Next: 7.37, §6 Total Up: 7.37 Previous: 7.37, §4 Eigenfunctions

7.37, §5 Solve

Expand all variables in the problem for u in a Fourier series:

Put into P.D.E. u_xx + u_yy = 0:

Because of the choice of the X_n:

Put into B.C. u_y(x,0) = p(x):

Put into B.C. u_y(x,1) = q(x):

Solve this O.D.E. and boundary conditions for C_n:

For :

O.D.E.:

Boundary conditions give for A_n and B_n:

Solve for A_n and B_n using Gaussian elimination and clean up:

For n = 0:

O.D.E.:

C₀ = A₀ + B₀ y

B.C.s:

Oops! This only works if

Also, A₀ remains unknown!

No problem! Students will explain and fix.

Next: 7.37, §6 Total Up: 7.37 Previous: 7.37, §4 Eigenfunctions