7.37, §5 Solve

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

Remember that the expression you find for the integrals in the bottom, , does not work for n=0, in which case it turns out to be 1.

Fourier-expand the PDE u_xx + u_yy = 0:

Because of the Sturm-Liouville equation in the previous section

giving the ODE

or substituting in the eigenvalue

Fourier-expand the BC u_y(x,0) = p(x):

Fourier-expand the BC u_y(x,1) = q(x):

Solve the above ODE and boundary conditions for u_n. It is a constant coefficient one, with a characteristic equation

Caution! Note that both roots are the same when n=0. So we need to do the n=0 case separately.

For the solution is

The boundary conditions above give two linear equations for A_n and B_n:

whic are best solved using Gaussian elimination. Rewriting the various exponentials in terms of sinh and cosh, the solution for the Fourier coefficients of u except n=0 is:

For n = 0 the solution of the ODE is

u₀ = A₀ + B₀ y

Put in the boundary conditions to get equations for the integration constants A₀ and B₀:

Oops! We can only solve this if

p₀ = q₀

Looking above for the definition odf those Fourier coefficients, we see we only have a solution if

Unfortunately, these two integrals will normally not be equal! Also, A₀ remains unknown. No problem! Students will explain and fix the problem.