A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems

Abstract

This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional 2α-order SturmLiouville problem where 1 2 < α ≤ 1. In this paper, we implement the reproducing kernel method RKM) to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at x = 1. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

Reproducing kernel functions and homogenizing transforms

“…For instance Cui et al [15] published a book about numerical analysis in the reproducing kernel space which is a comprehensive study. Syam et al [16] used the method to solve a class of fractional Sturm-Liouville eigenvalue problems. Jiang and Tian [23] reported the Volterra integro-differential equations of fractional order by the reproducing kernel method.…”

Adomian decomposition method for solving nonlinear fractional sturm-liouville problem

“…The linear and nonlinear Sturm-Liouville (S-L) problems have of great importance. Theory and algorithms of these problems are presented in [9][10][11][12][13][14][15][16][17][18]. The fractional specific problems are studied in [20], [21].…”

On Solutions of a Higher Order Nonhomogeneous Ordinary Differential Equation

“…For instance Cui et al [3] published a book about numerical analysis in the reproducing kernel space which is a comprehensive study. Syam et al [4] used the method to solve a class of fractional Sturm-Liouville eigenvalue problems.…”