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

Abstract

We discuss the solution of regular and singular Sturm-Liouville problems by means of High Order Finite Difference Schemes. We describe a code to define a discrete problem and its numerical solution by means of linear algebra techniques. Different test problems are proposed to emphasize the behaviour of the proposed algorithm.

Comparison between standard and non-standard finite difference methods for solving first and second order ordinary differential equations

“…The exact solution of (32) is . 3 In figure 7 we compared the standard and non-standard finite difference methods for 05 . 0  h with the power series solution of equation 39, we supposed that in equation 42, n changes from 1 to 1000.…”

“…Some famous equations such as Dynamic, Logistic, Lane-Emden and Airy equations have been provided. Solution of Airy equation which is a special case of Storm-Liouville equation [3] can't be displayed based upon primary functions. We use the series solutions' method to find the power series solution for this second-order linear differential equation and compare NSFD and FD methods with the power series solution.…”

Solving Dynamic Equation by the Non-Standard Finite Difference Method

“…The full details of these procedures are given by some studies [1][2][3][4][5][6]. The non-standard finite difference schemes have been developed as an alternative method for solving a wide range of problems such as the mathematical models of biology and chaotic systems [7][8][9][10][11][12][13][14].…”

Preface to the special issue NUMTA 2013

“…In this group of papers, the reader will find stochastic and deterministic methods developed for solving both global and local optimization problems as well as algorithms developed for dealing with multiobjective optimization. The third group of papers (see [2][3][4]) is dedicated to the numerical solution of differential problems. In particular, energy-conserving methods for Hamiltonian problems, multigrid methods for reaction-diffusion problems, and finite-difference methods for Sturm-Liouville problems.…”

“…It also considers numerical techniques and problems arising when natural phenomena are modelled on computers. The papers included in this special issue (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) have been carefully selected by the guest editors among the submissions reflecting the talks presented at the international conference ''Numerical computations: Theory and Algorithms (NUMTA)'' held in June 17-23, 2013, Falerna (CZ), Italy. The NUMTA 2013 has been organized by the University of Calabria, Rende (CS), Italy and the N.I.…”