9 Boundary Value Problems and Sturm–Liouville Theory

An initial value problem prescribes everything at one point and asks for the future. A boundary value problem prescribes conditions at two points, such as the ends of a vibrating string, the two faces of a heated slab, or the edge and centre of a drum, and asks for a solution in between. The theory is quite different: existence and uniqueness can fail, and the failure is the interesting part, because the special parameter values at which it fails are the natural frequencies, energy levels and critical loads of physics and engineering. Sturm (1836) and Liouville (1837) developed the theory that makes these eigenvalue problems tractable, and it explains why the Fourier series, Bessel expansions and Legendre expansions of the previous section exist.

Throughout, on a bounded interval \([a, b]\), \[ Ly = -\bigl (p(t)y'\bigr )' + q(t)y , \] with \(p\), \(p'\), \(q\) continuous and \(p > 0\) on \([a, b]\), and separated boundary conditions \[ B_ay = \alpha _1y(a) + \alpha _2y'(a) = 0, \qquad B_by = \beta _1y(b) + \beta _2y'(b) = 0 , \] with \((\alpha _1, \alpha _2) \neq (0, 0)\) and \((\beta _1, \beta _2) \neq (0, 0)\). Dirichlet conditions \(y(a) = y(b) = 0\) and Neumann conditions \(y'(a) = y'(b) = 0\) are special cases; the others are called Robin conditions.

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