期刊
JOURNAL OF COMPUTATIONAL PHYSICS
卷 252, 期 -, 页码 495-517出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2013.06.031
关键词
Regular/singular fractional Sturm-Liouville operators; Poly-fractonomial eigenfunctions; Fractal expansion set; Exponential convergence
资金
- Collaboratory on Mathematics for Mesoscopic Modeling of Materials (CM4) at PNNL
- Department of Energy
- AFOSR MURI
- NSF/DMS
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1216437] Funding Source: National Science Foundation
We first consider a regular fractional Sturm-Liouville problem of two kinds RFSLP-I and RFSLP-II of order nu is an element of (0,2). The corresponding fractional differential operators in these problems are both of Riemann-Liouville and Caputo type, of the same fractional order mu = nu/2 is an element of (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. (C) 2013 Elsevier Inc. All rights reserved.
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