Complex-valued fractional derivatives on time scales

We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given.Comment: This is a preprint of a paper whose final and definite form will appear in Springer Proceedings in Mathematics & Statistics, ISSN: 2194-1009. Accepted for publication 06/Nov/201

Preliminary results on the effects of orthopedic implant stiffness fixed to the cut end of the femur on the stress at the stump-prosthetic interface

A lot of trans-femoral amputation patients experience skin breakdown due to the pressures and shear stresses in the stump-prosthesis interface. In this study, a finite element model was employed to investigate the stresses at the stump interface in the case of an orthopedic implant fixed to the cut end of the femur. By changing the stiffness of this implant, we aim to see how the stiffness of this implant affects the stresses in the interface between the amputated limb and the prosthesis. To find out the effects of implant stiffness, five values for the elastic modulus, ranging from 0.1 to 0.5 Mpa, with an interval of 0.1 Mpa were employed in the implant structure of the FE model. Obtained results show that the implant played important role in reducing the stresses at the stump-prosthesis interface where the contact pressure did not exceed 53 Kpa and 17.3 Kpa for shear stress in the stiffer case of an implant, while the contact pressure in the case of femur without implant exceeded 79Kpa and 42 Kpa for shear stress. We also noted that the intensity of the contact pressure and the shear stress is proportional to the stiffness of the implant, as the greater the implant stiffness, the higher the peak of these stresses

Time-Fractional Optimal Control of Initial Value Problems on Time Scales

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler--Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann--Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book chapter with Springer International Publishing AG. Submitted 23/Jan/2019; revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial text overlap with arXiv:1508.0075

Variable order Mittag-Leffler fractional operators on isolated time scales and application to the calculus of variations

We introduce new fractional operators of variable order on isolated time scales with Mittag-Leffler kernels. This allows a general formulation of a class of fractional variational problems involving variable-order difference operators. Main results give fractional integration by parts formulas and necessary optimality conditions of Euler-Lagrange type.Comment: This is a preprint of a paper whose final and definite form is with Springe

The fuzzy Henstock–Kurzweil delta integral on time scales

We investigate properties of the fuzzy Henstock–Kurzweil delta integral (shortly, FHK Δ -integral) on time scales, and obtain two necessary and sufficient conditions for FHK Δ -integrability. The concept of uniformly FHK Δ -integrability is introduced. Under this concept, we obtain a uniformly integrability convergence theorem. Finally, we prove monotone and dominated convergence theorems for the FHK Δ -integral.publishe