@Alejandro Jakubi

1)  Thanks for the useful information regarding  parametric. 

2) You replaced the equation  sqrt(x-a) = x  with  the equation  x-a = x^2.  But  in the real domain, these equations are not equivalent.

@Carl Love  In my code the specific function  f  was used rather than arbitrary function as in your code. I think the questioner himself according this example will write the function that interests him.

@mahmood180  Your example for  tj=1:

HybrFunc(4, 3, 1):

for j from 0 to 2 do

seq('b'[i,j]=b[i,j](t), i=1..4);

od;

Download the text (not an image) of the code, so we can check its accuracy.

@Stephan   

1) See the help on  piecewise

2) Add the line

applyop(x -> lhs(x)/L < rhs(x)/L, {1,3}, Mf(xi));

Animation quality can be slightly improved:

PDE := diff(u(x, t), t$2) = diff(u(x, t), x$2):

IBCs := u(x, 0) = sin((1/2)*Pi*x)*exp(-x), (D[2](u))(x, 0) = 0, u(0, t) = 0, u(4, t) = 0:

Sol := pdsolve({PDE}, {IBCs}, numeric):

Sol:-animate(t = 0 .. 20, frames = 300, thickness = 2, numpoints = 3000, labels=[x, u(x, t)]);

@Markiyan Hirnyk  There are infinitely many matrices  A  and  B  satisfying the condition  A.B = C . This is seen from the direct solution:

restart;

C:=Matrix([[8, 2, -2], [2, 5, 4], [-2, 4, 5]]):

A:=Matrix(3,2, symbol=a):

B:=Matrix(2,3, symbol=b):

solve(Equate(A.B,C)):

assign(%):

'A'=A; 'B'=B;

simplify(B.A);

@Joe Riel  of siderals is  Equate(A.B, C)

@Markiyan Hirnyk   I think that this is a natural way to solve such problems.

@masoud moeini 

restart;

alias(z(t) = exp(t)+a+sqrt(exp(t)-a+b)):

A(t):=ln(z(t)):

diff(A(t), t);

@rlopez  Thank you for an elegant solution to the problem! I am amazed at your deep knowledge of the package.

@c4meleon 

Your loop does not make a single step, because  xnew-xold<0 . In addition, at each step, do nothing.

Corrected code:

restart:

 xold:=1.5:

 f:=x->tan(x)-x-1:

 divis:= evalf(subs(x=xold,f(x))/subs(x=xold,diff(f(x),x))):

 xnew:=xold-divis:

 dp:=7:

 while abs(xnew-xold)>5.0*10.0^(-(dp+1)) do

  divis:= evalf(subs(x=xnew,f(x))/subs(x=xnew,diff(f(x),x))):

  xold:=xnew: xnew:=xnew-divis:

od:

 xnew;

Your task is similar to the nonlinear programming problem  http://en.wikipedia.org/wiki/Nonlinear_programming . In Maple  these problems solves  Optimization  package. Give specific wording of your problem. 

@Markiyan Hirnyk  Read carefully my original question. Error does not appear immediately, and if second time to run the code with changing the parameters.

@Axel Vogt  I wrote that the error only occurs with repeated use of the procedure with modified  arguments. But if you run the procedure again with the same arguments (after error), the error disappears. Errors are the same in Maple 12 (classic) and M16 (standard).