Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6933565 | Journal of Computational Physics | 2013 | 23 Pages |
Abstract
We first consider a regular fractional Sturm-Liouville problem of two kinds RFSLP-I and RFSLP-II of order νâ(0,2). The corresponding fractional differential operators in these problems are both of Riemann-Liouville and Caputo type, of the same fractional order μ=ν/2â(0,1). We obtain the analytical eigensolutions to RFSLP-I & -II as non-polynomial functions, which we define as Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLP-I & -II. Subsequently, we extend the fractional operators to a new family of singular fractional Sturm-Liouville problems of two kinds, SFSLP-I and SFSLP-II. We show that the primary regular boundary-value problems RFSLP-I & -II are indeed asymptotic cases for the singular counterparts SFSLP-I & -II. Furthermore, we prove that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials. In numerical examples, we employ the new poly-fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.
Keywords
Related Topics
Physical Sciences and Engineering
Computer Science
Computer Science Applications
Authors
Mohsen Zayernouri, George Em Karniadakis,