A Stepsize Variation Strategy for the Solution of Regular Sturm-Liouville Problems

Reprint of Variable-step finite difference schemes for the solution of Sturm–Liouville problems

“…We mention Initial Value Problems [4], Boundary Value Problems [5,6,11,12] and Sturm-Liouville problems [7][8][9][10]. In all these cases the proposed formulae show good stability properties and the developed codes turn out to be competitive with the existing software, in particular when the problem to be solved is stiff and the solution has a slope different from that of its derivatives.…”

“…Matrix M has a band structure (bandwidth depends on the order of the method) with some additional elements on the first and last rows, because of the initial and final methods (see [8,9] for more details).…”

“…ð9Þ it is convenient, as suggested in [8], to define initial and final methods also involving the values y 0 0 % y 0 ðaÞ and y 0 n % y 0 ðbÞ. Eq.…”

“…Based on this idea, in [7,8] we illustrate a different approach able to solve regular and singular SLPs classifying the endpoints of the interval as regular or singular, limit points (LP) or limit circles (LC), oscillatory (LCO) or nonoscillatory (LCNO). The underlying methods approximate the derivatives of SLPs separately, therefore the second order differential equation is not transformed into an equivalent first order system; this means lower computational cost and easier stepsize selection strategy.…”

The Sturm-Liouville eigenvalue problem - a numerical solution using the Control Volume Method

“…One of the most common approaches to a numerical solution of the considered problem is the finite difference method [3,[6][7][8] where each derivative is discretized at each grid point with an adequate difference scheme. Apart from the finite difference method, several analytical ones, such as the variational or decomposition methods, are proposed to find an approximate solution.…”