|Boundary conditions (2 each)→||ϕ(0) = ϕ(L) = 0||(0) = (L) = 0|
|Eigenvalues λn||2; n = 1,2,…||2; n = 0,1,2,…||2; n = 0,1,2,…|
|Eigenfunctions||sin||cos||sin and cos|
|Series||f(x) = ∑ n=1∞Bn sin||f(x) = ∑ n=0∞An cos|
|Coefficients||Bn = ∫ 0Lf(x)sindx|
Furthermore, this infinite series converges to ∕2 for a < x < b (if the coefficients an are properly chosen).
where the BCs may somewhat simplify this expression.
|# of solutions||Orthogonality of forcing func w/homog soln (weight = 1): ∫ abf(x)ϕh(x)dx|
|ϕh = 0 (λ≠0)||1||0 (orthogonal)|
|ϕh≠0 (λ = 0)||∞||0 (orthogonal)|
|ϕh≠0 (λ = 0)||0||≠0 (not orthogonal)|
to simplify things. This leads to many equations (one for each eigenfunction coefficient an(t)) in t using Fourier’s trick, with one side equaling the generalized Fourier series of Q(x,t), which is just a general function of t, qn(t).
where the BC’s were
if = c, the equation simplifies to = 0.
if = c(f,x,t), the equation simplifies to = Q(f,x,t).