Eigenvalue asymptotics for a fractional boundary-value problem

This letter presents a result concerning eigenvalue approximation of a boundary-value problem with the Caputo fractional derivative. This approximation is derived by the use of the asymptotic (for large x   and λλ) form of the exact solution. The growth order of the eigenvalues is given and it is shown that their number is finite. Moreover, a simple method of estimating the size of the spectrum is proposed. The issue of a finite number of eigenvalues is a very peculiar and characteristic feature of differential equations with fractional order derivative. The paper is concluded with a numerical verification that our approximations are very accurate. This shows that the devised formulas can be readily used in applications of fractional boundary-value problems.