2,331 research outputs found
Variational methods for fractional -Sturm--Liouville Problems
In this paper, we formulate a regular -fractional Sturm--Liouville problem
(qFSLP) which includes the left-sided Riemann--Liouville and the right-sided
Caputo -fractional derivatives of the same order , . We introduce the essential -fractional variational analysis needed
in proving the existence of a countable set of real eigenvalues and associated
orthogonal eigenfunctions for the regular qFSLP when associated
with the boundary condition . A criteria for the first eigenvalue
is proved. Examples are included. These results are a generalization of the
integer regular -Sturm--Liouville problem introduced by Annaby and Mansour
in [1]
Fractional Sturm-Liouville eigenvalue problems, II
We continue the study of a non self-adjoint fractional three-term
Sturm-Liouville boundary value problem (with a potential term) formed by the
composition of a left Caputo and left-Riemann-Liouville fractional integral
under {\it Dirichlet type} boundary conditions. We study the existence and
asymptotic behavior of the real eigenvalues and show that for certain values of
the fractional differentiation parameter , , there is a
finite set of real eigenvalues and that, for near , there may be
none at all. As we show that their number becomes infinite and
that the problem then approaches a standard Dirichlet Sturm-Liouville problem
with the composition of the operators becoming the operator of second order
differentiation
A Finite Element Method for the Fractional Sturm-Liouville Problem
In this work, we propose an efficient finite element method for solving
fractional Sturm-Liouville problems involving either the Caputo or
Riemann-Liouville derivative of order on the unit interval
. It is based on novel variational formulations of the eigenvalue
problem. Error estimates are provided for the finite element approximations of
the eigenvalues. Numerical results are presented to illustrate the efficiency
and accuracy of the method. The results indicate that the method can achieve a
second-order convergence for both fractional derivatives, and can provide
accurate approximations to multiple eigenvalues simultaneously.Comment: 30 pages, 7 figure
Variational methods for the solution of fractional discrete/continuous Sturm-Liouville problems
The fractional Sturm–Liouville eigenvalue problem appears in many situations, e.g.,
while solving anomalous diffusion equations coming from physical and engineering applications.
Therefore to obtain solutions or approximation of solutions to this problem is of great importance.
Here, we describe how the fractional Sturm–Liouville eigenvalue problem can be
formulated as a constrained fractional variational principle and show how such formulation can
be used in order to approximate the solutions. Numerical examples are given, to illustrate the
method
A fractional analysis in higher dimensions for the Sturm-Liouville problem
In this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.publishe
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
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