Variational Approach for Solving Regular N-Dimensional Fractional Sturm-Liouville Problems of Order Between (0, 1]

Abstract

Here, we consider the regular N-dimensional fractional Sturm-Liouville problems (N-D FSLPs) of order

\nu =({\nu }_1,{\nu }_2,\cdots ,{\nu }_N)\in (0,1]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nu =(\nu _1,\nu _2,\cdots ,\nu _N)\in (0,1]$$\end{document}

. The FSLP is defined using a fractional version of the gradient operator in terms of left and right  Caputo fractional derivatives (CFDs). We establish the essential properties of the eigenvalues (EVs) and eigenfunctions (EFs) of the FSLP using the principle of fractional variational calculus. To validate our theoretical study, we take the fractional Laplace equation as an example and demonstrate that the EVs exist for fractional order

\nu \in (0,1]

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\nu \in (0,1]$$\end{document}

.

Keywords

Fractional variational analysis / Fractional Sturm-Liouville problem (FSLP) / Fractional Calculus of variations (FCOVs) / 26A33 / 34A08 / 34B24 / 49R05

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