Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

Abstract

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

Newton-Kantorovich Method for Solving One of the Non-Linear Sturm-Liouville Problems

“…Also, many interesting theories regarding the non-linear eigenvalue problems of the Sturm-Liouville Type are presented by Kurseeva, Moskaleva, and Valovik 2 in 2019 including deriving solvability results, asymptotics of positive and negative eigenvalues, and also applications were given. Moreover, among other recent works that have to be mentioned here is that presented by He and Yang 3 in 2019 and by Al-Khaled and Hazaimeh 4 in 2020, wherein the work of He and Yang, the existence of positive solutions for systems of non-linear Sturm-Liouville differential equations with weight functions was studied, while in the work of Al-Khaled and Hazaimeh a comparative study between a modified version of the variational iteration method and the Sinc-Galerkin method was presented to solve non-linear Sturm-Liouville eigenvalue problem. In this paper, the Newton-Kantorovich method is applied to approximate the solution for one of the non-linear Sturm-Liouville problems that are the problem: −𝑦 ′′ (𝑥) + 𝑦 2 (𝑥) = 𝜆 𝑦(𝑥); 𝑦(𝑥) > 0, 𝑥 ∈ 𝐼 = (0,1) subject to the boundary conditions 𝑦(0) = 𝑦(1) = 0 where 𝜆 > 0 is an eigenvalue parameter.…”

General Scheme of Modeling of Longitudinal Oscillations in Horizontal Rods

“…where x ∈ (x 0 ; x n ), t ∈ (0; +∞) Denote the product λ(x) ∂u ∂x by u [1] and let's call it quasi-derivative. Write down the general boundary conditions: p 11 u(x 0 ; t) + p 12 u [1] (x 0 ; t) + q 11 u(x n ; t) + q 12 u [1] (x n ; t) = ψ 0 (t) p 21 u(x 0 ; t) + p 22 u [1] (x 0 ; t) + q 21 u(x n ; t) + q 22 u [1]…”

“…Such a problem is called a problem of eigenvalues and eigenfunctions. It is also called the task of Shturm Liouville [1,9,11]. The Fourier method is an accurate method of solving these problems.…”

On Modified Second Paine–de Hoog–Anderssen Boundary Value Problem

“…The following brief literature review is by no means exhaustive. While a comparison between the Sinc-Galerkin technique and differential transform method reveals that the latter is more efficient than the former [22], a comparative study between the Sinc-Galerkin technique and variational iteration method indicates that the former is better than the latter when dealing with singular problems [23].…”