where
where
Boundary conditions (2 each)→ | ϕ(0) = ϕ(L) = 0 | (0) = (L) = 0 | |
Eigenvalues λ_{n} | ^{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}^{∞}B_{n} sin | f(x) = ∑ _{n=0}^{∞}A_{n} cos | |
Coefficients | B_{n} = ∫ _{0}^{L}f(x)sindx | ||
Furthermore, this infinite series converges to ∕2 for a < x < b (if the coefficients a_{n} are properly chosen).
where the BCs may somewhat simplify this expression.
# of solutions | Orthogonality of forcing func w/homog soln (weight = 1): ∫ _{a}^{b}f(x)ϕ_{h}(x)dx | |
ϕ_{h} = 0 (λ≠0) | 1 | 0 (orthogonal) |
ϕ_{h}≠0 (λ = 0) | ∞ | 0 (orthogonal) |
ϕ_{h}≠0 (λ = 0) | 0 | ≠0 (not orthogonal) |
to simplify things. This leads to many equations (one for each eigenfunction coefficient a_{n}(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, q_{n}(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).