Expand all variables in the problem for u in a Fourier series:
Put into P.D.E. uxx + uyy = 0:
Because of the choice of the Xn:
Put into B.C. uy(x,0) = p(x):
Solve this O.D.E. and boundary conditions for Cn:
For :
O.D.E.:
Boundary conditions give for An and Bn:
Solve for An and Bn using Gaussian elimination and clean up:
For n = 0:
O.D.E.:
C0 = A0 + B0 y
B.C.s:
Oops! This only works if
No problem! Students will explain and fix.