the error in summation , How can this be corrected?

restart;

U[0] := x^2;

for k from 0 to 1 do U[k+1] := sum(U[s]*(diff(U[k-s], x)), s = 0 .. k) end do;
U[0]*(diff(U[0], x));
                                 3
                              2 x 
U[1];
                               0

How can the value of uu(.1, .1, .1)  as a number, and what is the error when drawing?

restart;
uu := proc (x, y, t) options operator, arrow; e^(.2*t)*(e^(-x)+e^(-y)) end proc;

evalf(uu(.1, .1, .1));
                              2.  
                             -----
                              0.08
                             e    
plot3d(uu(x, y, .1), x = 1 .. 2, y = 1 .. 2);
Warning, expecting only range variables [x, y] in expression e^.2e-1*(e^(-x)+e^(-y)) to be plotted but found name e

how can revision of error in 

restart;
u0 := proc (x) options operator, arrow; 1+2*x end proc; h := 0;
for k from 0 to 5 do U := proc (k, h) options operator, arrow; eval((diff(u0(x), [`$`(x, k)]))/factorial(k), x = 0) end proc end do;
m := h+1;
for k from 0 to 5 do U := proc (k, h) options operator, arrow; (sum(sum(U(r, h-s)*U(k-r, s), s = 0 .. h), r = 0 .. k)+(k+1)*U(k, h))/m end proc end do;
U(0, 1);
Error, (in U) too many levels of recursion

How to draw the given data?

Download aaa.mw

How to solve a system of partial equations with boundary conditions.I used this formula

restart;
sys_Pde := diff(V(x, t), x, x) = 0, diff(T(x, t), x, x) = 0;
          
BC := eval(diff(V(x, t), x), x = 1) = 0, eval(V(x, t), x = 0) = 1+cos(w*t), eval(T(x, t), x = 0) = 0, eval(T(x, t), x = 1) = 1;
pdsolve({sys_Pde});
   {T(x, t) = _F1(t) x + _F2(t), V(x, t) = _F3(t) x + _F4(t)}
pdsolve({BC, sys_Pde});

Error, (in PDEtools:-ToJet) found functions to be rewritten in jet notation, {V(1, t)}, having different dependency than the indicated in [V(x, t)]