Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 4: 1398 - 1418

DOI: https://doi.org/10.1007/s42967-023-00340-3

ORIGINAL PAPERS

Variational and Numerical Approximations for Higher Order Fractional Sturm-Liouville Problems

  • Divyansh Pandey ,
  • Prashant K. Pandey ,
  • Rajesh K. Pandey
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  • 1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, Uttar Pradesh, 221005, India;
    2. Department of Mathematics, School of Advanced Science and Languages, VIT Bhopal University, Kothrikalan, Sehore, Madhya Pradesh, 466114, India

Received date: 2023-07-25

  Revised date: 2023-09-22

  Accepted date: 2023-10-03

  Online published: 2024-01-29

Supported by

Authors are thankful to the reviewers for their comments incorporated in the revised version of the manuscript.

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Abstract

This paper is devoted to studying the variational approximation for the higher order regular fractional Sturm-Liouville problems (FSLPs). Using variational principle, we demonstrate that the FSLP has a countable set of eigenvalues and corresponding unique eigenfunctions. Furthermore, we establish two results showing that the eigenfunctions corresponding to distinct eigenvalues are orthogonal, and the smallest (first) eigenvalue is the minimizer of the functional. To validate the theoretical result, we also present a numerical method using polynomials \Phi_j(t)=t^{j+1}(1-t)^2 \text { for } j=1,2,3… as a basis function. Further, the Lagrange multiplier method is used to reduce the fractional variational problem into a system of algebraic equations. In order to find the eigenvalues and eigenfunctions, we solve the algebraic system of equations. Further, the analytical convergence and the absolute error of the method are analyzed.

Key words： Fractional variational analysis; Fractional Sturm-Liouville problem (FSLP); Calculus of variations

Cite this article

Divyansh Pandey , Prashant K. Pandey , Rajesh K. Pandey . Variational and Numerical Approximations for Higher Order Fractional Sturm-Liouville Problems[J]. Communications on Applied Mathematics and Computation, 2025 , 7(4) : 1398 -1418 . DOI: 10.1007/s42967-023-00340-3

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