873 research outputs found
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
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 matrix method for fractional Sturm-Liouville problems on bounded domain
A matrix method for the solution of direct fractional Sturm-Liouville
problems on bounded domain is proposed where the fractional derivative is
defined in the Riesz sense. The scheme is based on the application of the
Galerkin spectral method of orthogonal polynomials. The order of convergence of
the eigenvalue approximations with respect to the matrix size is studied. Some
numerical examples that confirm the theory and prove the competitiveness of the
approach are finally presented
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
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