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

doi:10.1007/s42967-023-00340-3

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1398-1418.doi: 10.1007/s42967-023-00340-3

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

Divyansh Pandey¹, Prashant K. Pandey², Rajesh K. Pandey¹

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

收稿日期:2023-07-25 修回日期:2023-09-22 接受日期:2023-10-03 出版日期:2024-01-29 发布日期:2024-01-29
通讯作者: Rajesh K.Pandey,E-mail:rkpandey.mat@iitbhu.ac.in E-mail:rkpandey.mat@iitbhu.ac.in
作者简介:Divyansh Pandey, E-mail:divyanshpandey.rs.mat19@iitbhu.ac.in;Prashant K.Pandey, E-mail:prashantkp92@gmail.com
基金资助:
Authors are thankful to the reviewers for their comments incorporated in the revised version of the manuscript.

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

Divyansh Pandey¹, Prashant K. Pandey², Rajesh K. Pandey¹

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:2023-07-25 Revised:2023-09-22 Accepted:2023-10-03 Online:2024-01-29 Published:2024-01-29
Supported by:
Authors are thankful to the reviewers for their comments incorporated in the revised version of the manuscript.

摘要/Abstract

摘要： 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.

关键词: Fractional variational analysis, Fractional Sturm-Liouville problem (FSLP), Calculus of variations

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

中图分类号:

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.

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

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

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