Computing the sinp function via the inverse power method

Abstract

-In this paper, we discuss a new iterative method for computing sin p . This function was introduced by Lindqvist in connection with the unidimensional nonlinear Dirichlet eigenvalue problem for the p-Laplacian. The iterative technique was inspired by the inverse power method in finite dimensional linear algebra and is competitive with other methods available in the literature.2010 Mathematical subject classification: 65D20; 65L15; 35P15; 35P30; 35J92.

Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions

“…Since v µ,q = 0 = v µ,q on ∂Ω the inequalities in (11) mean that v µ,q and v µ,q are, respectively, suband supersolutions for (7). Moreover, v µ,q and v µ,q are ordered, that is v µ,q v µ,q in Ω.…”

“…In the one-dimensional case the first eigenpair (λ p , e p ) is explicitly determined by solving the corresponding ODE boundary value problem. If Ω = (a, b), then λ p = (π p / (b − a)) p−1 and e p = (p − 1) −1/p sin p (π p (x − a) / (b − a)), where π p := 2(p − 1) 1/p 1 0 (1 − s p ) −1/p ds and sin p is a 2π p -periodic function that generalizes the classical sine function (see [11,40]).…”

Inequalities for Eigenfunctions of the P-Laplacian

“…Numerous other authors, see e. g. [2], [3,4], [5], [6], [7] and the bibliographies of these papers, have extended this work in various directions including the study of generalized hyperbolic functions and their inverses. Our goal here to study these p-trigonometric and p-hyperbolic functions and to prove several inequalities for them.…”

One-dimensional singular problems involving the p-Laplacian and nonlinearities indefinite in sign

“…[7], Section 6.3; and for the precise definition and further properties of sin p , see e.g. [15] and [3], Section 2).…”