@tomleslie 

Thank you, for your comments. 

According to your code, the solution  ( analytic) from 10 is close to the numeric solution. 
I tested both with the solution obtained from finite difference ( Crank Nicholson finite difference). 
But, at endpoints, there is small oscillation...I feel that there is someting wrong ( in my code) but all matrices are correct, rhs is correct, CG is corrected. 

@Carl Love 

Thank you for this remark. I appreciate always your remarks.
If I change the time step and spacestep in my code, the solution behaviors change comppletely...moreover, 

I tested both at t=0.49, there is a strange phenomena at the end of interval...
All matrices introduced are correct....
But, I cannt understand what happen...

New_code.mw

@Carl Love 

Thank your for your comments
When I plot using your line code, I get the x-axis from 1 to 11 

But, U(x,t), is function with x from 0 to 1 
How, can I get my original  x-axis limits

@Zeineb 

problem_convergence.mw

@Kitonum 

Thank you for your remark 

I appreciate your remark

I applied the CGM to a sparse , symmetric matrix but I get divergence when I solve a linear system 

@dharr 

Thank you for your remark 
I applied the same method to solve  A U[n+1]=B U[n]+M, at each iteration, but there is divergence of the methdod

problem_convergence.mw

I get divergence 

I hope find an answer thank you

@vv 

Thak you I agree with you. 
But, f(x) and Natural spline are two different functions, I must have two different curve, why in plot give me only one curve 

@Carl Love 

Thank you, now its working. 
How can  I use series in Unapply, so that I can display all peicewise functions. If I have two or three for four I can modify by hand but If I have more how can I use many peices in piecewise with unapply

splineProb.mw

Thank you

@Scot Gould 

I would like to print the entries of G, by using print(G) nothing printed.

I think,  nn is not updated in the loop

I tried 

copy or format to copy the entries I get a bad shape of table ( copy and paste in powerpoint ) 

Maybe can i take the table from maple as figure, so that I can insert the figure in powepoint 

@Carl Love 
Thank you very much for your remarks, I appreciate all  remarks proposed, your remarks added in the code , but I get 
Error, (in Matrix) this entry is too tall or too short: tab_err

integ_R.mw

thank you

@dharr 

I moved that part to display tab_err before return, but unfortunattely no  tabl_err displayed  and the command plot does not working ( no figure obtained)

integ_R.mw

please see the modified code 

trap.mw

Now, it okay, there is no problem I get the same result. 

@acer 

I appreciate your answer thank you

@acer 

thank you. Now, i get the solution with the modified code.