Cartesian coordinates problem

Question

Assignment:

Let €:= {(x,y) : 0 <  x < 1, 0 < y < 1}.

(1) Let u(x,y) satisfy the PDE

      u_{xx  +}  u_{yy =} 0_,  

      u(0, y) = 0, u(1, y) = 0,  0 < y < 1;     

      u(x, 0) = 0, u(x,1) = f(x), 0 < x <1.

(2) If f(x) = x, 0 < x < ½; and f(x) = ½, ½ < x <1; then what does u(x,y) from problem (1) look like?

Provide complete and step by step solution for the question and show calculations and use formulas.