Abstract
This article deals with a special case of the Sturm–Liouville boundary value problem (BVP), an eigenvalue problem characterized by the Sturm–Liouville differential operator with unknown spectra and the associated eigenfunctions. By examining the BVP in the Schrödinger form, we are interested in the problem where the corresponding invariant function takes the form of a reciprocal quadratic form. We call this BVP the modified second Paine–de Hoog–Anderssen (PdHA) problem. We estimate the lowest-order eigenvalue without solving the eigenvalue problem but by utilizing the localized landscape and effective potential functions instead. While for particular combinations of parameter values that the spectrum estimates exhibit a poor quality, the outcomes are generally acceptable although they overestimate the numerical computations. Qualitatively, the eigenvalue estimate is strikingly excellent, and the proposal can be adopted to other BVPs.
Highlights
The Sturm–Liouville boundary value problem (BVP) is an active research area in mathematics and its applications [1,2]
Since finding eigenvalues and eigenfunctions is a central aspect of the Sturm–Liouville problem, our motivation is to provide an alternative for estimating the former without solving the original modified second Paine–de Hoog–Anderssen (PdHA) BVP
We utilize MATLAB’s finite boundary conditions, and initial guess to solve the corresponding with an unknown difference code bvp4c for the latter, accompanied by call functions of the system of the parameter
Summary
The Sturm–Liouville boundary value problem (BVP) is an active research area in mathematics and its applications [1,2]. An efficient numerical method for estimating eigenvalues and eigenfunctions of the Caputo-type fractional Sturm–Liouville problem based on the Lagrange polynomial interpolation was proposed [28] When it comes to applications, the Sturm–Liouville problem features an abundance of them in applied mathematics and physics. Sturm–Liouville BVP emerges naturally as an immediate consequence of implementing the method of separation of variables For these vibration problems, the eigenvalues determine the frequency of oscillation, whereas the associated eigenfunctions correspond to the shape of the vibrating waves at any point in time [29]. Since finding eigenvalues and eigenfunctions is a central aspect of the Sturm–Liouville problem, our motivation is to provide an alternative for estimating the former without solving the original modified second PdHA BVP.
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