diff(f1(x,t),x)=2*f3(x,t)+3*f1(x,t)-f2(x,t),
            diff(f2(x,t),t)=-2*f4(x,t)-3.2*f1(x,t)+f2(x,t),
            diff(f3(x,t),x)=-3*f3(x,t)+3.2*f4(x,t)-0.045*f1(x,t),
            diff(f4(x,t),t)=f3(x,t)-f4(x,t)];

>sol:=pdsolve(E1,[f1,f2,f3,f4]);

@Kitonum 

Hi Dera Kitonum

I apologize for the mistake in typing. Thank you for your complete response. I resubmitted the modified form of the problem. Thanks for the guidance again.

@mmcdara 

Hi Dear.

I apologize for the mistake in typing. The corrected problem has been replaced.

@Carl Love 

Thank you dear Carl
A is also a column vector and I have forgotten the "T" superscript.

A and H(t) can have finite number of elements(say, n).

@Rouben Rostamian  
thanks in advance

@Kitonum 
Thanks in advance

Thanks in advance

@mahmood1800 

Hi

Sorry, i have not seen the "=" sign. ok. so we have the constraint as "A=B+C", and if we change it to "A-B-C=0", then we could have following:

>restart;
> with(Optimization):
> with(plots):
>   obj := (c0-x)^2+(c1-(2/3)*x-(1/3)*y)^2+(c2-(1/3)*x-(2/3)*y)^2+(c3-y)^2;
>  cnsts := 32*x+19*y - 15*c0+18*c1+15*c2+3*c3;
>            Minimize(obj, {cnsts=0});
where we have 

      [0., [c0 = 0., c1 = 0., c2 = 0., c3 = 0., x = 0., y = 0.]]

but i am not sure that this is true.

@Kitonum 

Dear Kitonum
thanks in advance. the program is now works correctly. that is so great.
hope you the best.
mahmood

@Kitonum 

Dear Kitonum
thanks in advance. But this reduce the same polynomials for all the subintervals. this is not true.
hope you the best

@vv 
 thanks a lot

this is the corret code...

Best wishes

@vv 

Hi and thanks,

My main problem is the procedure that produce the b_n,m(t)'s correctly. The Fourier coefficient associated to the restriction of x(t) will be found under some constraints later.