Editing Heat equation (section)

=== Generalizing the solution technique ===
The solution technique used above can be greatly extended to many other types of equations. The idea is that the operator ''u<sub>xx</sub>'' with the zero boundary conditions can be represented in terms of its [[eigenfunction]]s. This leads naturally to one of the basic ideas of the [[spectral theory]] of linear [[self-adjoint operator]]s.

Consider the [[linear operator]] Δ''u'' = ''u<sub>xx</sub>''. The infinite sequence of functions
: <math> e_n(x) = \sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right)</math>
for ''n'' ≥ 1 are eigenfunctions of Δ. Indeed,
: <math> \Delta e_n = -\frac{n^2 \pi^2}{L^2} e_n. </math>

Moreover, any eigenfunction ''f'' of Δ with the boundary conditions ''f''(0) = ''f''(''L'') = 0 is of the form ''e''<sub>''n''</sub> for some ''n'' ≥ 1. The functions ''e''<sub>''n''</sub> for ''n'' ≥ 1 form an [[orthonormal]] sequence with respect to a certain [[inner product]] on the space of real-valued functions on [0, ''L'']. This means
: <math> \langle e_n, e_m \rangle = \int_0^L e_n(x) e^*_m(x) dx = \delta_{mn}</math>

Finally, the sequence {''e''<sub>''n''</sub>}<sub>''n'' ∈ '''N'''</sub> spans a dense linear subspace of ''L''<sup>2</sup>((0, ''L'')). This shows that in effect we have [[diagonal matrix|diagonalized]] the operator Δ.