To Lakeland,about Fourier and the problem.
This type of problem seems to be a typical application of FEM (finite element
method). That is, you try to find a solution as a sum of well known functions
with arbitrary coefficients, but FEM use a good basis for this problem, the
linear problem of calculating the coefficient has a coefficient matrix with
special properties (tridiagonal,etc.)
About Fourier, I don't know what is p but perhaps is
(1) the number of coefficients or
(2) a term to make a change in the interval
[-L,L] -> [-pi/2,pi/2]
In any case, I think that the usual formulas for the coefficients of Fourier
are trivial to program in Maxima.
b(n):= 1/L * integrate(f(x)*cos(n*x),x,-L,L);
a(n):= 1/L * integrate(f(x)*sin(n*x),x,-L,L);
f(x)= a0/2 + sum a(n) sin(n*w*x) + sum b(n)*cos(n*w*x) with w=%pi/L