Half-range solutions of indefinite Sturm-Liouville problems
We study diffusion equations of Sturm-Liouville type, i.e. boundary value problems of the form \[ w(\mu)(\partial \psi /\partial x)(x,\mu)=\partial /\partial \mu (p(\mu)(\partial \psi /\partial \mu)(x,\mu))-q(\mu)\psi (x,\mu)\quad (0<x<\infty,\quad \mu \in I) \] \[ \psi (0,\mu)=\phi +(\mu)\;if\;w(\mu)>0;\quad \int_{I}| \psi (x,\mu)|^ 2| w(\mu)d\mu =O(1)\quad (or\quad o(1)\quad or\quad O(x))\quad as\quad x\to \infty \] where \(\psi\) (x,\(\mu)\) satisfies self-adjoint boundary conditions of the differential operator \(h\to -(ph')'+qh\) whose spectrum is contained in the set \(\{\) \(0\}\cup [\epsilon,\infty)\) for some \(\epsilon >0\). I is a finite or infinite interval and w(\(\mu)\) is an indefinite weight function with finitely many sign changes. We develop two equivalent methods of constructing explicit representations for the solutions of the problem. The first method is based on an eigenfunction expansion whereas the second uses a Wiener-Hopf-type integral equation and factorization. Some illustrative examples are also worked out.
diffusion equations of Sturm-Liouville type
scientific article; zbMATH DE number 4004754
scientific article