Fractional Order Differential Operator Research Articles

A Problem on the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator

A Problem on the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator

Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation

Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion—typically exp[−( bD)]—where D is the apparent diffusion coefficient (mm 2/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model—exp[−( bD) α ], where α is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch–Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch–Torrey equation is replaced by a Riemann–Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.

A formula for the powers of fractional order operators

We derive none some explicit formula for the power of fractional order (differential and integral) operators.

The eigenvalues of a boundary value problem for a fractional-order differential operator

The eigenvalues of a boundary value problem for a fractional-order differential operator

A problem on the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator

A problem on the zeros of the Mittag-Leffler function and the spectrum of a fractional-order differential operator

Completeness of the system of eigenfunctions of a fractional-order differential operator

Completeness of the system of eigenfunctions of a fractional-order differential operator

The use of fractional-order derivatives for determination of the time-dependent rate constant

The time-dependent rate constant for diffusion-limited reactions is derived by the method of fractional-order differential operators.