Variational integrator for fractional Euler–Lagrange equations

Abstract

International audienceWe extend the notion of variational integrator for classical Euler-Lagrange equations to the fractional ones. As in the classical case, we prove that the variational integrator allows to preserve Noether-type results at the discrete level

Theory of discrete fractional Sturm–Liouville equations and visual results

“…Definition 8. [18,19] We define the integration by parts formula for G-L delta fractional difference operator, u, v is defined on {0, 1, ..., n}, then…”

Variational methods for the solution of fractional discrete/continuous Sturm–Liouville problems

“…Table 2 and Figure 1 present the solution to problem (2-14)- (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) for N = 4, h = 1 and different values of α's. By Theorem 2.5 the first eigenvalue λ 1 of (2-16) is the minimum value of J on N 1 k=1 y 2 k = 1 and the first eigenfunction of (2-16) is the minimizer of this problem.…”

“…, N − 1. As N → ∞, that is, as h → 0, the solutions of system (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) 3B. Sturm-Liouville problem.…”

Application of the Rayleigh-Ritz method to solve a class of fractional variational problem

“…This is also a fundamental problem in finding solution of variational differential equations [3], [17], [21]. Recently many researchers have focused their attention on the numerical solution of equations containing integral or differential operators [5], [6], [8], [9], [7], [10], [11], [12], [13], [14], [27].…”