Computation of Eigenvalues of Sturm-Liouville Problems using Homotopy Perturbation Method

Abstract

In this paper, a recently proposed analytical approximation technique, namely Homotopy Perturbation Method (HPM) is presented to compute eigenvalues of second and fourth order Sturm-Liouville problems. These types of equations appear in engineering applications such as vibration, elastic stability problems, which are great importance in engineering practice and real world applications. The presented technique is formulated and then applied to number of Sturm-Liouville problem cases for which solutions were obtained numerically. An easy to use feature of the method is promising for applying to such boundary value problems.

Exponentially convergent symbolic algorithm of the functional-discrete method for the fourth order Sturm–Liouville problems with polynomial coefficients

“…In this article a new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the fourth order Sturm-Liouville problem (see numerical algorithm from Section 7). We consider the eigenvalue problem on a finite interval [0, X] in the Hilbert space L 2 (0, X) for the fourth order ordinary differential equation (1) obtained with other existing numerical techniques [16,17,18,19,20,21].…”

“…This test example (1), (2), (59) was considered in [16,17,18,19,20] in which the following methods were applied: Adomian decomposition method (ADM) [16], variational iteration method (VIM) [17], homotopy perturbation method (HPM) [18] homotopy analysis method (HAM) [19], extended sampling method (ESM) [20]. Let us compare the behavior of the absolute errors [16,17,18,19,20] as the index number n of eigenvalue increases.…”

“…In Table 2 these absolute errors are given. They were calculated using the exact eigenvalues from Table 1 and the approximations from [16,17,18,19,20].…”

“…Section 8 illustrates the theoretical results by numerical examples. In Examples 1 and 2 the numerical results obtained with the FD-method are compared with the numerical test results obtained with other existing numerical techniques [16,17,18,19,20,21].…”

“…n,M(j+1)−p−3 , p = 0, 1, ..., M (j + 1) − 4, j = 0, 1, 2, ... Substituting the representation (18) in the boundary conditions (7), we obtain the nonhomogeneous system of linear algebraic equations for the coefficients…”