A Discrete Algorithm to the Calculus of Variations

TL;DR

This paper evaluates Guseinov's discrete algorithm for calculus of variations, demonstrating its effectiveness in approximating solutions to classical problems like brachistochrone and Mania with Lavrentiev's phenomenon.

Contribution

It provides a numerical analysis of Guseinov's algorithm, comparing its performance with other methods on key variational problems.

Findings

Guseinov's method yields better solutions in most tested cases.

The algorithm effectively approximates solutions to classical variational problems.

Comparison shows advantages over existing methods.

Abstract

A numerical study of an algorithm proposed by Gusein Guseinov, which determines approximations to the optimal solution of problems of calculus of variations using two discretizations and correspondent Euler-Lagrange equations, is investigated. The results we obtain to discretizations of the brachistochrone problem and Mania example with Lavrentiev's phenomenon are compared with the solutions found by other methods and solvers. We conclude that Guseinov's method presents better solutions in most of the cases studied.