FD-method for Sturm-Liouville problems. Exponential rate of convergence (Q2742784)

For the Sturm-Liouville problem \(u''(x)+[\lambda-q(x)]u(x)=0\), \(x\in (0,1)\) sufficient conditions for the exponential convergence of the functional-discrete method (FD-method) are obtained both for Dirichlet conditions and for conditions of the third kind. Explicit estimates of accuracy of the FD-method for all eigenvalues and all eigenfunctions are constructed. Using these estimates the authors obtain two-sided estimates for the exact eigenvalues and estimates for the remainder terms of the classical asymptotic formulas. Sufficient conditions of the exponential convergence of the FD-method for periodic conditions and generalization of the classical asymptotic expansions for the solution of the Sturm-Liouville problem are obtained.