@Rouben Rostamian  

I really appreciated with your comments and suggestions.Best wishes to all. Thanks

@Rouben Rostamian  

pls How can I code this with Maple. Thanks

The pde

$$
\eqalign{
  & u_t  =  - u_x  + 2 + t + x  \cr 
  & u(0,x) = f(x)  \cr 
  & u(t,0) = \alpha (t)  \cr 
  & where  \cr 
  & f(x) = 1 + x;\alpha (t) = 1 + t; \cr} 
$$
The iteration is given below

$$
\eqalign{
  & u^{[k + 1]} (t,x) = 1 + x - \int\limits_0^t {(u^{[x]}  + 2 + s) + x} )ds  \cr 
  & u^{[k + 1]} (t,x) = u^{[k + 1]} (t,x) - (u^{[k + 1]} (t,0) - (1 + t)) \cr} 
$$
 

Thanks,for the response.I constructed new Implicit Runge-kutta,from zeros of Gengebaur orthogonal polynomials.I want used the methods and solve a systems of ordinary differential equations and compared it with existing method,For examples those methods constructed with zeros of Legendre polynomial.Using Maple Code.Thanks