Fourth-order Sturm Liouville Problems Research Articles

A new application of the homotopy analysis method: Solving the Sturm–Liouville problems

In this paper, the homotopy analysis method (HAM) is applied to numerically approximate the eigenvalues of the second and fourth-order Sturm–Liouville problems. These eigenvalues are calculated by starting the HAM algorithm with one initial guess. In this paper, it can be observed that the auxiliary parameter ℏ, which controls the convergence of the HAM approximate series solutions, also can be used in predicting and calculating multiple solutions. This is a basic and more important qualitative difference in analysis between HAM and other methods.

Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs

We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem: , , , where and are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters is also studied.

An efficient technique for finding the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, we will develop a numerical technique for finding the eigenvalues of fourth-order non-singular Sturm–Liouville problems. We used the variational iteration methods as a basis for this technique. Numerical results and conclusions will be presented. Comparison results with others will be presented.

Fliess series approach to the computation of the eigenvalues of fourth-order Sturm-Liouville problems

This paper is an extension to our work on the computation of the eigenvalues of regular fourth-order Sturm-Liouville problems using Fliess series. The purpose here is twofold. First, we consider general self-adjoint separated boundary conditions. Second, we modify the algorithm presented in an earlier paper to ease considerably the computation of the iterated integrals involved.

Convergence of regular approximations to the spectra of singular fourth-order Sturm–Liouville problems

We prove some new results which justify the use of interval truncation as a means of regularising a singular fourth-order Sturm–Liouville problem near a singular endpoint. Of particular interest are the results in the so-called lim-3 case, which has no analogue in second-order singular problems.

Oscillation theory and numerical solution of fourth-order Sturm—Liouville problems

A shooting method is developed to approximate the eigenvalues and eigenfunctions of a fourth-order Sturm-Liouville problem. The main tool is a miss-distance function M(A), which counts the number of eigenvalues less than λ. The method approximates the coefficients of the differential equation by piecewise-constant functions, which enables an exact solution to be found on each mesh interval. In order to calculate M(A) for the approximate problem, certain oscillation numbers N L and N R must be computed. These consist of sums of nullities (or rank deficiencies) of 2x2 matrices obtained from the solutions of the approximate differential equation. Although these solutions can be found explicitly, the calculation of N L and N R is non-trivial, and is obtained by using certain properties of M(A).